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The tent map, the horseshoe and the pendulum

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Title:
The tent map, the horseshoe and the pendulum the geometry of chaos control
Creator:
Starrett, John
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Denver, CO
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University of Colorado Denver
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English
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iv, 102 leaves : illustrations ; 29 cm

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Chaotic behavior in systems ( lcsh )
Chaotic behavior in systems ( fast )
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bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )

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Bibliography:
Includes bibliographical references (leaf 102).
Thesis:
Submitted in partial fulfillment of the requirements for the degree, Master of Science, Applied Mathematics
General Note:
Department of Mathematical and Statistical Sciences
Statement of Responsibility:
by John Starrett.

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|University of Colorado Denver
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|Auraria Library
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Full Text
THE TENT MAP, THE HORSESHOE AND THE PENDULUM:
THE GEOMETRY OF CHAOS CONTROL
by
John Starrett
B.S. Mathematics
Metropolitan State College of Denver, 1994
A thesis submitted to die
University of Colorado at Denver
in partial fulfillment
of the requirements for the degree of
Master of Science
Applied Mathematics
1997


This thesis for the Master of Science
Applied Mathematics
degree by
John Starrett
has been approved
by
Randall Tagg
Weldon Lodwick
5-6-7?


Starrett, John (M.S., Mathematics)
The Tent Map, the Horseshoe and the Pendulum: the Geometry of Chaos
Control
Thesis directed by Associate Professor William Briggs
ABSTRACT
When the space of a dynamical system is continually stretched and folded, a
horseshoe structure may develop. The horseshoe is a guarantee of chaotic
behavior, and we study the suspension of a horseshoe map as a model for
chaotic dynamics of a periodically driven system. Consideration of the
dynamics of a suspended horseshoe and its perturbation leads to a general
geometrical approach to the control of chaos in low dimensional periodically
driven systems by the method of stable subspace targeting.. We show how
OGY control is a special case of the general method, and show why in
experimental situations certain modifications must be made to OGY in order
for it to work. We develop and implement two new types of chaos control
based on these considerations, control by time proportioned perturbations
(TPP) and control by capture and release (CR).
This abstract accurately represents the contents of this candidates thesis. I
recommend its publication.
Signed
William Brings
111


CONTENTS
1 Introduction....................................................1
2 Chaotic Maps and their Control................................3
2.1 The tent map...............................................3
2.2 The bakers map...........................................12
2.3 TheLozi map...............................................17
3 The Horseshoe.................................................35
3.1 The Smale horseshoe.......................................35
3.2 The Henon map and its horseshoe structure................43
3.3 The suspension of the horseshoe...........................46
3.4 The control of the horseshoe..............................50
4 The Dynamics of the Pendulum..................................54
4.1 Folding the horseshoe.....................................54
4.2 Bifurcation and orbit structure...........................67
5 Control of Continuous Time Systems............................72
5.1 Control by OGY............................................72
5.2 General geometric picture of control......................77
5.3 Control by capture and release............................84
5.4 Control by time proportioned perturbations................88
6 Appendix A....................................................99
7 Appendix B...................................................100
8 References...................................................102
iv


1. Introduction
Chaotic solutions exist for many systems, from maps like the logistic
map to continuous time systems described by differential equations, to real
world systems. Many interesting and important physical systems have chaotic
dynamics over certain parameter ranges, and it is sometimes possible to
control these systems to obtain improved performance. A prime example is
the control of a multimode laser[I], where the stability regime of a YAG
laser has been extended by an order of magnitude by the application of small
perturbations to the DC bias level of the laser pump.
Following the publication of results of Ott, Grebogi and Yorke[2]
(OGY) on the control of chaotic systems, experimental control of several
physical systems was reported[3]. OGY control works by applying a small
perturbation to a system parameter based on information taken from a map,
which, in an experimental system, is some sort of surface of section (SOS)
map. The control signal remains constant over the period between mappings,
and is recalculated at each SOS. The goal of the perturbation is to place the
system state on the stable subspace of an unstable periodic orbit (UPO) in
one iteration of the map. Experimentalists found that although strict
application of the OGY algorithm was suitable for some simple systems, in
order to achieve tight control it was sometimes necessary to delay the
application of control, and that sometimes the control perturbation must be
turned off before the next SOS was taken.
I show why delay of perturbation and change of perturbation length
are sometimes necessary in an experimental situation, and extend OGY
control to the continuous time systems by means of stable subspace
targeting. I develop these results by looking at the geometry of the
suspension of the Smale horseshoe, a simple geometrical construction
exhibiting chaotic dynamics. By considering the interaction of a suspended
horseshoe and its perturbation, I show how a continuous spectrum of control
rules based on perturbation magnitude, control on time, and delay time may
be established. The spectrum of limit cycle control will be made plausible
from a geometrical point of view, and later developed as a numerical
procedure for control of continuous time periodically forced experimental
systems.
1


In the final two chapters of this report, I develop the mathematics
necessary for the implementation of two new continuous-time chaos control
schemes, control by capture and release (CR) and control by time
proportioned perturbations. I then use these control schemes to control a
numerical model of the vertically forced pendulum with damping.
2


2. Chaotic Maps and their Control
2.1. The tent map
Consider the chaotic tent map, a difference equation defined by:
n+l for 0 < xn < ^ (2.1)
xn+1 = 2(1 - xn) for ^ < xn < 1. (2.2)
This map can be used to describe the stretching and folding of a one
dimensional elastic band of length one. Each time the map is iterated
the band stretches to twice its length and folds into itself. Points in
[0, |] will double in magnitude and map to points in [0,1], while points
in (i l] will be stretched out to twice their magnitude and folded into
[0,1]. Figure 1 illustrates the result of this mapping.
Figure 2.1.1. The tent map takes the unit interval linearly into itself.
The diagonal line in the graph of Figure 2.1.2 is the identity line
xn+i = xn and its intersection with the tent gives the fixed points |
and 0. We can trace the forward iteration of a point in the unit interval
by the method of graphical analysis: Start with an initial value x0 (the
seed) on the xn axis and draw a vertical line from this point to the
3


graph of the function. Then draw a horizontal line from the function to
the identity line. The xn coordinate of this point is X\. We can iterate
the map by repeating this process, producing what is known as a web
diagram.
Figure 2.1.2. The web diagram illustrates graphically the result of iter-
ation of the map.
The tent map is two to one except at 1, which has only \ as its first
preimage. Every x in [0,1] of the form x = p, n integer, eventually
maps under forward iteration to the fixed point x = 0. Solving xn+\ =
xn shows that the point x = | is the only other fixed point. There are,
however, an infinite number of periodic points of the form x = where
p and q are integers and q is odd, like the period two points | and |.
It can be shown that any fraction in lowest terms whose denominator
contains an odd number as a factor is a preimage of a point in a periodic
4


orbit. There are periodic points of all periods, and the set of irrationals
in [0,1] make up the set of points with chaotic orbits.
Definition 2.1. (Taylor and Toohey)[4] A dynamical system on a topo-
logical space is chaotic if every pair of non-void open subsets share a
periodic orbit.
Since points of the form q odd are periodic points and dense in
the unit interval, we can prove the map is chaotic there.
Theorem 2.2. The tent map is chaotic on [0,1].
Proof. Consider the open balls B(£j,Xj) and B(£k, xk), Xj, Xk arbitrary
points in (0,1), and £j chosen so that the balls are subsets of the unit
interval. Let B(ei,Xi)n and B(£iy be the nth forward and nth
backward iterate of B(si,Xi) respectively. Under the tent map there
will be an n such that B(ej,Xj)n completely covers B(£k,xk), and such
that a preimage B(£k,xk)-n C B(sj,Xj) of B(£k,xk) is a single open
interval. Iterate B{£k,xk) forward m times until it completely covers
B(£j,Xj), and so that a preimage B(£j,Xj)_rn C B(£k,xk) of B(£j,Xj)
is a single open interval. There is then a continuous mapping from
B(£k,Xk)~n to B(£k,Xk)m and, by Brouwers fixed point theorem for
one dimension, there is a fixed point x* for the n + m times iterated
map Fn+m : B(£k,xk)-n -* B(£k,xk)m. But B(£k,xk)-n C B(£j,Xj),
so x* £ B(£j,Xj). Thus there is an n + m periodic point x* whose orbit
is in common with both B(£j,Xj) and B(£k,xk), and the tent map is
chaotic.
Irrational numbers are the points in chaotic orbits, and the fact
that any irrational number r will map to another irrational, and that r
will never be repeated gives an intuitive feeling for the nature of these
chaotic orbits. The set of all periodic points that participate in orbits
of period n can be computed exactly. Say that xn, the nth iterate of
Xo, is in bin 0 if it lies in [0, |], and that it is in bin 1 if it lies in (|, 1].
We can construct a doubly infinite string of symbols ...s_2S-i.SoSiS2---,
where sn is the bin number of xn, that represents the history of a point
5


x0 under the map The left or right shift of the binary point corre-
sponds to forward and backward iteration of rc0, For example, the orbit
of | is .§,§,§,§,§,§ so its symbol sequence is ...001001.001001....
Shifting the binary point to the left or right produces the backward or
forward bin sequence, called the itinerary. Each itinerary is unique and
corresponds to one particular orbit, be it periodic or chaotic.
Now suppose we wanted to construct all orbits of period 2. There
are four possible strings that repeat after two cycles:...0000.0000...,
...1111.1111..., ...0101.0101... and...1010.1010.... If x is the initial value
and it is in bin 0, its value is to be multiplied by 2. If x is in bin 1 its
value is to be subtracted from 1 and then multiplied by 2. Thus the
period 2 point with orbit represented by ...0101.0101... has initial value
x = 2(1 (2x)) = |. The period 2 orbit represented by ...1010.1010...
starts om = 2(2(1 x)) = If we compute the points corresponding
to...000.000...and ...111.111... we get 0 and | respectively, the period
one points. Points of higher period are computed in the same way, by
compounding the algebraic operators in 2.1 based on the itinerary.
All periodic points of the tent map are unstable. Irrational points,
no matter how close to a point in a periodic orbit, will move exponen-
tially away from the rational points of this orbit under the action of
the map. This exponential departure, along with our ability to make
small perturbations in a control parameter will allow us to control the
chaos.
To implement control we need an accessible parameter that will
change the action of the map in a uniform way. If we change the height
of the tent smoothly we change the mapping of points in [0, |] and (|, 1]
in a continuous fashion, so we write the equations of the tent map as
f xn+1 = (2 + S)xn for 0 < xn < \ , .
\ Xn+1 = (2 + <5)(l -xn) for | < xn < 1
with 8 set initially to 0. Now we can control the height of the tent by
changing 8. If some iterate x of the map were to land near the preimage
of a periodic point x*, we could perturb <5 in a way that would cause
the next iterate to land on x*. Figure 2.1.3 illustrates the technique
6


applied to a period 2 point |.
Figure 2.1.3. The tent map is controlled around an unstable period two
orbit by changing the height of the tent.
An iterate of a chaotic orbit falls within e of |, the preimage of |.
We increase or decrease 6 depending on which side of | x falls, and the
next iterate lands squarely on Then we reset 6 to 0, and the orbit
is now periodic with period 2. Because the map is ergodic (for any e
ball around some point x and for any point xk in this interval, there
is an N such that \xk+N x\ < e), we can be assured of an iterate of
any point in a chaotic orbit eventually coming as near as we like to the
periodic point of interest. Suppose we want to control about a periodic
point x > and the orbit has landed in the control region B(e, £_i),
the epsilon ball around the preimage of x. The control parameter 8 is
computed as follows:
7


If x > | is the target point and x^x the current iterate, we want
(2 + <5)(1 x) = 2(1 x_i) (2.4)
so
S = -2- (2-5)
Using this <5 for one iteration gets us on the periodic orbit, and we then
reset 8 to 0. A similar calculation will get us to the periodic point if
xk < i
Just as we can direct the orbit of the tent map from any initial
condition to any periodic (or chaotic) orbit by a one-time application
of a perturbation of the tent height, we can direct a trajectory of the
tent map along any sequence of n periodic orbits, coming as close as
we like to any point in [0,1], as periodic points of the tent map are
dense in [0,1]. We merely compute the sequence of perturbations 8k
necessary to take the orbit to the target points xtk, k = 1,2...n and the
controlled tent map becomes
xk+i = (2 + 8k)xk for 0 < xk < ^ (2.6)
xk+1 = (2 + <5fc)(l xk) for ^ < xk < 1. (2.7)
While we can vary the height of the tent to take any point in [0,1]
to any other in one iterate, the size of the perturbation 8 required to
get from xtk to xtk+1, goes to oo as Xtk 0 or xtk > 1.
We specify a maximum perturbation <5max which determines the size
of our control region B(s, x_i), the £ ball around _1. This point is the
preimage of the periodic point we are targeting. There is a trade-off:
if we want to use only a small perturbation, it may take a long time
before the orbit lands in the control region, especially if the preimage of
the point we are trying to control is near 1 or 0. If we want to control
quickly we can consider the whole unit interval our control region and
force the chaotic orbit onto the target point in one iteration, but at the
expense of needing a possibly infinite 8. In a physical control situation,
8


there will certainly be a limit to the size of (5max, but here we choose to
limit ourselves to small control perturbations for the sake of elegance.
We can reduce the size of the perturbations necessary to get from
one place to another by using the sensitivity to initial conditions of a
chaotic map. Under iteration the map will come as close as we like to
any point, so by computing a clever sequence of corrections that takes
us on side-trips between target points, we can minimize some sort of a
cost function. If we are currently stuck in a periodic orbit and want to
go to another, we apply a small perturbation for one iterate of the map
to knock us out of the periodic orbit and onto a chaotic one. Then we
rely on the chaos of the system to eventually bring us into the control
region around a point in the periodic orbit of interest.
Suppose we were limited to very small perturbations. Are there
faster ways of getting from one region of the map to another, riding its
dynamics, so to speak? We know the dynamics of the tent map well:
any interval entirely inside [0,1/2] or (1/2,1] will be expanded by a
factor of two and mapped one to one into the unit interval. Segments
containing A have sets of points that map two to one into the interval
because of the folding due to the change in dynamics at |. Backward
iteration of an interval is up a binary tree: at each branch we have two
possibilities for our previous position. The interval also shrinks by a
factor of two, so the nth backward iterate of a line segment of length
L is a set of 2n segments of length 2~nL distributed in some fashion
over the interval.
Now if we had a target point x* and we iterated B (e, rrl-J backwards
n times, we would have 2n small regions that map into our target region,
the ball around the preimage of x*. The forward iterates of a point x
chosen at random would have a far greater chance of landing within e
of one of the kth preimages of B(e, x*_x) than of landing in B(£,x*_i).
Of course, for each fc, the preimage of B(e,a;!.1) requires a different e,
depending on the distance of the preimage from since for a given
maximum allowable perturbation the region of controllability shrinks
as we approach 0 or 1. Figure 4 shows the range of possible preimages
of a target point x under the restriction of the control parameter to
<5min < <5 < <5max- We can see that points x E [rrmin, :cmax] can all be
9


made to map to x by the appropriate choice of <5, but points arbitrarily
close to 0 or 1 require arbitrarily large perturbations to map to x in
one iterate. We can fix <5 and choose £ to be the min of all e for a
limited set of preimages. When a forward iterate of x comes within
e of B(£,x*_i)~k, where the subscript k indicates the kth backward
iterate of B(e,x_i), we vary <5 to coax it into B(e:,£* j)_*+1 and let the
dynamics of the map take us to B(e, We then apply the control
again so that we land right on x*.
Figure 2.1.4. The height of the tent map can vary from <5min to (5max,
and the effect of this parameter change on the controlable region around
a preimage of x varies inversely with the distance of the preimage from
0 or 1.
Another way to increase our efficiency is to use the other periodic
orbits of the map as routes to the final destination x*. Locating periodic
points is quite simple, and by a judicious choice of a limited set of
periodic orbits we can cover most of the interval with epsilon balls
10


about each point in each orbit. We can vary 8 to cause any initial
point to move onto a periodic orbit from our set in a few iterations,
and by choosing our set of orbits so that each epsilon ball contains the
periodic point of its two neighboring balls, we can also jump from orbit
to orbit by varying 8. There will be a path through these orbits that
will lead us to x.
If we are controlling a physical system or simulating the tent map
on a real computer, then experimental noise or round-off error will drive
our computed orbit away from the true periodic orbit we axe trying to
reach, and the control would have to be reapplied whenever we drifted
too far from our goal.
Now we can control a tent map, one in which the periodic orbits
axe not too difficult to compute. What if we wished to control the
logistic map? Computing high period orbits, even period 5 for instance,
becomes very difficult. However, the OGY algorithm doesnt require
exact knowledge of the dynamics or the periodic points of the system
under study. All we need is an approximate location of the periodic
point we want to stabilize, and the approximate dynamics about that
point. We can iterate any map we like, or collect experimental data
from a physical system, and if there are periodic points and ergodicity,
we can use the OGY,method to stabilize the UPOs.
To locate periodic points of the logistic map we could use a Newtons
method to find the zeroes of the compounded map, but in preparation
for the control of a physical system for which we have no model, we try
the following method. Iterate the equation on a computer and store
several thousand iterates in an array. Then check for pairs of points
xn, xn+k for which \xn+k xn\ < e for some small e, and k is the prime
period of the point we are seeking. A linear least squares fit to the data
can be made, followed by a linear least squares fit to a line orthogonal
to the first. The intersection of these lines will be the center of gravity
of the data points, an approximation of the position of the periood k
point. As the map is ergodic, we can expect a chaotic orbit to approach
arbitrarily close to any periodic point as n approaches infinity.
Once the periodic point x* has been located, we can determine the
local dynamics by noting the rate of escape of points in the epsilon
11


neighborhood of x*. Then we apply the control in the same way as
before, reapplying when necessary.
Once we lock in to a periodic orbit using the control, we can deter-
mine the position of the periodic point x* more accurately by moving
x* experimentally and minimizing the average value of 8 required to
keep the control locked in. Once we have fine tuned the position of
the periodic point, we can recompute the local dynamics to get a more
accurate estimate. After only a few applications of this procedure, we
can fine tune the position of the periodic points and the local dynamics
to the limit of the accuracy of our machine. In this way we can control
any unimodal chaotic map with almost the same ease as we can the
tent map.
2.2. The bakers map
Now let us move up one dimension to a two-dimensional tent map. This
map is sometimes called the two-dimensional bakers map. Imagine a
square sheet of rubber that stretches without deformation. We stretch
the sheet until it is twice its length, then cut it in half and flop the
right half over the left, as illustrated in Figure 2.2.1.
Figure 2.2.1. This two dimensional transformation is sometimes called
the bakers map.
12


This action is equivalent to one iteration of the map
xn+i 2xn
^ Un+l = 2Vn
j n+1 ~ 2(1 Xn)
{ Vn+1 = 1 ~\Vn
for 0 < xn < |
for \ < xn < 1
(2.8)
The physical description of the map did not take into account the
delicate matter of fitting surfaces together after a fold, or the structure
of the left and right sides after an infinite number of iterates. We could
let the map fold in a two-to-one manner, but for reasons of symmetry,
we will take a different course. In order that the map not be two-to-
one, we require that the first iterate fit together like a puzzle, that is,
require the square to be open on the top right side and closed on the
left. The halves will fit together as in Figure 2.2.2.
Figure 2.2.2. Here we see the condition that allows the map to fit to-
gether in a one-to-one fashion: the left half of the top is closed and the
right half is open for all iterates.
The top of the square is now what was the bottom right, so in order
for the next iterate to fit together as did the last, the bottom of the
square must have had this structure:
Figure 2.2.3. The right hand side of the bottom of the square must have
the structure of the right half of this figure in order that the top will
always have the structure illustrated in Figure 2.2.2.
13


Furthermore, as the bottom left half of the square becomes the
entire bottom on the next iterate, and we require the top to fold on
itself in a one-to-one fashion each time, the bottom left half must have
a scaled down structure like that of the bottom right. The left half of
the bottom left half must also have this scaled down structure, as must
its left half, etc.
Hthtt H t!.........I------f..............I--------------1
Figure 2.2.4. The bottom of the square must have this structure overall
in order that the right half of the bottom will have the structure of the
right half of Figure 2.2.3 at each iterate.
A similar structure appears on the left and right sides upon succes-
sive iteration as a result of the Dedekind cut we must make at each
iteration. Referring to 2.8 we see that the first cut and fold makes
the top left boundary open and the bottom half closed, while the right
hand boundary is closed. The next iterate finds the right hand side the
same and the left hand side has a closed bottom half, while the top
half is closed on its upper portion and closed on its lower portion. Re-
peated iteration give the structure we required of the top and bottom
boundaries, and we can see that reverse iteration of the map gives the
same structure to the top and bottom edges of the square as forward
iteration gave to the left and right edges.
This map has the same dynamics in the x direction as forward it-
eration of the one-dimensional tent map, and the dynamics in the y
direction are the dynamics of reverse iteration of the one-dimensional
tent map. Reverse iteration of the tent map causes segments to con-
tract, and if we iterate the unit segment 0 < y < 1, x* where x* is
the x coordinate of a point in a periodic orbit, the segment will ap-
proach a periodic point as a limit. Take for example the unit segment
0 this segment to cycle in the x direction between |, | and |, while in
the y direction the segment will contract and approach y = |, | and |
in the limit. The points (§,§),(§,|) and (|, |) are therefore a period
three orbit.
14


Figure 2.2.5. This figure shows a period three orbit of the bakers map.
In order to control this map, we only have to control the x direction.
There is something new in this map: there is a contracting as well
as an expanding direction. Chaotic and periodic orbits are dense in the
unit square under this map, so we can still hop from orbit to orbit to
get where we want to go. Whats more, to implement OGY control,
we still only have to perturb 8, the x direction control parameter. The
contraction of the map in the y direction automatically brings us into
the two dimensional periodic orbit we are aiming for.
This map shares many of the dynamical properties of maps made
from dissipative chaotic three-dimensional flows. There are expanding
and contracting directions, unstable periodic orbits and chaotic orbits.
However, maps of more complex dynamical systems are usually not so
polite as to have their contracting and expanding directions perpendic-
ular to each other and linear. Furthermore, this map is not dissipative,
so it doesnt have the fractal structure of the continuous time systems
we will examine later.
We can build a three-dimensional dynamical system whose Poincare
section exhibits the dynamics of the tent map, and whose pseudo flow is
similar in behavior to general chaotic flows in three dimensions. First,
15


imagine a long skinny rectangular solid (see Figure 2.2.6). Squash it
gradually along its length so that at one end it is twice as wide as
it is high. Now split it up the middle from the flatter end almost to
the square end. Twist the split ends so that the two top faces come
together. Now form it into a loop, twist it 90 degrees and connect the
two ends.
1.squash
Figure 2.2.6. The bakers map can be easily suspended to give a smooth
pseudo-flow.
This construction gives a continuous evolution in three dimensions
between the steps pictured in Figure 2.2.1 of the two-dimensional tent
map. The periodic points are now continuous periodic orbits and
chaotic points are now chaotic orbits. Once again, the periodic and
chaotic orbits are dense in the space, which is now three-dimensional.
We can consider the mapping at the cross-section that is the unit square
to be the intersection of a flow with a plane perpendicular to the flow,
a Poincare section. A Poincare section of a three-dimensional flow is
used in most OGY control of real physical systems.
Controlling the three-dimensional bakers map is just as easy as con-
trolling the two-dimensional bakers map: we look at the two-dimensional
16


Poincare section of the flow, and change 8 in the same fashion as before
in order to coax the flow onto a periodic point of the two-dimensional
map. Then, every time the orbit pierces the Poincare plane, we check
for divergence from the periodic point and reapply the control as nec-
essary. In a physical situation or a real computer simulation this reap-
plication of the control is necessary because of noise and uncertainty of
the actual dynamics of the system.
2.3. The Lozi map
The Lozi [5] map is a two-dimensional piecewise linear map whose dy-
namics are similar to those of the more familiar Henon[6] map. The
Lozi map, like the Henon map, is an affine transformation that has a
chaotic invariant set for certain parameter combinations.
Definition 2.3. A set A is called an attractor of a map if whenever an
initial point Xq is chosen close enough to A, the distance between the
kth iterate x(k) and the set A goes to 0 as k * oo.
One formulation for the Lozi map is
xn+1 = (1 + p) + ayn + p \xn\ /2gv
Vn+l = ~Xn, '
where a determines the contraction, /3 the stretching and folding, and
p is a parameter that shifts a region to the left or right. A geometrical
view of the action of the Lozi map on a planar region is shown in Figure
17


2.3.1.
Figure 2.3.1. The action of the Lozi map may be broken down into
separate components.
The action of the map on the unit square may be broken down into
its components:
a. the original square is flipped about the horizontal axis by the xn
term,
b. the square is compressed to a rectangle by the ayn term,
c. the rectangle is folded into a chevron by b\xn\,
d. the chevron is shifted up by 1 + p,
e. the chevron is rotated about the origin by 90 due to the switching
of variables.
18


Figure 2.3.2. The basins of attraction of the Lozi map consist of the
set of points whose orbit under the map is asymptotic to a subset of
the plane. There are two attractors, the point at infinity, whose basin
is colored white, and the Lozi attractor, whose basin is shown in black.
The Lozi attractor itself is the white area inside the dark basin. Like the
Lozi attractor, the basin of attraction has a fractal structure.
Most points in the plane move further and further from the attractor
under repeated applications of the mapping. However the points in a
region called the basin of attraction collapse onto an attracting set (see
Figure 2.3.2), which in this case is a strange attractor (see Figure 2.3.3),
a fractal object that is an invariant set of the mapping.
19


Figure 2.3.3. This figure shows severed thousand iterates of an initial
condition inside the basin of attraction for the Lozi attractor.. The orbit
appears to be chaotic, and in fact, the Lozi map has been shown to have
a strange attractor with a hyperbolic structure (it is the union of an
infinite set of saddles).
These three deformations (stretching, folding and rotating) are re-
sponsible for the chaotic motion of individual points on the attractor.
Almost any two points that are initially nearby are stretched away from
each other, only to be folded and rotated back nearby, but in a different
layer. There is an infinite number of layers, and there are layers infin-
itesimally close to each other. Points jump in what appears to be an
irregular manner all over the attractor as the map is iterated. The Lozi
map has a set of stable manifolds of periodic points that is dense
in the basin of attraction, but as these sets are piecewise linear and
non-differentiable, they are not technically manifoldsJ
En
The state vector is the current point Xn -
Vn
path of the point in the plane under successive itera
An orbit is the
dons of the map.
20


We can write the Lozi map in matrix notation,
Xn+1 Vn+l = 1 + P 0 + p -1 a 0 Xn Vn for xn < 0 (2.10)
*Ui+1 Vn+l = ' 1+p' 0 + ' -p -1 a 0 Xn Vn for xn > 0
where two cases account for the absolute value in the original formula-
tion.
As the map is iterated for some initial condition X0 in the attractor,
the orbit will fall in the left or right half of the plane, designated L and
R respectively, and for each iterate, the map applied will depend on
whether Xn G L or Xn G R. The orbit of the path can be specified
to any degree of accuracy simply by knowing enough of its left-right
history. The itinerary of a point is its L-R history, written as a doubly
infinite sequence such as
...LLKRLLLKRL. LLRRLLLKRL...
The first digit to the right of the binary point is L, the current state of
the system. Upon forward iteration of the map the current state will
follow the itinerary. The itinerary specifies a periodic orbit uniquely in
this map, and a right or left shift of the binary point gives the forward
iterates or backward iterates of the. map.
A periodic point with period A; is a point X* whose image under k
mappings is again X*. For the set of parameters that make the Lozi
map chaotic, all of these periodic points are unstable. Upon iteration,
any state vector not precisely on a periodic point will wander away. The
Lozi attractor is the closure of the dense set of all periodic points. There
is an infinite number of periodic itineraries, and the periodic orbits that
they define fill a fractal region of space. There are also non-periodic
itineraries that correspond to chaotic orbits, but we can find a periodic
itinerary that matches the itinerary of the chaotic orbit to as many
places as we desire. Therefore,orbits whose itinerary is non-repeating
(chaotic) lie in the closure of the open set of periodic orbits.
21


Consider a 5-periodic point X* with itinerary
...LLKRLLLKRL. LLKRLLLKRL...
Which point exactly does this itinerary represent? To answer this ques-
tion, we rewrite the equations and in a more convenient form. Letting
*n = %n tt it P j [ 0 J P CX j) -i 1 1 i O' 0
(2.11)
gives us X,,n = U + LXn for Xn G L (2.12)
Xn+j = U T RXn for Xn E R,
or
Xn+1 = F(Xn).
Now we can easily express multiple iterates of the map in terms of its
itinerary. For example, five iterates of the initial system state Xn whose
forward itinerary is LLRRL are
Xn+1 = U + LXn
Xn+2 = U + LU T LLXn
Xn+3 = U + RU + RLU + RLLXn
Xn+i U + RU + RRU.+ RRLU + RRLLXn
Xn+5 U + LU + LRU + LRRU + LRRLU + LRRLLXn
= [I + L + LR + LRR + LRRL]U + LRRLLXn
= TU + KXn
where K, the full itinerary matrix, is the product of the matrices L and
R associated with the reverse of the itinerary (which is LRRLL above).
The partial itinerary matrix T is the sum of the identity I and the first
22


A: 1 partial products of K, which is I+ L + LR + LRR + LRRL above.
A periodic point of period k is a fixed point (period one point) of the
k times iterated map
Xn+k = TU + KXn.
Solving for the period k point X* we obtain
X* = [I-K^TU.
As an example, we can find the period-3 point whose itinerary is
LLR. We give p, a, and (5 simple rational values for which the Lozi
map has a strange attractor, p = 0, a = and /3 = |. A period-3 point
is a fixed point of the map
Xn+3 = U + RU + RLU + RLLXn
where
r -7 11 r 7 ii r 7 1 1 [ -343 -57 1
K = RLL = 4 2 -1 0 4 2 -1 4 -1 2 0 = -6tl 16 1 £Th
T
I RRL
' =69 ^3
t
. 4 2 .
1 0
0 1
+
zl I
4 2
-1 0
Solving for the period three point we have
- ' l 0 ' r -343 -57 1 -1 r -69 -3 1 ' 1 '
64 32 16 8
0 1 -41 -7 -11 1 0
_ L 16 8 J . L 4 2 J
-68
157
-412
471 J
Since the Lozi map is a well-defined geometrical transformation that
maps a region of the plane into the attractor by successive stretches
and folds, we expect characteristic rates of stretching and folding. For
the infinite number of unstable periodic points in the attractor, these
23


directions of stretching and folding are locally linear and vary over the
attractor.
Lets look at the characteristic directions and rates near a period
one point. From the point of view of the fixed point X* the state vector
is Xn = AXn+X*. The next iterate of the map is Xn+i = AXn+i +X*.
As is the usual practice for control of a two-dimensional map, we expand
in a Taylor series about the fixed point X* (this is unneccesary in the
linear case, but the idea will be used for nonlinear systems later). We
have
A Xn+1+X* =F(X*) +
r §L 1
dx dx
ag 22
- &y dy .
AX + 0(AXS
In the case of the Lozi map, the higher order terms in the Taylor series
vanish. Since X* is the fixed point of the once iterated map F(X*) =
X*, we have
AXn+i
r 2* §L 1
dx dx
22 dg
. dy dy .
AXn = AAXn,
(2.13)
where A is the Jacobian of the map.
Here is a simple example using the parameter values p = 0,a =
and (3 = Because of the absolute value in the Lozi map, the Jacobians
will be different in the left and right halves of the plane:
Al
J(h(x,y)9i.(x,y)) =
*(1+ & + ?*) + ^ + H
r ^ df I
dx 22
. dy dy .
&(-*)
£(-*>
7 1 -
4 2
-1 0
Ar = J{fR{x,y),gR{x,y)) =
r 9/ df -]
dx
£2
. dy
%
dy .

*(-*)
&(-*)
24


=7_
4
-1
1 '
2
0
Suppose that the state vector is X
. Then
' 5 ' ' 4 ' ' 1 '
II * 1 II 11 i34 - 13 -2
. 13 . . 13 . . 13 .
The Jacobian AR applied to AX gives us the change from the point
of view of the fixed point. This change must equal the change given
by the Lozi map applied to Xn, when viewed from the fixed point. We
should have
AIn+i = ARAXn = U + RXn-X*
Indeed, we get
' -7 1 ' ' 1 ' 5 '
4 2 -1 0 X 13 . = . 13 .
Ar
A X
AXn+l
and
1 ' -7 1 ' JL ' ' 4 ' ' 5 '
1 0 + 4 2 -1 0 i34 = 52 -5
13 - . 13 . . 13 .
u
R
Xn
AXn+l
Usually the Jacobian is used this way to estimate the local linearized
dynamics near the fixed point of a nonlinear system, but the Lozi map
is already linear (except for the affine part
0
), so the Jacobian
exactly describes the local dynamics near a periodic point.
Near a fixed point the direction in which the map stretches space
is called the unstable direction and the direction in which the trans-
formation contracts space is called the stable direction. The stable
and unstable directions correspond to the stable and unstable sets near
the fixed point. These directions are given by the eigenvectors of the
Jacobian A. The eigenvalues associated with the eigenvectors give the
amount of stretch or compression of the space along these characteristic
directions. The fact that the Lozi attractor has a hyperbolic structure
25


guarantees that all of the unstable periodic points on the Lozi attractor
are saddles, that is, they have attracting and repelling directions.
+4 4-
4

Figure 2.3.4. This figure illustrates four different types of saddles. The
numbers next to the points indicate the initial point and its next three
iterates. The upper left saddle, a hyperbolic saddle, has eigenvalues
0 < Xs < 1, Xu > 1. The upper right saddle, a flip-out saddle, has
eigenvalues 0 saddle, with eigenvalues 1 < As < 0, Aw > 1. The bottom right saddle
is a flip saddle with eigenvalues 1 < As < 0, Xu < 1.
We designate p as the control parameter and replace it with a func-
tion p(x,y) that will change the dynamics of the Lozi map for one
iteration whenever the system state comes within a certain distance of
the periodic point we wish to stabilize. This temporary change in the
dynamics is calculated so that the system state will land on the stable
manifold of the periodic point on the next iterate. We have chosen
p = p(x,y) to be our parameter because of the specific way a change
in p affects the local dynamics. To calculate the perturbation that will
direct the orbit onto the stable manifold we need to know something of
the local dynamics of the periodic point we want to stabilize.
-M-
Hyperbolic Saddle
4------------
Flip Out Saddle
Hr
-M-
-H-
Flip In Saddle
Flin
26


There is an attractor for a = p = 0. Let us stabilize the
period one point whose itinerary is ...RRR.RRR... Earlier we found
Xn+i = U + RXn,
so for the periodic orbit ...RRR.RRR...
+
En+l
Vn+1
1 + 0
0
=1 I
4 2
-1 0
Xn
Un
The fixed point of this once iterated map is found by solving
X* = [I-K^TU,
where T = I and K =
zlL i
4 2
-1 0
. Numerically the fixed point is
T 1 0 ' 1 1 2 - -1 1 0 ' ' 1' 4 13
[ 0 1 -1 0 0 1 0 -4 13 .
We will perturb the dynamics near this point to steer the orbit onto
the stable manifold.
The local dynamics are determined by the eigenvalues \u, As and
the eigenvectors eu,e3 of A. For reasons that will become apparent
shortly we will also need the left eigenvectors fu and fs. These vectors
are orthogonal to eu and es and are normalized such that
fu' ' &S fs ' O II
fu ' < II 3 * 1
27


fs
*
K
\
\
i
fu
Figure 2.3.5. The right and left (contravariant) eigenvectors are shown.
The eigenvalues of Ar are Xu = ~7~^, As = -~7~^v/^7, while the
associated eigenvectors are
Ci;
7+%/tf
8
1
7\Zl7
8
1
,fs =
T7
17-7
2v/i7
, /u
-4
\/l7+7
2VT7
We also need to know how the position of the fixed point, its associated
stable and unstable manifolds, and thus the local dynamics, change
upon a parameter perturbation. For example, if we change p from 0 to
| then the fixed point X* changes from
' 4 ' ' 9 '
a to a
. 13 . . 26 .
, for a change
AX =
L 26 J
. Therefore the change in the location of the fixed point
4
a
L 13
with change in parameter p is -§^X*
Let dp be some small change in p. Then the location of the per-
turbed fixed point upon this change will be X* +6p-^X* (see Figure
28


2.3.6).
Figure 2.3.6. The perturbation of the fixed point is shown above.
For convenience of notation we write g = so that 8pg = 8p
#-x*. P
dp
In one iteration a phase point Xn evolves to Xn+i. Relative to the
fixed point X* these points may be represented by AXn = Xn X*
and AXn+i = Xn+1 X*. Our goal is to push the state variable
onto the stable manifold of the desired periodic orbit in one iteration
under a perturbation of the dynamics. Once on the stable manifold,
the perturbation is turned off and the natural dynamics will draw the
orbit into the fixed point. Figure 2.3.7 illustrates how this process looks
schematically.
29


Figure 2.3.7. The schematic above illustrates the basic principle of OGY
control. In 2.3.7a, we see the current system state with an arrow point-
ing to its next iterate. In 2.3.7b the system has been perturbed by an
amount calculated to send the system state to the stable manifold of the
unperturbed system in one iterate of the map. In 2.3.7c the perturba-
tion has been turned off, as the system state is now on the unperturbed
stable manifold. The stable dynamics will henceforth evolve the system
state toward the periodic orbit as in 2.3.7d.
We will vary the parameter p to achieve this perturbation, so we
write the local dynamics from the point of view of the shifted fixed
point. Using a prime (') to indicate the quantities in the perturbed
system, the state vectors are now
AX'n = AXn 6pg
and
30


A^n+i AXn+1 Spg.
We can treat the dynamics at the shifted fixed point as we did in
2.13 and write
ax;+1 = a'ax;,
where A! is the Jacobian at the perturbed fixed point. Assuming a
small perturbation Sp, we may approximate A' by A and write
AXn+i Spg = A[ AXn Spg}. (2-14)
In this case, the parameter change affects only the side-to-side shift of
the attractor and not the matrix A, so A! = A.
A state vector that is on the stable manifold is orthogonal to the
left unstable eigenvector. Therefore the requirement that the next state
vector be on the stable manifold of the unshifted fixed point can be
written
fu AXn+1 = 0.
We must now express bp as a function of the eigenvectors and eigen-
values of the system and current position AXn. Rewriting 2.14 as
AI+i = Spg + A[AXn bpg] (2-15)
and dotting both sides of this equation with fu yields
0 = fu Spg + fu A[AXn Spg}.
We wish to solve for Sp, so to this end we write A in terms of its
components via the representation (see Appendix A)
A = A uelfu + K^fs (2.16)
Substituting 2.16 into 2.15, we obtain
0 = Spfu g T fu [Aueu/u T A5esy3] [AAn Spg}
31


bpfu 9 T \_^ufu &ufu T Asfu &sfs\ [A An &P0\
= fipfu 9 + [Kful [AAn Spg]
bpfu 9 I" ^ufu AXn ^u^pfu 9
Thus
Sp
Au fu A-Xjj
Au 1 fu-9
(2.17)
which gives the control perturbation we need based on the current
system state as seen from the unperturbed fixed point and the precal-
culated quantities A, fu and g.
Lets try this scheme on the example discussed above. Substituting
the values of \u, fu, AX and g we found earlier into 2.17, we obtain
-7-vd7 8 4 VV7-7 y/\7 2y/l7 AXn
-7-Vrr 8 1 4 Vl7-7 yi7 2\/l7 \A ^1 L13 13J
Whenever an iterate of the map enters a small box around the fixed
point (whose size is limited by the extent of the local stable manifold),
we change the value of the parameter by bp for one iteration. Suppose
an iterate of the map lands on Xn = [A) 5 so AXn = Xn X*.
Then our control rule says we need to apply a perturbation
7\/l7 +17 '4 y/17 7 -1 2'
32 y/ff 2y/Yf 13 13
A quick calculation shows that
-17- 15y/l7
104\/l7
1 -17-15y/l7
io4vTr
0
r -7 1 ] r 3 1 39-^17-17
+ 4" 2 1 0 1 1 1 rtCi|2 l
)
32


which, when translated to the fixed point, is on the stable manifold
of X* [^7j, . The control process for this example is depicted in
Figure 2.3.8 below.
Figure 2.3.8. This figure illustrates the shifting of the manifolds nec-
essary for control of the Lozi map, along with the measured quantities
necessary to calculate the proper shift.
An iterate of the map Xn lands near the fixed point X* (its man-
ifolds are the dashed lines). The correction is applied, shifting the
fixed point (and its manifolds) to X* +8pX* (solid lines). Under these
dynamics the point Xn maps to Xn+i The correction is turned off,
and the system state is on the stable manifold of the unshifted fixed
point. The unperturbed dynamics now act on the system state under
successive iterations to carry it along the stable manifold into the fixed
point.
We can check this by dotting our answer minus the fixed point with
the unstable left eigenvector:
33


39VT7 17 4 -3 -4 1 1
104 Vl7 13 14 13 y/VT 2 y/Y7
7y/V7 17 1 1 1
104\/l7 13 y/Tf 2y/l7
We can control higher periodic points similarly. Control of a single
point in a periodic orbit guarantees control of the other points in the
orbit. The method of control is essentially the same as one would use
to control the Henon map, except that the control would have to be
applied continually to the Henon map to account for the non-linearity
of the stable and unstable manifolds associated with the fixed points.
Formulation of the OGY control rule is not difficult, even for physi-
cal systems for which no mathematical model is known. A phase space
may be built from a time series (a list of successive measurements of
some property of the system) by the method of delay coordinate em-
bedding [7], and the fixed points and local linearized dynamics may be
determined by a combination of the methods of closest approach least
squares fit, and Newtons method.
3.4


3. The Horseshoe
3.1. The Smale horseshoe
We have seen that piecewise linear chaotic systems are easy to con-
trol. The analysis of the dynamics of nonlinear systems, both maps
and flows, is made simpler by observing the correspondence of their
dynamics with those of an abstract chaotic structure, the Smale horse-
shoe. The horseshoe is a fundamental chaotic object. The dynamics of
the invariant set of the Smale[8] horseshoe map mimic, in abstract, the
dynamics of a generic chaotic map in the plane; that is, the horseshoe
contains unstable periodic orbits and chaotic orbits and their stable and
unstable sets, as does the generic chaotic map. We will investigate the
dynamics of the horseshoe, and show that there are similar dynamics
in the Henon map, and that a continuous interpolation between the
invariant set of the horseshoe map (along with its stable and unstable
structures) and its first iterate, provides a model for the dynamics of a
chaotic flow in less abstract systems. We begin with a few definitions
from Devaney[9].
A system is structurally stable if every nearby system has essentially
the same dynamics. Structural stability is an important property for
a chaotic system to have if we wish to control its unstable periodic or-
bits (UPOs). To control a chaotic system we must perturb a system
parameter, and we require that the perturbed system have similar topo-
logical structure to the unperturbed system. It would be impossible to
use capture-and-release control (CR), for instance, if the UPO we are
trying to stabilize disappeared upon a small perturbation (see section
5.3).
Definition 3.1. Let f and g be two maps. The C distance
between f and g is given by
Mf>9) =sup \ f{x)-g(x)\.
xeR
The Gr distance is given by
dr(f,9) =sup (|/(z) g(x)\, If'(x) g'{x)\,..., \fr(x) gr(x)|)
eK
35


Definition 3.2. Let f : A A and g : B * B be two maps.
Then f and g are said to be topologically conjugate if there exists
a homeomorphism h : A B such that h f = g h.
Definition 3.3. Let f : J J. Then f is said to be CT struc-
turally stable on J if 3e > 0 such that whenever dr(f,g) < e for
g : J J, it follows that f is topologically conjugate to g.
The classic Smale horseshoe is formed as follows. Consider the sta-
dium shaped region D in Figure 3.1.1 below. Squash D in the vertical
direction by a factor of 6 < ^ and stretch it in the horizontal direction
by a factor of p > 2. Bend the region over in the shape of a horseshoe
and lay the horseshoe over D in the manner shown. Now repeat this
process, stretching and folding the horseshoe again by the same factors
and laying it in the stadium.
Figure 3.1.1. The Smale stadium consists of a rectangular central region
fitted with semi-circular endcaps.
36


Figure 3.1.2. The Smale stadium, squashed, bent, and placed back inside
itself.
Call the action of the map F. There is a unique attracting fixed point
p in D\ since F is a contraction mapping, and as D% is mapped to D\,
all points in D\ U tend to p under iteration, that is, lim Fn(q) =
n*oo
p for all q E Di U _D2. Furthermore, any point r E S whose image is
not in S for all n obeys lim Fn(r) = p. We are interested in the points
n>00
s E S such that Fn(s) E S for all n.
Consider the two segments of S that are mapped back inside S by
the horseshoe map F. Call these Hq and Hi, and their preimages Vo
and V\ (see Figure 3.1.3). Since F : S * S is a linear map, it preserves
horizontal and vertical lines in S. The width of Vo and V\ are p and
the height of Hq and Hi are 6, and if h is any horizontal line segment
in S whose image under F is also in S, then the length of F(h) is ph.
Likewise if v is any vertical line segment in S whose image under F is
37


also in S then the length of F(v) is 6v.
Figure 3.1.3. The intersection of S and F(S) gives H0 and Hi, and
the preimages of these horizontal strips are Vo and Vi.
Suppose Fn(s) G S Vn > 0. Then s must be in Vo U Vi, F(s) G
VoU Vi, F2(s) G VoUVi,..., for all points not in VoUVi map to
Thus, we have that s G F~n(VoU Vi) for all n > 0. The inverse image of
any vertical strip of width w in Vo or Vi that extends from the bottom
to the top of S' is a pair of strips of width -w, one in Vi and one in Vo,
that extend from the bottom to the top of S. The inverse image F~l of
Vo U Vi is a set of four rectangular strips of width \w, two in Vo and
two in Vi (see Figure 3.1.5), the inverse image of i7'_1(Vo U Vi) is a set
of eight vertical strips of width -tiw, etc. Therefore lim F~n(V0 U Vi)
is the product of a Cantor set with a vertical interval. Any point s G S
such that Fn(s) G S Vn > 0 must be in this set which we label A+.
By the same type of reasoning, we see that if a point s G S such that
F~n(s) G S Vn > 0, then it must belong to a product of a Cantor set
with a horizontal interval, and we label this set A_. Any point s G S
38


such that Fn(s) E S Vn must be in the intersection A = A+ D A_.
Figure 3.1.4. The second iterate of the Smale horseshoe map is shown
above.
Figure 3.1.5. Shown is the intersection of the horizontal and vertical
strips after the first two iterates of the map. In the limit the intersection
is a Cantor set.
Just as we did with the tent map, we may define a symbol sequence
on A. The sequence is doubly infinite and is written as
...S2, S-j.So, Si, S2,(3-1)
where the Sj are 0 or 1 depending on which vertical strip Vo or Vj s is in
at the jth forward iterate of the map, and the S-j are 1 or 0 depending
39


on which horizontal strip s is in on the jth iterate of the maps inverse.
The sequence ...s_2, S-i-So, si, s2, uniquely defines a point in A, and
the left or right shift of the binary point gives the backward or forward
iteration of that point respectively. This shift map is a model for the
dynamics of s under F restricted to A.
As F is topologically conjugate to the shift on the symbol sequence,
we can define a metric on F by
<*[.]= E <3-2)
z=OO
where (s) = ...s_2, S-i-So, Si, s2,.
Definition 3.4. Consider a set Q and a mapping F : Q Q.
Two points pi and p2 are forward asymptotic if Fn(pi), Fn(p2) 6
Q V n > 0 and lim \Fn(pi) Fn(p2)| = 0.
71>00
Definition 3.5. Two points p\ and p2 are reverse asymptotic if
Fn(pi), Fn(p2) e Q V n < 0 and lim \F~n{Pi) F"n(p2)| = 0.
n>00
Points in any vertical segment in A+ are forward asymptotic, and
points in A_ are reverse asymptotic. We can now formally define the
stable and unstable set of a point s in A.
Definition 3.6. The stable set Ws of s is the set of points t that
axe forward asymptotic to s, or Ws(s) = {t ] ]Fn(t) Fn(s)\ > 0
as n * 00} and the unstable set Wu of s is the set of points t that
are reverse asymptotic to s, Wu(s) = {t \ \F~n(t) F-n(s)| > 0
as n > 00}.
Consider a fixed point s = ...111.111... £ A. Its stable set contains
not only the vertical segment l3 in which it resides, but also any segment
l that maps into ls. Thus the stable set of s consists of UkF~k(ls). The
unstable set of s G A is different in form. Let lu be the horizontal
40


segment in which s resides. Forward iteration of the map will stretch
and fold lu, giving the structure in Figure 3.1.6.
Figure 3.1.6. The stable and unstable sets of the point s are shown in
this figure.
Periodic points are dense in the horseshoe, and orbits are topologically
transitive and have sensitive dependence on initial conditions. These
conditions are the signature of chaos, and a system that can be shown
to have a horseshoe, can be proven to be chaotic, at least on a subset of
its attractor, according to the definition of Devaney[9, Robert Devaney,
Chaotic Dynamical Systems, Addison-Wesley, 1989].
Definition 3.7. f : J * J is said to be topologically transitive
if for any pair of open sets U,V E J there exists k > 0 such that
Definition 3.8. / : J J has sensitive dependence on initial
conditions if there exists 6 > 0 such that, for any x E J and any
neighborhood N of x there exists ay E N and n > 0 such that
\fn(x)-fn(y)\>6.
Definition 3.9. Let V be a set. f : V > V is said to be chaotic
on V if
41


1. / has sensitive dependence on initial conditions
2. / is topologically transitive
3. periodic points are dense in V
Devaneys definition is equivalent to that of Taylor and Toohey (Def.
2.9) when V is a uniform Hausdorff space. We can see that the horse-
shoe is chaotic by considering the symbolic dynamics.
Density of periodic points: We must exhibit an orbit that con-
verges to an arbitrary point s = ...s_2, S-i-So, $1, s2,... Let r =
...So...sn, So--.sn.So...sn, so-..sn... be the sequence that repeats the
first n symbols of s. Then d[r, s] < and r > s.
Topological transitivity: We must exhibit a point that comes arbi-
trarily close to every other point in A. Consider the symbol string
s* = ...Oil 010 001 000,11 10 01 00, 1 0.0 1,00 01 10 11, 000 001
010 Oil... formed by concatenating all possible permutations of
strings of length k, k = 1, ...,n (the commas merely delimit the
groups of permutations of strings of length k). Then for some
shift a* the itinerary will agree with that of any point in A to the
precision we desire.
Sensitive dependence on initial conditions: Let r,s £ A and have
identical symbol strings in the first n places to the right of the
binary point and for the first m places to the left. Then iteration
forward or backward will eventually shift out the identical strings
and the orbits will diverge.
To sum up the action of the horseshoe map, we note:
The horseshoe mapping of the Smale stadium has an invariant
set A that is the product of two Cantor sets.
Dynamics on A are chaotic under the map.
42


There are unstable periodic orbits of all periods, and periodic
points have stable sets that consist of the vertical line segment
in which they reside and the set of all segments that map to the
vertical segment.
The stable and unstable sets of points in A are orthogonal.
3.2. The Henon map and its horseshoe structure
The horseshoe structure arises naturally in dissipative chaotic maps of
the plane and Poincare maps of chaotic flows. Consider the Henon map
xn+1 = a + byn-xl
Vn+l = n
where a and b are parameters. The Henon map is invertible, dissipative
and has chaotic dynamics for certain values of a and b. A few thousand
iterates of the Henon map are shown in Figure 3.2.7.
Figure 3.2.7. Several thousand iterates of the Henon map are shown for
parameters a 2, b = \. This figure and Figure 3.2.8 were produced
by the program Dynamics by James Yorke.
43


This map has a fixed point X* at approximately (.922, .922) and it
also has a set of points that are forward asymptotic to X*. This set
of points consists of the points in the intersection of the stable and
unstable manifolds of X*. Figure 8 shows the Henon attractor and part
of the stable manifold Es of the period one point (fixed point) X*.
Figure 3.2.8. Shown is the stable manifold of a period-one point of the
Henon map and the Henon attractor.
The intersection of the unstable manifold Eu of X* with its stable
manifold is the analog of the set of points in the intersection of the
stable set Ws and the unstable set Wu of a period one point in A, the
invariant set of the Smale horseshoe. This set Ax* is defined by Ex. n
Ex. = Ax*.
Let us follow a segment of the stable manifold Esx. as it is iterated
forward. Consider a segment lQ G Es with endpoints Xa,Xb G Ax*
and containing no other points of Ax*. Call the set of all segments so
defined Lx*- As only two of the points of lo are in Ax*, the segment
must map only to other segments like itself, that is, the forward orbit
44


of Iq is in Lx*. Under the action of the map Xa and Xb will approach
X* arbitrarily closely, and the length of Iq > 0 as n > oo.
As in the horseshoe, the unstable manifold of the Henon attractor
A is the same for periodic points of all periods and for points in chaotic
orbits. Each periodic point has its own stable manifold, and the same
argument as above can be made for each of these stable manifolds,
namely, any point in a segment L 6 LXi, where X) is the jth point
in the periodic orbit i will approach X1- arbitrarily closely as n * oo.
The set Sp of all stable manifolds for all periodic points in the Henon
attractor is dense in the basin of attraction of the attractor. The closure
of the set Sp is the entire basin, and the set that closes Sp is the set of
stable manifolds Sc of the chaotic orbits in A.
Definition 3.10. Let A be an invariant set for a discrete dynamical
system defined by f : Rn > Rn. A hyperbolic structure for A is a
continuous invariant direct sum decomposition T\Rn = E\ E^ with
the property that there are constants (7>0,0 1. if v e E%, then \Df~n(x)v\ < CXn |v|;
2. if v e E, then \Dfn(x)v\ < CXn \v\.
The invariant set A of the Smale horseshoe has a hyperbolic struc-
ture. Essentially, a hyperbolic structure implies that all the periodic
points are of saddle type. The Henon map may be shown to have a
horseshoe structure; that is, it may be shown to contain a chaotic set
and unstable points of all periods. This set is a product of Cantor sets,
and lives in the unstable manifold of the Henon attractor. However, we
do not know whether all the points that appear to be in the attractor
are part of a hyperbolic structure (there are good reasons to suppose
that they are not [10]). This means that what appears to be chaotic mo-
tion over the entire connected unstable manifold of the Henon attractor
may instead be a chaotic transient preceding asymptotic approach to
any one of an infinite number of stable periodic orbits of arbitrarily
high period. We take the view that although we dont know for sure
whether the apparent chaotic evolution of an orbit over an attractor is
45


really chaotic or just a chaotic transient, for the purpose of develop-
ing a control strategy there is no difference. Whether we have a true
chaotic orbit or one of very long period, our goal is to stabilize low pe-
riod unstable orbits, and to navigate through the attractor along these
orbits. In either a simulation on a computer or in a physical experi-
ment, chaotic orbits and orbits of very large period will appear to be
the same.
3.3. The suspension of the horseshoe
Our purpose in considering the horseshoe map is to use its suspension
as an abstract representation of the dynamics of the flow of a chaotic
dynamical system. The dynamics of a Poincare map of a flow determine
the dynamics of the flow itself; that is, if the Poincare map has a
periodic point of period n, the flow has a periodic orbit of period n,
and if the Poincare map has a chaotic orbit, the flow has a chaotic orbit.
The stable and unstable manifolds of the Poincare map of a chaotic flow
act very much like the stable and unstable sets of the horseshoe, and
with this in mind, we suspend the horseshoe in a way that forces the
stable and unstable sets of the suspension to act like the stable and
unstable manifolds of a flow. Consideration of the geometry of this
suspension leads to the construction of a general framework in which
control by stable subspace targeting takes place.
We provide some definitions:
Definition 3.11. Let ip be a flow on a manifold M with vector
field X and suppose that E is a submanifold of M of codimension
1 that satisfies 1 2
1. Every orbit of p meets E for arbitrarily large positive and
negative times.
2. If x E E then X (a;) is not tangent to E.
Then E is said to be a global cross-section of the flow.
46


Definition 3.12. Let y e E and r(y) be the least positive time
for which P(v) = V G S-
As with the bakers map, the horseshoe can be suspended to give
a three-dimensional flow. While the bakers map required some care
to ensure that it was not two to one, the suspension of the horseshoe
map is straightforward, and its generalization can model the actual
dynamics of a physical system rather well.
Consider the physical description of the iterative process that leads
to the Smale horseshoe. We take a stadium D composed of a central rec-
tangular region S and two endcaps D\ and D2, stretch it linearly in the
horizontal direction, while squashing it linearly in the vertical direction,
fold it over and reinsert it into its original boundaries in such a way that
only points in S get mapped back into S. This insures a structurally
stable dynamical system consisting of the points s £ A, and the map-
ping F restricted to A. The itinerary S(s) = (...s_2, S-1.S01 Si, s2, )
is a doubly infinite sequence of symbols 1 and 0 that give a record of
the travels of s under F. Take A and its stable and unstable sets W3
and Wu and call their union G. We cross G with the circle C to get
G X C, and stretch and fold G X C over itself in such a way that after
one circuit of the circle vertical lines in Vq and V\ have mapped into Vo
47


and V\.
Figure 3.3.1. We may connect the line segments in the stable set of the
horseshoe thereby changing them to manifolds.
48


Figure 3.3.2. The ends of the stable sets are connected in this figure
to make the stable sets into stable manifolds. When the horseshoe is
stretched and folded, the manifolds unbend as the straight segments join.
The unstable set is a differentiable manifold, and we can make the
stable set a differentiable manifold by including in it the paths traced
by points in the right hand side of the top edge of A+ as it stretches and
folds over to rejoin the left hand side, as well as the paths traced out
by the complimentary points on the bottom of A+ (see Figures 3.3.1
and 3.3.2).
We can fold other horseshoes in continuous time that have proper-
ties superficially similar to the apparent stretching and folding of other
dynamical systems. We emphasize that just because the stretching and
folding seems similar, there is no guarantee that a horseshoe structure
persists through an infinite number of stretch and fold operations. We
have already seen the similarity between the stable and unstable sets
of the Smale horseshoe and those of the Henon attractor. In section
4.1 we will look at the stretching and folding of the pendulum attrac-
tor, and compare the pendulum dynamics to those of the suspended
horseshoe. Depending on the values of parameters in the pendulum
equation, the pendulum may have a strange attractor corresponding to
chaotic motion that includes excursions over the top of the pivot, or
chaotic motion for which the pendulum bob never goes over the top.
In either case, there exist unstable periodic orbits whose manifolds are
flip saddles, like the flip saddles of the Smale horseshoe. In the course
49


of one system cycle, the manifolds associated with a period one orbit in
either the pendulum or the suspended Smale horseshoe undergo a half
twist before rejoining, forming a Mobius band.
The rotation of the manifolds of the flip saddle are what make pos-
sible capture and release control, and a knowledge of the movement of
the manifolds as an attractor goes through its cycle of stretching and
folding allow us to design general control rules beyond the map based
rules of OGY.
3.4. The control of the horseshoe
Like the bakers map, the horseshoe map can be controlled. We imagine
that we have two parameters that control the geometry of the horseshoe;
8 < | which controls how much the stadium is flattened, and p which
controls the stretching. We fix 8 and allow p to vary so that the position
of the stable set Ws of A shifts position as p varies. The change in
position of the stable set is continuous with change in p, as long as
the endcaps Di, D2 do not map to the central rectangular region S.
Suppose an iteration of the horseshoe map brings a system state s
within e of the state s* the state we wish to control. There will be
a parameter value p(s*,s) for which the dynamics of the map carry s
to L* the vertical segment containing s*, in one iteration. Successive
iteration of the map brings any point in l3* as close to s* as we like.
Figure 3.4.1 below is a sequence showing how ls> approaches s* when s*
is the period one point ...1,1,1.1,1,1.... The points shown are members
50


of the invariant set of the mapping.
Figure 3.4.1. Here, the orbit of a line ls in the stable set of a period
one point with itinerary ...111.111... is shown under three iterations of
the horseshoe map.
In the same way we can target a preimage of l3-, and again, suc-
cessive iteration will bring the segment as close as we like to s*. If our
perturbation is limited to a small e, we may choose a set of periodic
orbits {s1; S2, ...sn}, each of which we can target by the method above.
We can restrict the number of preimages of lSk we will consider in our
targeting procedure, noting that the accuracy of our calculation of the
51


position of the preimages of l3k decreases as we go further back in time.
If we choose a sufficient number of orbits, an e region around any verti-
cal line in the stable set of one particular orbit will contain a member of
the stable set of another orbit in the set {si, s^, ...sn}. Then, by thread-
ing our way through the preimages of the l3k, k = 1, ...n we can move
from one orbit to another in an efficient fashion.
We can define an operator that evolves local behavior continuously
in the horseshoe. Consider the suspension of the horseshoe G x C, that
is, the union of the stable and unstable sets of the invariant set A of
the horseshoe mapping F crossed with the circle C. Define a periodic
folding operator P(t), t = tmod2n such that P(r)(G X C) gives the
Poincare section of Gx C at time r. Then P(0)(G xG) P(2n){G x C)
and P(t)(G X C) = P(r 2n7r)(G X C). We noted earlier that a piece
4 of the stable set A of the horseshoe maps to lk+i after one iterate,
and to 4+n after n iterates. The discrete time operator P(2ir) will be
designated P, and Pn designates n applications of P. Therefore P(2ir)
Ik = Plk = 4+1, and Pnlk 4+n- Similarly, the unfolding operator
P_1(r), r = tmod27T is defined so that P(r)(Gx C) is the Poincare
section of G x C at time r.
The operator P(t) takes an initial segment l and evolves it smoothly
from one iterate of the horseshoe map to the next. If the initial segment
Iq was in the stable set of a period one point, for instance, and l0 was
inside the attractor, the map would take Iq to 4, the segment one layer
in and on the other side of the saddle. When the continuous time
operator is applied to Iq, as r goes from 0 to 2n, P(t)Iq traces out a
twisting ribbon that becomes narrower each cycle as it approaches the
periodic orbit. P{t)Iq may even braid the ribbon. The fold operator
evolves periodic orbits along their flow, and the set of periodic orbits
is an infinite set of intertwined links. The orbits of a continuous time
dynamical system can link and knot in exceedingly complex ways. It
has recently been shown, for instance, that the set of periodic orbits
of the Lorenz attractor link in every possible way (the set of links
of periodic orbits contains representatives of every tame knot isotopy
class) [11].
Let A(p) be the invariant set of the suspension of the horseshoe,
52


where the dependence of A on the stretching parameter p E 6 < 0 < 6
has been made explicit, and let A(0) be the unperturbed system. The
bound 6 is such that the end-caps of the stadium never map into S. Then
A(P) is the set of interlinked periodic orbits of all periods, and P(r)A(0)
is the continuous time system in which points s £ A get mapped to
their images Ps E A at r = 2mr, n = 1, 2,.... We choose to stabilize
a periodic point s*. Suppose we monitor the evolution of an initial
condition s E A(0) until it is close enough to ls-, the vertical segment
in which s* resides, so that a perturbation p(s, s*) E 6 < 0 < 6 for one
iteration would serve to place s in ls*. We apply this perturbation for
one iterate and then turn it off. The system state s is then in ls*, and
it approaches s* as t * oo. This is the discrete time control described
above. We will look later at the meaning of control in continuous time
from the perspective of control of the suspended horseshoe.
The set A has a hyperbolic structure, so all of its periodic points
under F are saddles. It is not difficult to see that a segment l 6 Ws
is flipped over upon iteration if it is in the right side of S, and retains
its orientation upon iteration if it is in the left half of S. This implies
that the local stable sets of periodic orbits whose itineraries contain n
ls experience n flips, so a saddle will be a flip saddle if n is odd, and a
regular saddle if n is even.
Note that even though a periodic point of a Poincare section may
not have a flip saddle, if its itinerary contains a 1 then the continuous
time periodic orbit has a flip. Consider the n periodic orbit Pns* =
P(0)s* = s* where sk = 1 for some integer k right half of S, and so must flip upon iteration or during the continuous
time evolution from P(2'K(k l))s* to P(2ir(k 1) + 27t)s*. Later we
will show how P twists perturbed saddles relative to each other in time,
allowing us to establish a framework in which to formulate control by
capture and release.
53


4. The Dynamics of the Pendulum
4.1. Folding the horseshoe
Chaos in the driven pendulum may be viewed as resulting from a
stretching and folding of an invariant subset of the full phase space.
This stretching and folding appears to produce an attractor with a
horseshoe structure in the Poincare section. If we start with a carefully
chosen set of initial conditions in the x, y phase plane and evolve them
forward by ut we expect to find that they are stretched, folded and
mapped inside the boundaries of the initial set. This horseshoe struc-
ture is the hallmark of chaos, and it guarantees periodic points of all
periods as well as those of infinite period. The pendulum equation has
not to my knowledge been shown to have a horseshoe, but its seemingly
chaotic behavior and suggestive folding seem to indicate that it likely
does. Although the classic Smale horseshoe preserves line segments,
linearity is not a necessity for the formation of a chaotic invariant set.
Recall that the horseshoe map / has an invariant Cantor set A such
that
(a) A contains a countable set of periodic orbits of arbitrarily long
period.
(b) A contains an uncountable set of unbounded nonperiodic motions.
(c) A contains a dense orbit.
Moreover, any sufficiently C1 close map / has an invariant Cantor
set A with f\j^ topologically equivalent to /|A.
We therefore proceed with confidence constructing horseshoes whose
dynamics we liken to the dynamics of the vertically driven pendulum..
We establish a framework for the allowable types of horseshoes with
the following definitions, lemmae and a theorem.
Definition 4.1. A vertical curve x = v(y) is a curve for which
0 < v(y) < 1, |v(yi) v(y2)\ < n \yi y2\ inO for some 0 < fj, < 1.
54


Definition 4.2. A horizontal curve y = h(x) is a curve for which
0 < h(x) < 1, \h(xi) h(x2)| < p\x\ x?\ in 0 < Xi < X2 < 1
for some 0 < p < 1.
Definition 4.3. A vertical strip V is defined by
V = {{x,y)\x where V\(y) < V2(y) are non-intersecting vertical curves.
Definition 4.4. A horizontal strip H is defined by
H = {(x,y)\y [hi(x), h2(x)}] x G [0,1]}
where hi(x) < /^(rr) are non-intersecting horizontal curves.
Definition 4.5. The width of a vertical or horizontal strip is defined
as
d(V) =max \v2(y) Vi(y)\ d(H) =max Ih2(x) hAx)]
ye [0,1] e[o,i]
Lemma 4.6. If V1 D V2 D V3... is a sequence of nested vertical (or
horizontal) strips and if d(Vk) * 0 as k > 00 then d= V is
a vertical (or horizontal) curve.
Lemma 4.7. A vertical curve v(y) and a horizontal curve h{x) inter-
sect in precisely one point.
Hypothesis 1: Let be the set (1, 2, and let Hi, Vi for i be
disjoint horizontal and vertical strips and let f(Hi) = Vi,i 6 .
Hypothesis 2: / contracts vertical strips and /-1 contracts horizontal
strips uniformly. Let Vi,V2 £ Vi be any two vertical curves bound-
ing a vertical substrip V[ C Vj. Then f(V?) D Vj is a vertical strip
and
55


for some v E (0,1) and i,j E &. Similarly, letting hi,h2 Hi be
any two horizontal curves bounding a horizontal substrip H- C
Hi, then f^1(Hli)r]Hj is a horizontal strip and d(f~1(H/i)DHj) <
vd{Hj)
Theorem 4.8. [10]If f is a two-dimensional homeomorphism satisfy-
ing Hypotheses 1 and 2 then f possesses an invariant set A, topologically
equivalent to a shift a on ]T), the set of bi-infinite sequences of elements
ofG.
This theorem shows that a wide range of different nonlinear foldings
can produce horseshoe structures, and if, in addition, the map / is
a Cr diffeomorphism with r > 1, then (with a couple of additional
assumptions) A is hyperbolic.
Numerical evidence indicates that there are stable and unstable
manifolds of periodic points in the Poincare section of the driven pendu-
lum that act like the stable and unstable manifolds of A. The stretching
and folding of the pendulum attractor can be mimicked by a stretching
and folding of the manifolds of the suspension of a suitable horseshoe
map.
Let us consider how the dynamics of the vertically driven pendulum
equation (see Appendix B for a physical derivation) stretch and fold a
subset of the phase space with the passage of time. We examine the
dynamics of the system of first order ODEs
x = y
y = py sin:r(l Acosut)
where x is the angular velocity of the pendulum shaft, y the angular
acceleration, p the damping factor, A the amplitude of the driving term
and u the drive frequency. We will examine the dynamics in the phase
plane of x and y.
The first ODE x = y guarantees that a phase point (x, y) moves in
the x direction in direct proportion to the value of the y component.
Points in the upper half plane will move to the right and points in the
lower half plane will move to the left. In the second ODE, the py
56


term forces points to move towards the x axis, and the periodic part
of sinrc(l cos ut) reinforces and counteracts the py term by causing
points to move away from the x axis. Since the effect of the periodic
term is modulated by sin re, points near x = | and x = | will be
driven upward and downward with more vigor than those near 7r or
0. This periodic action produces a fold that is pulled from left to right
on the upper half plane, and from right to left on the lower half plane.
Figure 4.1.1 is a plot of y = sin x(l cos ut). In this plot, the y axis is
vertical, the x axis goes from upper left to lower right, and the t axis
goes from front to back.
Figure 4.1.1. The plot above shows the periodic term responsible for the
stretching dynamic of the pendulum equation.
The sequence in Figure 4.1.2 shows the evolution of the Poincare
section of the pendulum through one period of the drive cycle. The
stretching and folding is clearly evident. As the attractor stretches and
folds, the stable manifolds of the unstable periodic orbits stretch and
57


fold also.
Figure 4.1.2. The sequence above shows 16 successive Poincare sections
of the vertically driven pendulum when the damping is large enough so
that the bob never goes over the top.
We can picture the stretching and folding in terms of horseshoes,
but with an important difference. The Smale horseshoe construction
maps part of the phase space out of the region of interest, resulting in
an invariant set that is a product of Cantor sets. The basin of attraction
of the pendulum equation is the whole phase space, and the folds at
the ends of the attractor (see Figures 4.1.2 and 4.1.4) get mapped back
into the central region corresponding to S in the Smale stadium. These
internal folds are continually stretched out and folded deep into the
interior of the attractor. Since no region is mapped out of the space,
as in the Smale horseshoe, the pendulum attractor is the product of a
Cantor set and a curve, rather than the product of two Cantor sets. If
58


the pendulum contains a horseshoe structure, it is non-linear horseshoe,
and it is most likely a product of Cantor sets imbedded in the attractor
rather than the attractor itself.
We may construct pendulum-like horseshoes and make an analysis of
their dynamics. In some of the gross features, the pendulum horseshoes
have similar dynamics to those of the pendulum. In the horseshoe
of Figure 4.1.3, a symbol sequence of three symbols would suffice to
describe the dynamics on the invariant set, a Cantor set formed by
the intersection of the nestings of three vertical and three horizontal
strips (as the Smale horseshoe was formed of two). There is a fixed
point in the center of the rectangular region, and the outer layers get
mapped successively closer to the horizontal segment in the center of the
rectangular region. Note the similarity to the folding of the sequence
of Poincare sections of the pendulum. Furthermore, there are two fixed
points in the end caps. If the mapping included a rotation by 180 the
fixed points in the end caps would be periodic of period two, as in the
pendulum attractor.
Figure 4.1.3. This particular folding resembles the folding of the pen-
dulum attractor, and the action of the folding on the invariant set is
topologically conjugate to a shift on three symbols.
59


Figure 4.1.4. The invariant set of the pendulum horseshoe.
In the mapping of Figure 4.1.5, we can see that successive iterations
will eventually map the end cap into the linear body of the attractor.
This situation is more similar to the dynamics of the pendulum and
other real world attractors. There are folds within folds within folds,
and the dynamics become extremely difficult, if not impossible, to an-
alyze using symbolic dynamics. Nevertheless, if this folding generates
a horseshoe structure, then whether the remainder of the attractor is
populated with periodic orbits of extremely high period, or infinite pe-
riod orbits, the horseshoe contributes chaotic dynamics on its invariant
60


set, and long chaotic transients for orbits that pass nearby.
Figure 4.1.5. This stretching and folding is like that of the pendu-
lum, but because the folds are eventually mapped into the body of the
attractor, hyperbolicity cannot easily be established. The inclusion of the
endcaps in the body of the attractor essentially destroy the nested se-
quences of vertical strips, as the folds are neither vertical nor horizontal
and could create tangencies between the stable and unstable manifolds.
When tangencies exist, there exists the possibility for an infinite number
of stable orbits in the attractor.
When we look at a sequence of Poincare sections of the pendulum
taken at successive phases of the forcing term, we see a stretching and
folding of the stable and unstable manifolds that bears a striking re-
semblance to the stretching and folding of the stable and unstable sets
of the suspended horseshoe. Specifically, line segments l whose end-
points are in the invariant set Ap of the pendulum are mapped to other
segments with endpoints in the set Ap. We may imagine the continuous
time evolution of a segment ls as the unfolding of a ribbon that writhes
through the torus in which the pendulum solution lives, eventually find-
ing its way in between layers of the attractor, getting closer and closer
61


to the periodic orbit while becoming narrower and narrower.
A segment ls in the stable manifold of a periodic orbit X* eventually
comes as close as we like to X*, and it is this fact that we count on to
make control by stable subspace targeting possible. We seek to perturb
the system so that beginning with any state X0 we can find a pertur-
bation or sequence of perturbations so that X evolves to ls 6 Ex., a
line segment in the stable set of X*. The dynamics of the evolution of
l3 then assure that the orbit will approach that of X* asymptotically.
The following sequence of pictures shows a line segment in the stable
set of the straight down steady state of the pendulum as it evolves
through one drive cycle. Notice how the segment, originally outside
the attractor is folded into the inside of the attractor. Thereafter,
its fate is to go deeper and deeper into the interior, alternating sides
as it approaches the steady state. The stable straight down state is
the center manifold of a flip saddle. It is easy to see the rotation of
the manifolds in this picture. The figures are in sequence of increasing
drive phase angle from top left to bottom right, and the final frame
the attractor is in its original position, except that the stable manifold
that started as a loop is now folded into the attractor. If we viewed the
entire stable manifold of the unstable steady state, we would also see
the loop that replaces the one folded inside.
62


Figure 4.1.3. These seven Poincare sections taken at equally spaced
intervals through the drive cycle illustrate how the stable manifold is
folded into the attractor during the course of a cycle.
63


The dynamics of the stable manifold above are repeated over the
entire body of the attractor for all the periodic orbits, that is, pieces
of stable manifolds outside the attractor unfold and are compressed
inside the attractor. Once inside they migrate deeper and deeper
into the fractal layers, sometimes approaching a periodic point asymp-
totically, and sometimes mapping back out, only to be swallowed up
again.
When we fold a horseshoe in the pattern of the folding of a pendu-
lum attractor, we find that the dynamics along the stable and unstable
sets of the horseshoe seem to mimic some of the dynamics of the verti-
cally driven pendulum if the suspension includes a half twist to make
a Mobius band.
Figure 4.1.4. The pendulum horseshoe above corresponds to an attrac-
tor that has no points surrounding the origin. We can see that points
near the origin get stretched out and there is no reinjection of the points
in the fold to the central region. In a physical pendulum, this condition
would correspond to an attractor with minimum period two. The at-
tractor consists of a folded region that coils through the torus once in
two drive cycles, piercing the Poincare plane in two separate places. If
we were to watch a movie of the evolution of the attractors Poincare
section with change in drive phase, we would see two folded regions that
circled the origin, changing places once per cycle. Physically, this cor-
responds to a pendulum that always crosses the straight down position
with positive velocity.
64


Figure 4.1.5. The pendulum horseshoe above corresponds to a pendulum
attractor that has points near the origin. The folds reinject an orbit that
has been stretched out of the central region back into the region near the
origin. This situation corresponds to a pendulum that can have zero
velocity at the origin. There is then a path to the steady state along a
stable manifold of the attractor.
Figure 4.1.6 shows the unstable manifold of the pendulum attractor
along with a portion of the stable manifolds of three periodic points.
Notice the similarity of the loops in the stable manifold to those of
the suspended horseshoe. The folding in Figure 4.1.6 corresponds to
the folding of the pendulum horseshoe in Figure 4.1.5, as there is a
65


reinjection region at 0,0.
Figure 4.1.6. Shown are the unstable manifold of all points of all peri-
ods (the Z shaped region) and a portion of the stable manifolds of three
separate periodic points. The stable manifolds of all the unstable pe-
riodic points are dense in the basin of attraction. The origin and the
period two points are all flip saddles.
The rotation of the manifolds of the flip saddle are what make pos-
sible capture and release control, and a knowledge of the movement of
the manifolds as an attractor goes through its cycle of stretching and
folding allow us to design general control rules beyond the map based
rules of OGY.
66


4.2. Bifurcation and orbit structure
The pendulum ODE has no analytical solutions in terms of simple func-
tions, but numerical integration of this equation reveals rich dynamics,
both chaotic and non-chaotic. Some analytic methods are useful in es-
tablishing approximate characteristics and locations of periodic orbits,
but the method of OGY is model-independent, and for the rest of the
paper we will assume that we have access to a data set only. Solutions
to the driven pendulum equation live naturally in a solid torus of rec-
tangular cross section. A slice through the torus perpendicular to the
minor axis reveals the familiar phase plane of the unforced pendulum.
This phase plane is now a Poincare section, the map made by taking a
cross section of a flow at some phase of the drive. Along the minor
axis of the torus we plot the drive phase, which is 2n- periodic. The
cross section is scaled in the vertical direction to fit the largest angular
velocity attained on the attractor, and is 27T-periodic in the horizontal
direction, corresponding to the periodic nature of over the top rotation.
Figure 4.2.1. The dynamics of the pendulum live inside a torus of
rectangular cross section. The left and right boundaries of the rectangle
are periodic, but the top and bottom must be unbounded.
As any one of the three parameters p, u>, or A is varied, and for
67


certain sets of fixed values of the other two parameters, the motion of
the pendulum goes through a series of period doubling bifurcations to
chaos. We will consider only the case of variation of the damping para-
meter p, as variation of the other parameters produce similar changes,
except in the details. With high enough damping (and in a specific
range of fixed values for u> and A), the steady state 6 = 0 of the pendu-
lum is straight down, even while being driven. This state is stable and
its basin of attraction is the entire space, save a set of measure zero
attracted to the unstable inverted state. As damping is decreased, a
new stable solution appears, a period-two orbit, where the pendulum
swings back and forth once every two drive cycles. There are now two
basins of attraction, one for each solution. Paired with this stable sub-
harmonic period-two solution is an unstable period-two solution. These
solutions appear together and split from each other as the damping is
decreased.
Hf- '
-+ -
Figure 4.2.2. The figure above illustrates the center, the straight down
stable state, and at the sides, the stable and unstable period two orbits.


Figure 4.2.3. The stable and unstable period two orbits have separated
as the damping decreases still more.
68


As the stable and unstable solutions separate, the basin of the sta-
ble straight down state shrinks. Finally, the basin of the straight down
stable state blinks out of existence, and the only stable solution is the
period-two orbit. The unstable period-two solution has merged with
the stable period one solution and the basin of the period two solution
is the entire space, except for a set of measure zero (the stable set of
the 0 = 0 solution). As damping is decreased still further, the symmet-
ric period-two solution bifurcates to two stable asymmetric period-two
orbits, where swings to one side alternate between two heights. The
symmetric period-two solution remains as an unstable orbit. The basins
of attraction of the two asymmetric states (which are mirror images)
are fractally entwined in each other. A section through the basin plane
would reveal a Cantor structure of rather high dimension. As the sta-
ble solutions separate from each other, the dimension of this fractal
decreases.
Upon further decrease in damping, the stable period-two solutions
bifurcate to period-four, period-four to period-eight, etc. The stable
states of the pendulum keep bifurcating, but with smaller and smaller
changes in the parameter until a point pcritl is reached where an infin-
itesimal change in parameter produces a state where only an orbit of
infinite period is stable. This is the onset of chaos.
As we decrease the damping still more, this attracting infinite period
orbit approaches tv in the 6 direction, which corresponds to the straight
up position. As the pendulum goes over the top, the attractor collapses
suddenly to two period one orbits, whirling continually over the top
either clockwise or counterclockwise. This sudden collapse of a chaotic
attractor is called a crisis, and it occurs whenever the basin of attraction
of a chaotic orbit collides with the basin of a stable periodic orbit as
a parameter is varied. More generally, a crisis occurs whenever basins
collide.
We continue to decrease the damping and witness another period
doubling bifurcation, but now as pcHt2 is approached we have a stable
infinite period orbit that can go over the top. As damping is decreased
still further, the attractor increases in size, apparently occupying more
of the phase space. It is this change in the size of the attractor, along
69


with the attendant shift in the position of the unstable periodic orbits
therein embedded that makes OGY control possible.
Figure 4.2.4. The plot above is a bifurcation diagram. Damping is
decreasing from left to right, and angular position is along the vertical
axis.
The structure of this attracting orbit is quite complex, yet embedded
in this structure are the ghosts of all its previous behaviors, the unstable
periodic orbits. These periodic orbits in sum determine the structure of
the attracting set. When the attracting set is chaotic and has a fractal
structure, it is known as a strange attractor. As an orbit moves through
phase space, it is attracted to a saddle orbit along the stable manifold,
only to be driven off along the unstable manifold as it approaches the
saddle. This occurs continually as the orbit moves through the varying
70


influences of the saddles, which are dense in the attractor,
Figure 4.2.5. The Poincare section of the 'pendulum in the non-over-
the-top mode.
Figure 4.2.6. The Poincare section of the pendulum in the over-the-top
mode.
71


5. Control of Continuous Time Systems
form the vector ^(0, p) =
which we denote where the
5.1. Control by OGY
Theoretically there is an infinite number of unstable saddles in the
attractor of the chaotic pendulum, but in practice the lower period
ones are easiest to control. The dynamics in the Poincare plane is all
we need to establish control by the method of Ott, Grebogi and Yorke,
and we can use the Poincare map to establish the linearized dynamics
about any periodic point of interest.
Now let us suppose we want to stabilize one of these periodic orbits,
and all that we have is a data set of length N composed of the successive
positions 6 and velocities 6 taken at some phase (j) of the drive. We
6i{(f>, p)
Oi((f),p)
dependence on the phase (j) at which the Poincare section is taken, and
the damping p is made explicit. For this section, however, we will ignore
the dependence on and p.
How do we locate these unstable periodic orbits from the data alone?
For the sake of demonstration, let us suppose we want the pendulum
to whirl clockwise over the top, a period one orbit of the flow, or equiv-
alently a fixed point of the Poincare map. There is only one periodic
orbit of this form embedded in the attractor, and since we are limited
to the preexistent unstable limit cycles by the nature of the algorithm,
this is the orbit for which we must settle.
Define some small distance e and locate all pairs of points £J+1)
in the data set whose Euclidean distance apart is less than £ (if we were
seeking points of period n we would look for pairs of points (£., £j-+n)).
We locate the center of mass of these points, which is a good approx-
imation to the position of £0, the period one point we seek. Now let
us compute the linearized dynamics near this point. We seek a 2 X 2
matrix A that when applied to some point near the periodic point
gives the next iterate. One especially simple way is as follows: For the
unperturbed system, write
[£j+ij£fc+i] >
72


where the column vectors are two experimental points near the
saddle and £j+1, £fc+1 are their first iterates. Solving for A we get
[£j+i>£fc+i] [£;?>£*;]
We can average the As obtained from several instances of close en-
counter, but there are two possible sources of error.
Our assumption of local linearity must hold throughout the do-
main we choose as close encounters, otherwise our local transition
matrix A will be in error.
If an iterate of the Poincare map approaches the fixed point too
closely, we may obtain an inaccurate estimate of the dynamics
due to the error in our approximation of the location of the fixed
point. Clearly, the set of points used to estimate the fixed points
location should be close to the true fixed point if we want an
accurate estimate of the fixed point.
Some familiarity with the dynamics is necessary to make an intel-
ligent choice, and in practice, I used my judgement to determine the
size of the data set used to locate the fixed point, and in the choice of
close encounters used to obtain A.
As in the case of control of the Lozi map, we need the eigenvalues
Xu As and the eigenvectors eu ea of A. For convenience, we normalize
the lengths of these vectors so that they are unit vectors. We also need
the left, or contravariant, eigenvectors fu and f3. These vectors are
orthogonal to eu and es and will be defined by
fu &S
fu e
U
fs-eu = 0
fs-ea = 1.
Now consider what happens as a parameter of the system (damp-
ing, in this case) is varied slightly. As mentioned above, the attractor
changes size with variation of damping, and the location of the fixed
points moves with the attractor. Let p be the damping parameter and
73


bp be some small change in p. Then the location of the new fixed point
upon this change will be bp g
During one drive cycle a phase point ^ evolves to £i+1. Relative to
the fixed point £0 these points may be represented by
A £i+i ~ &>
We are interested in varying the parameter p to achieve a change
in the next iterate, so we write the equation to express the dynamics
from the point of view of the shifted fixed point. The state vectors are
now
A& bpg
A£;+i bpg]
so
A&+! bpg = i4[A& bpg], (5.1)
where we have assumed that the local linearized dynamics of the per-
turbed system are near enough to those of the unperturbed system so
that the use of A for both the unperturbed and the perturbed system
is justified.
If the parameter p is allowed to vary, the position of the unstable
limit cycles in the attractor will vary, as will the position of the
attractor itself in the phase space. Our goal is to push the state variable
onto the stable manifold of a desired periodic orbit. Once there, the
dynamics will draw us into the fixed point.
The requirement that the next system state be on the stable mani-
fold of the unshifted fixed point may be written as
fu A£n+1 = 0
We desire an expression equating bp to a function of the eigenvalues
and eigenvectors of A and the current position. Rewriting 5.1 as
74


A£i+1 = 8pg + A[A& 6pg\
and dotting by fu yields
0 = fu-Spg + fu- A[A£i 8pg). (5.2)
We wish to solve for 8p, the perturbation of the damping that puts
us on the stable manifold, so to this end we write A in terms of its
components via the transformation
A = A uelfu + Xaejfa. (5.3)
Substituting 5.3 into 5.2, we obtain
0 = 8pfu g T fu [Au&ufu T As 6s/a] [A£j 8pg\
0 = 8pfu 1 9 T [Kfu &ufu T Asfu^sfs] [A^j 8pg\
0 8pfu 9 "b Aufu A^j Au8pfu g] j
so that
8p =
Xu U-AZi
xu -1 fu-g
which is the control law we seek.
Whenever an iterate of the Poincare map enters a small box around
the fixed point (determined by either a physical restriction of an ex-
perimental parameter or by the limit to the approximation of linearity
near the fixed point), we change the value of the parameter by 8p for
one Poincare cycle. The shifted and unshifted attractors are shown in
75


Figure 5.1.1 below.
Figure 5.1.1. In these panels, the lobes containing the period one orbit
are shown. The damping in the left panel is greater than that in the
right panel. As the lobes contract or expand, the location of the fixed
point changes.
Figure 5.1.2. This figure shows a computer screen image of the Poincare
section of the attractor and the data extracted from the dynamics.
76


Figure 5.1.3 below shows a plot of the time series of control of a
physical pendulum by the method of Ott, Grebogi and Yorke[12].
Figure 5.1.3. The picture illustrates the control of a physical pendu-
lum by the OGY method. Control is alternated between period one and
period two.
5.2. General geometric picture of control
We have seen how the stretching and folding of an attractor brings
points in the stable manifold of a periodic orbit closer and closer to the
orbit upon each fold. We have also seen that some manifolds twist as the
attractor stretches and folds, specifically in the sequence of 16 Poincare
sections of the pendulum attractor. We can use this knowledge to gain
control of a chaotic system in less than one fundamental period, that
is, we can establish control before the next SOS iteration is taken. The
simplest of these procedures is control by capture and release (CR),
which works only for flip saddles.
Flip saddles arise in period doubling bifurcations. The illustrations
below show how a flip saddle is formed.
77


Figure 5.2.1. The stable period one orbit before bifurcation.
Figure 5.2.2. The stable period one orbit has become unstable, bifurcat-
ing to stable period two.
The first illustration shows a period one orbit and its piercing of
a SOS map. A parameter is changed until the period one orbit loses
stability, and a stable period two appears. One of the stable directions
of the period one orbit has become unstable and is now serving also
as a stable manifold of the period two orbit. Note that a half twist in
the common manifold is necessary in order to form a period two orbit.
Paths near the periodic orbit will follow the twisting manifolds and
thus alternate sides in the surface of section map.
Now suppose we move in a frame that follows the period two orbit.
For this orbit to bifurcate to period four, it must undergo a half twist
in this moving frame. As the orbit bifurcates, the now unstable period
78


one and period two orbits must still be connected by their common
manifold. For a new half twisted manifold to form connecting the
period four orbit and its unstable progenitor, the manifold connecting
the unstable period two and period one orbit must also fold, not once,
but an infinite number of times. This scenario is responsible for the
folded fractal structure of the strange attractor. Thus, a bifurcation
to period four in a continuous time system implies an infinitely folded
manifold. A cross section through this manifold will have a fractal
structure.
Suppose we perturb a period one flip saddle, and look at the rela-
tionship between the perturbed and unperturbed manifolds. As both
the perturbed and unperturbed manifolds undergo a half twist over the
course of a drive cycle, the stable manifolds (or their linearized ideal-
izations) must coincide at some point. Imagine the hands of two clocks
side by side. One can see that at some time during the hour, the hands
are colinear.
Figure 5.2.3. This figure show an idealized period one flip orbit and its
manifolds.
We simplify the picture somewhat if we agree to move in the refer-
ence frame of the unperturbed manifold. Then the perturbed manifolds
79


will be seen to orbit around us without twisting. Compare the right
hand side of Figure 5.2.4 with the sketch in Figure 5.2.5.
Figure 5.2.4. The perturbed and unperturbed manifolds of a flow are
shown on the left, and their Poincare section on the right.
Figure 5.2.5. In this figure we are moving in the frame of the un-
perturbed manifolds. The perturbed manifolds will be seen to orbit us.
The actual orbit will be more elliptical than circular in the case of the
pendulum.
80


Capture and release control works by perturbing the system in such
a way as to place the current system state on the stable manifold of the
perturbed system, and waiting until the perturbed and unperturbed
manifolds are nearly colinear (we are seldom blessed with linear man-
ifolds, so they can only be approximately colinear). At this point the
perturbation is turned off, and the system evolves along the stable man-
ifold of the unperturbed system. No knowledge of the eigenvalues of
the system is needed to implement CR control. We need only discover
how long it takes for the manifolds to become colinear, but we do need
a continuously adjustable control parameter.
no control
control on

control off
Figure 5.2.6. This figure illustrates the method of capture and release.
The system is perturbed so that the current state is on the stable mani-
fold of the perturbed periodic point. The perturbation remains in effect
until the perturbed and unperturbed stable manifolds are most nearly
colinear.
At this point we point out that the effective use of OGY control
requires that the movement of the stable manifold be transverse to the
direction of the unstable manifold. It is easily seen that one could
81


choose a Poincare section at a phase where the perturbed and unper-
turbed stable manifolds are colinear. Should this be the case, control
by OGY is impossible, as OGY requires that the perturbed and un-
perturbed manifolds separate so that the perturbed dynamics in the
unstable direction force the system state onto the unperturbed stable
manifold by cycles end. It may be that in experiments where it was
found necessary to delay the activation of control until sometime be-
tween SOS maps that this situation was encountered. Experimentalists
sometimes have no choice as to when SOS maps are taken due to the
exigencies of the experimental apparatus or type of system under study.
j
no control
£=0
coritrol on
£=0
£=90
£=180
£=270

iz
360
control off
£=360
Figure 5.2.7. Successful OGY control. The change in manifold position
with perturbation is such that there is a change in the unstable direction.
The perturbed unstable dynamics act on the system state in such a way
as to place the system state on the stable manifold of the periodic orbit
in one system cycle.
82


-s
s
s
s
S S
Unsuccessful OGY control-no movement along eu
Figure 5.2.8. Unsuccessful OGY control. The change in manifold po-
sition with perturbation is such that there is no change in the unstable
direction. The perturbed unstable dynamics act on the system state in
such a way as to move the system state away from the stable manifold
of the periodic orbit.
From the schematic of OGY control, we can see that a little larger
perturbation could drive the system state onto the stable manifold of
the unperturbed system in less than one fundamental period. Figure
5.2.9 illustrates this idea. Control by time proportioned perturbation
and control by capture and release will be analyzed more thoroughly
and implemented in the next two sections.
83


*
4
no control
*=0
f=0
^=145
control off
*=?
Figure 5.2.9. This schematic of control by time proportioned pertur-
bations shows the system being controlled before the next SOS map is
taken.
5.3. Control by capture and release
Control by capture and release (CR) works only for flip orbits. CR con-
trol requires knowledge of the dynamics of the system between Poincare
maps, specifically, the change in fixed point with control perturbation,
and the point in time at which the stable manifolds of the perturbed
and unperturbed system coincide. Figure 5.3.1 is a plot of the sta-
ble and unstable manifolds of a period-one over-the-top orbit of the
84


pendulum at 16 Poincare phases and at three different damping levels.
Figure 5.3.1. The figure above shows the perturbed and unperturbed
manifolds of a period one orbit of the vertically driven pendulum. The
phase space is periodic at the left and right sides, and only the upper
half plane is shown. The orbit moves from left to right.
In each group of three the middle manifolds are the unperturbed
ones, with a damping level pQ = 0.23 in units of actual damping to
small angle approximation critical damping. The left members of the
trios are at an increased damping level p = 0.25, and the right members
have a damping of p = 0.21. In the 6th full group of three from the left,
the unstable manifolds are nearest coincidence, and in the 11th group
of three the stable manifolds are nearest coincidence. In capture and
release control we capture the system state at Poincare section 6, where
the perturbation gives the greatest change in the unstable direction, and
release the system at Poincare section 11, where the stable manifolds
are almost colinear. The system then progresses to the center manifold
85


along the stable direction.
Figure 5.3.2. The perturbed and unperturbed left and right stable and
unstable eigenvectors of the period one orbit are shown.
Let -§^Xq = g be the change in fixed point x0 with change in parame-
ter p, and 8p be a small parameter perturbation. Our goal is to find a
8p that will perturb the fixed point xq at time tc by 8pg so that the sys-
tem state Ax lies in the stable manifold of Xq + 8pg. The system state
Ax is now captured by the stable dynamics of the perturbed orbit. We
leave the control on for a time tr until x lies in e3, the stable manifold
of the unperturbed system. The control is turned off and the system
state is released into the custody of the unperturbed stable manifold
es. The system will now evolve along es to the center manifold.
Suppose we have a continuous time chaotic system that we can
sample at several different Poincare phases (f)^i = 1,2,...,n over the
course of the fundamental system period. Using standard techniques,
we establish the location of the fixed point Xo in the Poincare map
the local linearized dynamics Axn+i = Axn near the fixed point, the
stable and unstable right and left normalized eigenvectors es,eu, fs,fu
of A, and the change in fixed point with respect to parameter change
9 = x§p We also compute the time tc at which the unstable manifolds
of g and Xq coincide (es = 0) and the time tr at which the stable
86


manifolds of g and x0 coincide (eu = 0). We choose the Poincare phase
(f>i at which we take data as close as we can to tc to maximize the effect
of the parameter perturbation. The control-on time is therefore fixed at
tT, the time it takes for the manifolds to rotate from unstable coincident
to stable coincident.
Let dp be some small change in p. As the system parameter p
varies, the attractor and the periodic orbits move in state space. We
will assume that for small parameter change dp the fixed points move
but the local dynamics do not change. The location of the new fixed
point with change dp is dp-j^xo = dpg. We will choose dp so that the
perturbed stable manifold contains the current system state Axn =
xn Xq. This condition is satisfied when
fu {dpg Axn) = 0
or
dp =
fu Axn
fu 9
If we apply this correction dp the phase point will be on the per-
turbed stable manifold until the correction is turned off at tr. The
system will then evolve along the unperturbed stable manifold into the
fixed point.
Near a saddle, the rate of departure of an initial condition Ax along
the unstable direction eu is a function of the distance from the stable
manifold. Thus, perturbing a system so that the new stable manifold
contains the current system state Ax results in a stable system so long
87


as the perturbation remains.
Figure 5.2.3. This picture is a time series of the pendulum being con-
trolled by the method of capture and release. The control is alternated
between a period one clockwise and a period one counterclockwise rota-
tion.
Figure 5.3.3 shows a time series of a session of control by capture
and release. The control was alternately applied to period one clockwise
and period one counterclockwise rotation. The 6 scale runs from 7r
to 7T. The control parameter is velocity dependent damping, with a
control off level of .23 normalized units. The control perturbation is
shown in gray, and its scale runs from .21 to .25. Circles indicate
when the control was turned on and off.
5.4. Control by time proportioned perturbations
The method of Ott, Grebogi and Yorke (OGY) may be used to control
a low- dimensional continuous time chaotic system when the flow may
be sampled at some fundamental period to obtain a surface of section
88


(SOS) map. Then, small parameter perturbations may suffice to sta-
bilize the system around one of its unstable periodic orbits (UPOs).
Using OGY control, we would apply a perturbation, whose size is de-
pendent on the value of the system state and the left eigenvector of the
stable manifold, for one iterate of the map. The system state will lie
on the stable manifold after one fundamental period.
Frequently the only parameter available in a physical system is ca-
pable of only a few discrete states, and sometimes only two; either on
or off. Control using this type of parameter is called bang-bang control.
An air conditioner or a thermostat controlled gas heater are two types
of systems that use bang-bang control. When the temperature in your
house gets too low, the mercury switch in the thermostat is tripped and
the furnace goes full on until the temperature rises above a preset level,
at which time the furnace is turned off. If we can vary the parameter
only by a fixed amount, then we must resort to schemes other than
standard OGY.
If we can reconstruct some of the dynamics between surface of sec-
tion maps we can use time proportioned perturbation (TPP), where a
fixed perturbation is applied for less than one fundamental period. We
show how to implement TPP control when we have access to several
SOS maps within one fundamental period. In fact, TPP needs access to
several SOS maps only for the learning stage. Then, once the dynamics
of the system have been reconstructed linearly in the region near the
behavior of interest, we monitor only one surface of section map.
The goal of time proportioned perturbation is to apply a fixed per-
turbation p in order to direct the system state onto the stable manifold
in less than one fundamental period of the system. Usually, the stable
and unstable eigenvectors es and eu rotate about the UPO during the
course of a system cycle, and the angle between them changes period-
ically. Furthermore, the distance between the perturbed and unper-
turbed orbits is periodic, and we must take these factors into account
89


when building our control rule.
Figure 5.4.1. Shown are a perturbed and an unperturbed orbit and the
important features we need to build a rule for control by time propor-
tioned perturbations.
Suppose we have access to n surface of section maps equally spaced
in time during one fundamental period T of a continuous time system
for both perturbed and unperturbed states. Then we can determine the
perturbed and unperturbed UPOs Xo(kAt) (k = 0,..., n 1), At =
and Xp(kAt), respectively, the left unstable eigenvector fo(kAt) of the
unperturbed orbit, and the transition matrix Bk taking a system state
X(kAt) near the unperturbed periodic orbit Xo(kAt) at SOS k to its
image X((k + l)At) at SOS k + 1. From the set of transition matrices
Bk we form the set of matrices Ak where Aq = B0, A\ = BiB0, A2 =
B2B1B0i etc. In other words, Ak takes an initial state X (0) and evolves
it to the system state at Poincare section k.
We now fit smooth periodic functions fu(t), Xo(t), Xp(t) and A(t)
to the data fu(kAt), Xo(kAt), Xp{kAt) and Ak. Note that A(t) is not
a periodic matrix in the sense that yl(0)x(0) = x(2tt) (it doesnt), but
the entries of the matrix are periodic.
Suppose we monitor SOS 0 and measure an initial system state X (0)
near the UPO Aq(0). We want X(t) to be on the stable manifold es(t)
90


at time t. The system evolves as
X(t) X0(t) = A(t)[X(0) Xo(0)].
When we turn on the perturbation p the system evolves as
X(t) Xp(t) = A(t)[X(0) Xp(0)}.
where we have assumed that A(t) does not differ significantly from
Ap(t), the dynamics of the perturbed system. Then
V(t) = A(f)[X(0)-Xr(0)]+Xp(t).
We want, at some future time t
/,(() [X(t) X0(t)} = U(t) (X(f)[X(0) J?p(0)] +Xp(t) X(()) = 0.
Rewriting
A(i)[A(0) Xp(0)] + Xp(t) X0(t)
as
A(t)[X(0) Xo(0) ~(XP(0) Xo(0))]+ Xp(t) X0(t)
'------------' '------V------' '----V-------'
£o(o) flp(o) ffp(t)
and grouping terms, we have finally
fu(t) (A(f)[£o(0) ^p(0)] + gp(t)) = 0,
the condition that X(t) is on the stable manifold es(t).
We are only interested in the first zero, so instead of using a New-
tons method to solve for the zeroes, we step time forward and take
products of the current and previous iterate. The first time tc for
which this product is negative is the t we want. The fixed control is
then applied for this time and turned off. At the next Poincare section
we take the data and determine the on time for the next control cycle.
The vertically driven parametric pendulum
x = y
V -PV ~ sin(rc)(l a cos(wt)
91


is chaotic for the fixed parameter values a 1.2, u = 1.5 and for
p G [0, 0.28]; we choose Co = 0.23 as our unperturbed parameter value.
We choose to operate the system with p G [0.21,0.25], and as we are
implementing control by time proportioned perturbations, we fix two
possible perturbations, pl = 0.21 and p2 = 0.25.
Data sets were generated by a 4 th order Runge-Kutta integrator.
Data were extracted from 48 data sets consisting of 4096 values for x
and y for each of 16 equally spaced Poincare phases, and at three values
of damping, p1 = 0.21, pQ = 0.23, and p2 = 0.25. A PB computer pro-
gram extracted the fixed points, the local linear map, its eigenvalues,
and its stable and unstable left and right eigenvectors. The local tran-
sition matrices, that is, the 16 matrices that evolve a system state Xk
from the kth Poincare section to the k +1 Poincare section were calcu-
lated and continuous periodic functions fitted to the sequence of entries
in the i,jth position of the local transition matrices for i, j = 1,2. These
four functions ai,i(i), &it2(t), a2)2(t) form the entries in
<*1,1 (*) <*l,2(f)
a2ji(t) a2<2(t)
The functions Oiii(t), ait2(t), a2ii(t) a2t2(t) appear in the table below.
In this particular case, the Poincare control plane was sampled at
phase ^ since at this value the change in parameter moved the fixed
point along the unstable manifold. Choosing this phase for control
then gave the greatest controllability. Since at some drive phase

periodic orbit crosses the periodic boundary, the function A(t) will be
discontinuous at t = ip. We avoid the problem by choosing to control in
less than ut 7r, hence, we actually fit A(t) to the values taken from
Poincare sections 7 to 15 and 0 to 1 for a total of 11 successive Poincare
sections.
<21,1 = 0.291063 + 0.793685 cos [t]
0.0905197 cos[2t] + 0.0296494 cos[3t]
0.0559318 sin[t] + 0.00286768 sin[2t] + 0.050346 sin[3t]
olj2 = 0.408048 0.220476 cos[t]
m =
92


0.0905197 cos[2£] + 0.0296494 cos[3£]
0.0559318 sin[t] + 0.00286768 sin[2i] + 0.050346 sin[3£]
a2ii = 1.29693 + 0.430001 cos[i]
+0.776662 cos[2£] + 0.0230557 cos[3£]
+0.786877 sin[i] 0.240939 sin[2i] 0.22432 sin[3£]
a2i2 = 0.309087 + 0.400585 cos[£]
+0.136693 cos[2£] + 0.0268144 cos[3£]
-0.13708 sin[t] 0.0935679 sin[2£] 0.00711615 sin[3£]
Shown in Figures 2a through 2d are plots of the data sets (connected
piecewise linear plot) from the four entries of the matrix A and the
functions used to approximate the data.
93


Figures 5.4.2a, 5.4.26, 5.4.2c, 5.4.2d show the fits to the data from the
entries of A plotted with the piecewise linear fit to the data in the
entries. From top to bottom are a^i, ali2, a2,i, a2<2.
The unstable periodic orbits for three levels of damping were com-
puted in a similar fashion, by fitting smooth periodic functions to the
7th through the 1st successive state vectors, forming the continuous
vector function Xo (t). The function Xo (t) is shown below.
x0 = 0.866553 1.97963 cosft]
0.494012 cos[2t] + 0.0414162 cos[3i]
-0.439859 sin[t] + 0.379237 sin[2t] + 0.131035 sin[3t]
y0 = 1.5033 + 0.371607 cos[t]
94


-0.0176539 cos[2£] 0.0344509 cos[3£] +
0.557474 sin[t] + 0.2615 sin[2t] + 0.0335449 sin[3£]
x~(t) = 0.707967 2.10011 cos[s]
0.222945 cos[2s] + 0.240454 cos [3s]
+0.0253629 sin[s] + 0.653105 sin[2s] + 0.105806 sin[3s]
Po = 1.49461 + 0.405109 cos[£]
+0.00936983 cos[2t] 0.0279653 cos[3£]
+0.553614 sin[t] + 0.275949 sin[2£] + 0.0387564 sin[3£]
x$ = 0.772507 2.00497 cos[£]
0.47772 cos[2£] + 0.0564898 cos[3£]
0.443086 sin[t] + 0.390533 sin[2£] + 0.130429 sin[3£]
y£ = 1.51107 + 0.333787 cos[£]
0.0459989 cos[2f] 0.0436186 cos[3£]
+0.56489 sin[t] + 0.248926 sin[2£] + 0.0278085 sin[3£]
Figure 5.4.3. This figure shows the fit to the x coordinate of the
periodic orbit data for the three different damping levels.
95


Figure 5.4.4. This figure shows the fit to the y coordinate of the
periodic orbit data for the three different damping levels.
The function fu(t), the left unstable eigenvector is shown below:
ff = 0.351576 + 0.50572 cos[t]
+0.0330918 cos[2t] + 0.0873002 cos[3t] -
0.640942 sin[t] + 0.186759 sin[2t] 0.051183 sin[3i]
/; = -0.288731 0.113669 cos[t]
-0.0912112 cos[2t] + 0.122428) cos[3t]
-0.819938 sin[t] + 0.193646 sin[21] 0.158725 sin[3£]
Figure 5.4.5. This figure shows the data and the fit to the x coordi-
nate of the left unstable eigenvector of the unperturbed orbit.
96


Full Text

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THE TENT MAP, THE HORSESHOE AND THE PENDULUM: THE GEOMETRY OF CHAOS CONTROL by John Starrett B.S. Mathematics Metropolitan State College of Denver, 1994 A thesis submitted to the University of Colorado at Denver in partial fulfillment of the requirements for the degree of Master of Science Applied Mathematics 1997

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This thesis for the Master of Science Applied Mathematics degree by John Starrett has been approved by Weldon Lodwick Date

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Starrett, John (M.S., Mathematics) The Tent Map, the Horseshoe and the Pendulum: the Geometry of Chaos Control Thesis directed by Associate Professor William Briggs ABSTRACT When the space of a dynamical system is continually stretched and folded, a horseshoe structure may develop. The horseshoe is a guarantee of chaotic behavior, and we study the suspension of a horseshoe map as a model for chaotic dynamics of a periodically driven system. Consideration of the dynamics of a suspended horseshoe and its perturbation leads to a general geometrical approach to the control of chaos in low dimensional periodically driven systems by the method of stable subspace targeting .. We show how OGY control is a special case of the general method, and show why in experimental situations certain modifications must be made to OGY in order for it to work. We develop and implement two new types of chaos control based on these considerations, control by time proportioned perturbations (TPP) and control by capture and release (CR). This abstract accurately represents the contents of this candidates thesis. I recommend its publication. lll

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CONTENTS 1 Introduction . . . . . . . .................................................................. 1 2 Chaotic Maps and their ControL. . . . . . . . . ....................... J 2.1 The tent map.. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . ................ 3 2.2 The baker's map ................... ............ 12 2.3 The Lozi map..... ................ ......... 17 3 The Horseshoe ............... 3 5 3 .1 The Smale horseshoe . . . . . . . . . . . . . . . . . ........ 3 5 3.2 The Hen on map and its horseshoe structure.. ........................ .43 3. 3 The suspension of the horseshoe . . . . . . . . . . . . . ..... .46 3.4 The control of the horseshoe.. . ....................... 50 4 The Dynamics of the Pendulum.. ............ 54 4. 1 Folding the horseshoe. . . . . . . . . . . . . . . . . ......................... 54 4.2 Bifurcation and orbit structure.. ....................... 67 5 Control of Continuous Time Systems... ..... 72 5.1 Control by OGY.. . .. . . . . . ... 72 5.2 General geometric picture of control.. ............ 77 5. 3 Control by capture and release . . . . . . . . . . . . . . . . . . ....... 84 5.4 Control by time proportioned perturbations.. .... ......... ...... 88 6 Appendix A...... . . . . . . . . . . ............. 99 7 Appendix B... .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .... I 00 8 References . . . . . . . . . . . . . ............ I 02 IV

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1. Introduction Chaotic solutions exist for many systems, from maps like the logistic map to continuous time systems described by differential equations, to real world systems. Many interesting and important physical systems have chaotic dynamics over certain parameter ranges, and it is sometimes possible to control these systems to obtain improved performance. A prime example is the control of a multimode laser[ 1], where the stability regime of a Y AG laser has been extended by an order of magnitude by the application of small perturbations to the DC bias level of the laser pump. Following the publication of results of Ott, Grebogi and Yorke[2] (OGY) on the control of chaotic systems, experimental control of several physical systems was reported[3]. OGY control works by applying a small perturbation to a system parameter based on information taken from a map, which, in an experimental system, is some sort of surface of section (SOS) map. The control signal remains constant over the period between mappings, and is recalculated at each SOS. The goal of the perturbation is to place the system state on the stable subspace of an unstable periodic orbit (UPO) in one iteration of the map. Experimentalists found that although strict application of the OGY algorithm was suitable for some simple systems, in order to achieve tight control it was sometimes necessary to delay the application of control, and that sometimes the control perturbation must be turned off before the next SOS was taken. I show why delay of perturbation and change of perturbation length are sometimes necessary in an experimental situation, and extend OGY control to the continuous time systems by means of stable subspace targeting. I develop these results by looking at the geometry of the suspension of the Smale horseshoe, a simple geometrical construction exhibiting chaotic dynamics. By considering the interaction of a suspended horseshoe and its perturbation, I show how a continuous spectrum of control rules based on perturbation magnitude, control on time, and delay time may be established. The spectrum of limit cycle control will be made plausible from a geometrical point of view, and later developed as a numerical procedure for control of continuous time periodically forced experimental systems. 1

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In the final two chapters of this report, I develop the mathematics necessary for the implementation of two new continuous-time chaos control schemes, control by capture and release (CR) and control by time proportioned perturbations. I then use these control schemes to control a numerical model of the vertically forced pendulum with damping. 2

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2. Chaotic Maps and their Control 2.1. The tent map Consider the chaotic tent map, a difference equation defined by: 1 Xn+I 2Xn for 0 :::; Xn :::; 2 (2.1) 1 Xn+l = 2(1 Xn) for 2 < Xn :::; 1. (2.2) This map can be used to describe the stretching and folding of a one dimensional elastic band of length one. Each time the map is iterated the band stretches to twice its length and folds into itself. Points in [0, 1 J will double in magnitude and map to points in [0, 1], while points in 1 J will be stretched out to twice their magnitude and folded into [0, 1]. Figure 1 illustrates the result of this mapping. 0 1/2 1 ==:=:" 0 1 Figure 2.1.1. The tent map takes the unit interval Nnearly into itself. The diagonal line in the graph of Figure 2.1.2 is the identity line Xn-l-l = Xn and its intersection with the tent gives the fixed points and 0. We can trace the forward iteration of a point in the unit interval by the method of graphical analysis: Start with an initial value x0 (the seed) on the Xn axis and draw a vertical line from this point tG the

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graph of the function. Then draw a horizontal line from the function to the identity line. The Xn coordinate of this point is x1 We can iterate the map by repeating this process, producing what is known as a web diagram. Figur-e 2.1.2. The web diagr-am illustrates gr-aphically the r-esult of iteration of the map. The tent map is two to one except at 1, which has only 1 as its first preimage. Every x in [0, 1] of the form x = ,[n, p, n integer, eventually maps under forward iteration to the fixed point x = 0. Solving Xn+l = Xn shows that the point x = is the only other fixed point. There are, however, an infinite number of periodic points of the form x = E, where q p and q are integers and q is odd, like the period two points and g. It can be shown that any fraction in lowest terms whose denominator contains an odd number as a factor is a preimage of a point in a periodic 4

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orbit. There are periodic points of all periods, and the set of irrationals in [0, 1] make up the set of points with chaotic orbits. Definition 2.1. (Taylor and Toohey){4} A dynamical system on a topo logical space is chaotic if every pair of non-void open subsets share a periodic orbit. Since points of the form q odd are periodic points and dense in the unit interval, we can prove the map is chaotic there. Theorem 2.2. The tent map is chaotic on [0, 1]. Proof. Considertheopen balls B(Ej, Xj) and B(Ek, Xk), Xj, :x;k arbitrary points in (0, 1), and E; chosen so that the balls are subsets of the unit interval. Let B(c;, x;)n and B(c;, x;) .. n be the nth forward and nth backward iterate of B(c;, x;) respectively. Under the tent map there will be ann such that B(cj,Xj)n completely covers B(Ek,xk), and such that a preimage B(Ek,Xk)-n C B(Ej,Xj) of B(Ek,xk) is a single open interval. Iterate B(c:k,Xk) forward m times until it completely covers B(c:j,xj), and so that a preimage B(Ej,Xj)-m C B(Ek,xk) of B(cj,Xj) is a single open interval. There is then a continuous mapping from B(Ek,xk)-n to B(c:k,xk)m and, by Brouwer's fixed point theorem for one dimension, there is a fixed point x' for the n + m times iterated map pn+m: B(Ek,xk)-n-> B(ck,xk)m But B(Ek,xk)-n C B(Ej,Xj), sox' E B(Ej, Xj) Thus there is ann+ m periodic point x' whose orbit is in common with both B(cj,Xj) and B(Ek,xk), and the tent map is chaotic. Irrational numbers are the points in chaotic orbits, and the fact that any irrational number r will map to another irrational, and that T will never be repeated gives an intuitive feeling for the nature of these chaotic orbits. The set of all periodic points that participate in orbits of period n can be computed exactly. Say that Xn, the nth iterate of x0 is in bin 0 if it lies in and that it is in bin 1 if it lies in We can construct a doubly infinite string of symbols ... s.2s_1.s0s1s2 . where Bn is the bin number of Xn, that represents the history of a J.?Oint 5

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x0 under the map The left or right shift of the binary point corresponds to forward and backward iteration of x0 For example, the orbit is .. so its symbol sequence is ... 001001.001001.... Shifting the binary point to the left or right produces the backward or forward bin sequence, called the itinemry. Each itinerary is unique and corresponds to one particular orbit, be it periodic or chaotic. Now suppose we wanted to construct all orbits of period 2. There are four possible strings that repeat after two cycles: ... OOOO.OOOO ... ... 1111.1111..., ... 0101.0101... and ... 1010.1010 .... If xis the initial value and it is in bin 0, its value is to be multiplied by 2. If x is in bin 1 its value is to be subtracted from 1 and then multiplied by 2. Thus the period 2 point with orbit represented by ... 0101.0101... has initial value x = 2(1(2x)) = The period 2 orbit repr<".sented by ... 1010.1010 ... starts on x = 2(2(1x)) = g. If we compute the points corresponding to ... OOO.OOO ... and ... 111.111... we get 0 and respectively, the period one points. Points of higher period are computed in the same way, by compounding the algebraic operators in 2.1 based on the itinerary. All periodic points of the tent map are unstable. Irrational points, no matter how close to a point in a periodic orbit, will move exponentially away from the rational points of this orbit under the action of the map. This exponential departure, along with our ability to make small perturbations in a control parameter will allow us to control the chaos. To implement control we need an accessible parameter that will change the action of the map in a uniform way. If we change the height of the tent smoothly we change the mapping of points in [0, and ( 1] in a continuous fashion, so we write the equations of the tent map as { Xn+I = (2 + 15)xn Xn+l = (2 + 15)(1Xn) (2.3) with 15 set initially to 0. Now we can control the height of the tent by changing 15. If some iterate x of the map were to land near the preimage of a periodic point x', we could perturb 15 in a way that would cause the next iterate to land on x'. Figure 2.1.3 illustrates the technique 6

PAGE 11

applied to a period 2 point D Figure 2.1.3. The tent map is controlled around an unstable period two orbit by changing the height of the tent. An iterate of a chaotic orbit falls within E the preimage We increase or decrease li depending on which side of g x falls, and the next iterate lands squarely on Then we reset b to 0, and the orbit is now periodic with period 2. Because the map is ergodic (for any E ball around some point x and for any point xk in this interval, there is an N such that lxk+N-xl < E), we can be assured of an iterate of any point in a chaotic orbit eventually coming as near as we like to the periodic point of interest. Suppose we want to control about a periodic point x > and the or bit has landed in the control region B ( E, x._1), the epsilon ball around the preimage of x. The control parameter 6 is computed as follows: 7

PAGE 12

If x > is the target point and x, .... 1 the current iterate, we want so (2 + 6)(1x) = 2(1 -x ...... 1 ) 6=21-x-1_2. 1-x (2.4) (2.5) Using this 8 for one iteration gets us on the periodic orbit, and we then reset 8 to 0. A similar calculation will get us to the periodic point if 1 Xk < z Just as we can direct the orbit of the tent map from any initial condition to any periodic (or chaotic) orbit by a one-time application of a perturbation of the tent height, we can direct a trajectory of the tent map along any sequence of n periodic orbits, corning as close as we like to any point in [0, 1], as periodic points of the tent map are dense in [0, 1]. We merely compute the sequence of perturbations {)k necessary to take the orbit to the target points x,k, k = 1, 2 ... n and the controlled tent map becomes 1 Xk+l = (2 + Ok)Xk for 0 :':: Xk :':: 2 Xk+1 = (2 + Ok)(1Xk) for < Xk :':: 1. (2.6) (2.7) While we can vary the height of the tent to take any point in [0, 1] to any other in one iterate, the size of the perturbation 6 required to get from x,k to x,k+J, goes to oo as .cr,k -+ 0 or x,k -+ 1. We specify a maximum perturbation Dmax which determines the size of our control region B(c, x_1), theE ball around x_1 This point is the preirnage of the periodic point we are targeting. There is a trade-off: if we want to use only a small perturbation, it may take a long time before the orbit lands in the control region, especially if the preirnage of the point we are trying to control is near 1 or 0. If we want to control quickly we can consider the whole unit interval our control region and force the chaotic orbit onto the target point in one iteration, but at the expense of needing a possibly infinite 6. In a physical control situation, 8

PAGE 13

there will certainly be a limit to the size of Dmax, but here we choose to limit ourselves to small control perturbations for the sake of elegance. We can reduce the size of the perturbations necessary to get from one place to another by using the sensitivity to initial conditions of a chaotic map. Under iteration the map will come as close as we like to any point, so by computing a clever sequence of corrections that takes us on side-trips between target points, we can minimize some sort of a cost function. If we are currently stuck in a periodic orbit and want to go to another, we apply a small perturbation for one iterate of the map to knock us out of the periodic orbit and onto a chaotic one. Then we rely on the chaos of the system to eventually bring us into the control region around a point in the periodic orbit of interest. Suppose we were limited to very small perturbations. Are there faster ways of getting from one region of the map to another, riding its dynamics, so to speak? We know the dynamics of the tent map well: any interval entirely inside [0, 1/2] or (1/2, 1] will be expanded by a factor of two and mapped one to one into the unit interval. Segments containing have sets of points that map two to one into the interval because of the folding due to the change in dynamics at Backward iteration of an interval is up a binary tree: at each branch we have two possibilities for our previous position. The interval also shrinks by a factor of two, so the nth backward iterate of a line segment of length L is a set of 2n segments of length 2-n L distributed in some fashion over the interval. Now if we had a target point x' and we iterated B(c:, x'_ 1 ) backwards n times, we would have 2n small regions that map into our target region, the ball around the preimage of x'. The forward iterates of a point x chosen at random would have a far greater chance of landing within c: of one of the kth preimages of B ( c:, x'_ 1 ) than of landing in B ( c:, x'_ 1). Of course, for each k, the preimage of B(c:, x'_1 ) requires a different c:, depending on the distance of the preimage from %, since for a given maximum allowable perturbation the region of controllability shrinks as we approach 0 or 1. Figure 4 shows the range of possible preimages of a target point x under the restriction of the control parameter to Omin ::; 8 ::; Omax We can see that points :r; E [xmin, Xmaxl can all be 9

PAGE 14

made to map to x by the appropriate choice of b, but points arbitrarily close to 0 or 1 require arbitrarily large perturbations to map to x in one iterate. We can fix b and choose E to be the min of all E for a limited set of preimages. When a forward iterate of x comes within E of B(E, .. k, where the subscript -k indicates the kth backward iterate of B(E, x_1), we vary b to coa.'C it into B(c, x:,1 ) ... k+l and let the dynamics of the map take us to B(E:, We then apply the control again so that we land right on x'. x:min x:max X Figure 2.1.4. The height of the tent map can vary from 8min to Dmax, and the effect of this pammeter change on the control able region around a preimage of x varies inversely with the distance of the preimage from 0 or 1. Another way to increase our efficiency is to use the other periodic orbits of the map as routes to the final destination x'. Locating periodic points is quite simple, and by a judicious choice of a limited set of periodic or bits we can cover most of the interval with epsilon balls 10

PAGE 15

about each point in each orbit. We can vary o to cause any initial point to move onto a periodic orbit fiom our set in a few iterations, and by choosing our set of orbits so that each epsilon ball contains the periodic point of its two neighboring balls, we can also jump from orbit to orbit by varying o. There will be a path through these orbits that will lead us to .1:. If we are controlling a physical system or simulating the tent map on a real computer, then experimental noise or round-off error will drive our computed orbit away from the true periodic orbit we are trying to reach, and the control would have to be reapplied whenever we drifted too far from our goal. Now we can control a tent map, one in which the periodic orbits are not too difficult to compute. What if we wished to control the logistic map? Computing high period orbits, even period 5 for instance, becomes very difficult. However, the OGY algorithm doesn't require exact knowledge of the dynamics or the periodic points of the system under study. All we need is an approximate location of the periodic point we want to stabilize, and the approximate dynamics about that point. We can iterate any map we like, or collect experimental data from a physical system, and if there are periodic points and ergodicity, we can use the OGY method to stabilize the UPOs. To locate periodic points of the logistic map we could use a Newton's method to find the zeroes of the compounded map, but in preparation for the control of a physical system for which we have no model, we try the following method. Iterate the equation on a computer and store several thousand iterates in an array. Then check for pairs of points Xn, Xn+k for which [xn+k-Xn[ < E for some small E, and k is the prime period of the point we are seeking. A linear least squares fit to the data can be made, followed by a linear least squares fit to a line orthogonal to the first. The intersection of these lines will be the center of gravity of the data points, an approximation of the position of the periood k point. As the map is ergodic, we can expect a chaotic orbit to approach arbitrarily close to any periodic point as n approaches infinity. Once the periodic point x' has been located, we can determine the local dynamics by noting the rate of escape of points in the eJ?silon 11

PAGE 16

neighborhood of x'. Then we apply the control in the same way as before, reapplying when necessary. Once we lock in to a periodic orbit using the control, we can deter mine the position of the periodic point x' more accurately by moving x' experimentally and minimizing the average value of {j required to keep the control locked in. Once we have fine tuned the position of the periodic point, we can recompute the local dynamics to get a more accurate estimate. After only a few applications of this procedure, we can fine tune the position of the periodic points and the local dynamics to the limit of the accuracy of our machine. In this way we can control any unimodal chaotic map with almost the same ease as we can the tent map. 2.2. The baker's map Now let us move up one dimension to a two-dimensional tent map. This map is sometimes called the two-dimensional baker's map. Imagine a square sheet of rubber that stretches without deformation. We stretch the sheet until it is twice its length, then cut it in half and flop the right half over the left, as illustrated in Figure 2.2.1. D 2 3 D Figure 2.2.1. This two dimensional transformation is sometimes called the baker's map. 12

PAGE 17

This action is equivalent to one iteration of the map 1 Yn+! -2Yn for 0 < Xn < { Xn+I = 2xn Xn+l: 2(1 Xn) for l < X < 1 Yn+l 1 -:;_Yn 2 n (2.8) The physical description of the map did not take into account the delicate matter of fitting surfaces together after a fold, or the structure of the left and right sides after an infinite number of iterates. We could let the map fold in a two-to-one manner, but for reasons of symmetry, we will take a different course. In order that the map not be two-to one, we require that the first iterate fit together like a puzzle, that is, require the square to be open on the top right side and closed on the left. The halves will fit together as in Figure 2.2.2. Figure 2.2.2. Here we see the condil'ion that allows the rnap to fit to gether in a one-to-one fashion: the left half of the top is closed and the right half is open for all iterates. The top of the square is now what was the bottom right, so in order for the next iterate to fit together as did the last, the bottom of the square must have had this structure: 1---------+-----------1-----1 Figure 2.2.3. The right hand side of the bottom of the square rnust have the structure of the right half of this figure in order that the top will always have the structure iU.ustmted in Figure 2.2.2. 13

PAGE 18

Furthermore, as the bottom left half of the square becomes the entire bottom on the next iterate, and we require the top to fold on itself in a one-to-one fashion each time, the bottom left half must have a scaled down structure like that of the bottom right. The left half of the bottom left half must also have this scaled dowu structure, as must its left half, etc. HIIHH .. +--tl---+----------------1 Figure 2. 2.4. The bottom of the square must have this structure overall in oTder that the Tight half of the bottom will have the structure of the right half of Figure 2.2.3 at each iterate. A similar structure appears on the left and right sides upon succes sive iteration as a result of the Dedekind cut we must make at each iteration. Referring to 2.8 we see that the first cut and fold makes the top left boundary open and the bottom half closed, while the right hand boundary is closed. The next iterate finds the right hand side the same and the left hand side has a closed bottom half, while the top half is closed on its upper portion and closed on its lower portion. Repeated iteration give the structure we required of the top and bottom boundaries, and we can see that reverse iteration of the map gives the same structure to the top and bottom edges of the square as forward iteration gave to the left and right edges. This map has the same dynamics in the x direction as forward iteration of the one-dimensional tent map, and the dynamics in the y direction are the dynamics of reverse iteration of the one-dimensional tent map. Reverse iteration of the tent map causes segments to con tract, and if we iterate the unit segment 0 :=; y :=; 1, x' where x' is the x coordinate of a point in a periodic orbit, the segment will ap proach a periodic point as a limit. Take for example the unit segment 0 :=; y :=; 1, x = Iteration of the two-dimensional tent map will cause this segment to cycle in the x direction between and while in the y direction the segment will contract and approach y = and in the limit. The points ( ( and ( are therefore a period three orbit.

PAGE 19

(2/9.8/9) (8/9.2/9] Figu:re 2.2.5. This figure shows a period three orb-it of the baker's map. In or-der to control this map, we only have to control the x direction. There is something new in this map: there is a contracting as well as an expanding direction. Chaotic and periodic orbits are dense in the unit square under this map, so we can still hop from orbit to orbit to get where we want to go. What's more, to implement OGY control, we still only have to perturb 6, the x direction control parameter. The contraction of the map in the y direction automatically brings us into the two dimensional periodic or bit we are aiming for. This map shares many of the dynamical properties of maps made from dissipative chaotic three-dimensional flows. There are expanding and contracting directions, unstable periodic orbits and chaotic orbits. However, maps of more complex dynamical systems are usually not so polite as to have their contracting and expanding directions perpendic ular to each other and linear. Furthermore, this map is not dissipative, so it doesn't have the fractal structure of the continuous time systems we will examine later. We can build a three-dimensional dynamical system whose Poincare section exhibits the dynamics of the tent map, and whose pseudo flow is similar in behavior to general chaotic flows in three dimensions. E'irst, 15

PAGE 20

imagine a long skinny rectangular solid (see Figure 2.2.6). Squash it gradually along its length so that at one end it is twice as wide as it is high. Now split it up the middle from the flatter end almost to the square end. Twist the split ends so that the two top faces come together. Now form it into a loop, twist it 90 degrees and connect the two ends. l.squash 1 2. fold over 180 degre;; Figure 2.2.6. The baker's map can be easily suspended to give a smooth pse1Ldo-flow. This construction gives a continuous evolution in three dimensions between the steps pictured in Figure 2.2.1 of the two-dimensional tent map. The periodic points are now continuous periodic orbits and chaotic points are now chaotic orbits. Once again, the periodic and chaotic orbits are dense in the space, which is now three-dimensional. We can consider the mapping at the cross-section that is the unit square to be the intersection of a flow with a plane perpendicular to the flow, a Poincare section. A Poincare section of a three-dimensional flow is used in most OGY control of real physical systems. Controlling the three-dimensional baker's map is just as easy as con trolling the two-dimensional baker's map: we look at the two-dimeneional 16

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Poincare section of the flow, and change 6 in the same fashion as before in order to coax the flow onto a periodic point of the two-dimensional map. Then, every time the orbit pierces the Poincare plane, we check for divergence from the periodic point and reapply the control as nec essary. In a physical situation or a real computer simulation this reap plication of the control is necessary because of noise and uncertainty of the actual dynamics of the system. 2.3. The Lozi map The Lozi[5] map is a two-dimensional piecewise linear map whose dy namics are similar to those of the more familiar Henon[6] map. The Lozi map, like the Henon map, is an affine transformation that has a chaotic invariant set for certain parameter combinations. Definition 2.3. A set A is called an at tractor of a map if whenever an initial point Xo is chosen close enough to A, the distance between the kth iterate x(k) and the set A goes to 0 ask-+ oo. One formulation for the Lozi map is Xn+l = (1 + P) + Ci.Yn + f31xnl (2.9) where a determines the contraction, {3 the stretching and folding, and p is a parameter that shifts a region to the left or right. A geometrical view of the action of the Lozi map on a planar region is shown in Figure 17

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2.3.1. Figure 2.3.1. The action of the Lozi map may be broken down into separate components. The action of the map on the unit square may be broken down into its components: a. the original square is flipped about the horizontal axis by the --Xn term, b. the square is compressed to a rectangle by the ayn term, c. the rectangle is folded into a chevron by b)xn), d. the chevron is shifted up by 1 + p, e. the chevron is rotated about the origin by 90 due to the switching of variables. 18

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Fig'are 2.3.2. The basins of attmclion of the Lozi map consist of the set of points whose orbit under the map is asymptotic to a subset of the plane. There are two allmctors, the point at infinity, whose basin is colored white, and the Lozi attractor, whose basin is shown in black. The Lozi atlmctor itself is the white arm inside the dark basin. Like the Lozi attra.ctor, the basin of attm.ction has a fractal structure. Most points in the plane move further and further from the at tractor under repeated applications of the mapping. However the points in a region calkx:l the basin of attracl'ion collapse onto an attracting set (see Figure 2.3.2), which in this case is a strange attractor (see Figure 2.3.3), a fractal object that is an invariant set of the mapping. 19

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Figure 2.3.3. This figure shows several thousand iterates of an initial condition inside the basin of attraction for the Lozi attractor .. The or-bit appears to be chaotic, and in fact, the Lozi map has been shown to have a strange attractorwith a hyperbolic str-ucture {it is the union of an infinite set of saddles). These three deformations (stretching, folding and rotating) are re sponsible for the chaotic motion of individual points on the at tractor. Almost any two points that are initially nearby are stretched away from each other, only to be folded and rotated back nearby, but in a different layer. There is an infinite number of layers, and there are layers infin itesimally close to each other. Points jump in what appears to be an irregular manner all over the at tractor as the map is iterated. The Lozi map has a set of stable "manifolds" of periodic points that is dense in the basin of attraction, but as these sets are piecewise linear and non-differentiable, they are not technically manifolds. The state vector is the current point Xn = [ Xn ] An orbit is the Yn path of the point in the plane under successive iterations of the map. 20

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We can write the Lozi map in matrix notation, [ ] ] [ Xn+l Yn+1 = where two cases account for the absolute value in the original formula tion. As the map is iterated for some initial condition X0 in the attractor, the orbit will fall in the left or right half of the plane, designated L and R respectively, and for each iterate, the map applied will depend on whether Xn E L or Xn E R. The orbit of the path can be specified to any degree of accuracy simply by knowing enough of its left-right history. The itinemTy of a point is its L-R history, written as a doubly infinite sequence such as ... LLRRLLLRRL LLRRLLLRRL ... The first digit to the right of the binary point is L, the current state of the system. Upon forward iteration of the map the current state will follow the itinerary. The itinerary specifies a periodic orbit uniquely in this map, and a right or left shift of the binary point gives the forward iterates or backward iterates of the map. A periodic point with period k is a point X' whose image under k mappings is again X'. For the set of parameters that make the Lozi map chaotic, all of these periodic points are unstable. Upon iteration, any state vector not precisely on a periodic point will wander away. The Lozi attractor is the closure of the dense set of all periodic points. There is an infinite number of periodic itineraries, and the periodic orbits that they define fill a fractal region of space. There are also non-periodic itineraries that correspond to chaotic orbits, but we can find a periodic itinerary that matches the itinerary of the chaotic orbit to as many places as we desire. Therefore,orbits whose itinerary is non-repeating (chaotic) lie in the closure of the open set of periodic orbits. 21

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Consider a 5-periodic point X' with itinerary .. LLRRLLLRRL LLRRLLLRRL ... Which point exactly does this itinerary represent? To answer this ques tion, we rewrite the equations and in a more convenient form. Letting X-n-[ Xn ] U = [ 1 + P] L = [ f3 CY] R = [ -j3 CY ] Yn 0 -1 0 1 0 (2.11) gives us U + LXn for Xn E L (2.12) Xn+l = U + RXn for Xn E R, or Now we can easily express multiple iterates of the map in terms of its itinerary. For example, five iterates of the initial system state Xn whose forward itinerary is LLRRL are Xn+l = U + LXn Xn+2 = U + LU + LLXn Xn+3 = U + RU + RLU + RLLXn XnH = U + RU + RRU + RRLU + RRLLXn Xn+s = U + LU + LRU + LRRU + LRRLU + LRRLLXn = [1 + L + LR + LRR + LRRL]U + LRRLLXn =l'U+KXn where K, the full itinerary matrix, is the product of the matrices L and R associated with the reverse of the itinerary (which is LRRLL above). The partial itinerary matrix T is the sum of the identity I and th first 22

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k1 partial products of K, which is I+ L + LR + LRR + LRRL above. A periodic point of period k is a fixed point (period one point) of the k times iterated map Solving for the period k point X' we obtain As an example, we can find the period-3 point whose itinerary is LLR. We give p, a, and (3 simple rational values for which the Lozi map has a strange attractor, p = 0, and (3 period
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directions of stretching and folding are locally linear and vary over the at tractor. Let's look at the characteristic directions and rates near a period one point. From the point of view of the fixed point X' the state vector is Xn = l::.Xn +X'. The next iterate of the map is Xn+l = l::.Xn+l +X*. As is the usual practice for control of a two-dimensional map, we expand in a Taylor series about the fixed point X' (this is unneccesary in the linear case, but the idea will be used for nonlinear systems later). We have [ 8! l::.Xn+r +X' = F(X') + &y In the case of the Lozi map, the higher order terms in the Taylor series vanish. Since X' is the fixed point of the once iterated map F(X') = x, we have 8f f::.Xn+ 1 = [ 8y where A is the Jacobian of the map. (2.13) Here is a simple example using the parameter values p = 0, a = and (J = Because of the absolute value in the Lozi map, the Jacobians will be different in the left and right halves of the plane: AL = J(h(x,y),gL(x,y)) = [ i] By 8y [ .f!..(-x) .f!..(-x) &y 8y [ i ] -1 0 ( ( ) ( )) [ ] J !R x,y ,gR x,y = 8 &y [); [ %x(l+ ix) gy ( -x) 24 .f!..(l + ly-Zx) ] ax .!!.. (:_ x) 4 Oy

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Suppose that the state vector is Xn = [ :5 : ] Then = Xn X' = [ ] [ ] [ ] The Jacobian AR applied to gives us the change from the point of view of the fixed point. This change must equal the change given by the Lozi map applied to Xn, when viewed from the fixed point. We should have Indeed, we get [ ] [ 1\ ] = [ ;2 ] -1 0 :::1 -5 .6.Xn+l and Usually the Jacobian is used this way to estimate the local linearized dynamics near the fixed point of a nonlinear system, but the Lozi map is already linear (except for the affine part [ ] ), so the Jacobian exactly describes the local dynamics near a periodic point. Near a fixed point the direction in which the map stretches space is called the unstable direction and the direction in which the transformation contracts space is called the stable direction. The stable and unstable directions correspond to the stable and unstable sets near the fixed point. These directions are given by the eigenvectors of the .Jacobian A. The eigenvalues associated with the eigenvectors give the amount of stretch or compression of the space along these characteristic directions. The fact that the Lozi attractor has a hyperbolic str\)cture 25

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guarantees that all of the unstable periodic points on the Lozi attractor are saddles, that is, they have attracting and repelling directions. 4 3 3 . 2 1 1 . . Flip Out Saddle 2 Hyperbolic Saddle 4 4. 3 2 1 1 Flip In Saddle ; Ftip Saddle 4 Figure 2.3.4. This figure illustrates four different types of saddles. The numbers next to the points indicate the init-ial point and its next three iterates. The upper left saddle, a hyperbolic saddle, has eigenvalues 0 < A8 < 1, Au > 1. The upper right saddle, a flip-out saddle, has eigenvalues 0 < As < 1, Au < -1. The lower left saddle is a flip-in saddle, with eigenvalues -1 < As < 0, Au > 1. The bottom right saddle is a flip saddle with eigenvalues -1 < As < 0, Au < -1. We designate p as the control parameter and replace it with a function p(x:,y) that will change the dynamics of the Lozi map for one iteration whenever the system state comes within a certain distance of the periodic point we wish to stabilize. This temporary change in the dynamics is calculated so that the system state will land on the stable manifold of the periodic point on the next iterate. We have chosen p = p(x, y) to be our parameter because of the specific way a change in p affects the local dynamics. To calculate the perturbation that will direct the orbit onto the stable manifold we need to know something of the local dynamics of the periodic point we want to stabilize. 26

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There is an at tractor for a = -;,7 (3 = p = 0. Let us stabilize the period one point whose itinerary is ... RRR.RRR .... Earlier we found so for the periodic orbit ... RRR.RRR ... [ .Tn+ 1 ] [ 1 + 0 ] + [ ] [ Xn ] Yn el 0 1 0 Yn The fixed point of this once iterated map is found by solving X*= [J-Kr1TU, where T = I and K = [ ] Numerically the fixed point is X' = [ [ n [ ]] -I [ ][ ] [ ] We will perturb the dynamics near this point to steer the orbit onto the stable manifold. The local dynamics are determined by the eigenvalues Au, As and the eigenvectors eu, e, of A. For reasons that will become apparent shortly we will also need the left eigenvectors fu and fs These vectors are orthogonal to eu and e8 and are normalized such that 27

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f, Itt ' ' ' ' ' ' J J J J t Figure 2.3.5. The right and left (contravariant) eigenvectors are shown. The eigenvalues of AR are Au = associated eigenvectors are es e-s [ lYTI ] [ 7-.Ji7 u-1 's-1 A = -7+/T! while the 8 8 _4_ .Ji7 .Ji7-7 2.Ji7 Ar7]. 2.Ji7 We also need to know how the position of the fixed point, its associated stable and unstable manifolds, and thus the local dynamics, change upon a parameter perturbation. For example, if we change p from 0 to then the fixed point X' changes from [ ] to [ ] for a change 13 26 ll.X = [ ] Therefore the change in the location of the fixed point with change in parameter p is gPX' = [ 1\ ] l3 Let lip be some small change in p. Then the location of the perturbed fJXed point upon this change will be X' +op8 %X' (see Figure 28

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2.3.6). ' ' ' Figure 2.3.6. The perturbation of the fixed point is shown above. tor convenience of notation we write g = ;;px so that opg = op x i!p In one iteration a phase point Xn evolves to Xn+J Relative to the fixed point X' these points may be represented by 6-Xn = Xn -X' and 6Xn1r = Xn+J X'. Our goal is to push the state variable onto the stable manifold of the desired periodic orbit in one iteration under a perturbation of the dynamics. Once on the stable manifold, the perturbation is turned off and the natural dynamics will draw the orbit into the fixed point. Figure 2.3. 7 illustrates how this process looks schematically. 29

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unshifted unstable anifold eu c stable manifold es d Fignre 2.3. 7. The schematic above illnstmtes the basic principle of OGY control. In 2.8.1a, we see the C1JT7'ent system state with an arrow point ing to its next itemte. In 2 .'3.1b the system has been pertnrbed by an amonnt calcnlated to send the system state to the stable manifold of the unpertnrbed system in one demte of the map. In 2.8. 7c the pertuTba tion has been tnmed off, as the system state is now on the nnpeTtnTbed stable manifold. The stable dynamics will henceforth evolve the system state towaTd the periodic orbit as in 2 .'3. 7 d. We will vary the parameter p to achieve this perturbation, so we write the local dynamics from the point of view of the shifted fixed point. Using a prime (') to indicate the quantities in the perturbed system, the state vectors are now and 30

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fl. 1 = Ll.Xn+l 8 pg. We can treat the dynamics at the shifted fixed point as we did in 2.13 and write Ll.X' = A' Ll.X' n+l n' where A' is the Jacobian at the perturbed fLxed point. Assuming a small perturbation 8p, we may approximate A' by A and write Ll.Xn+l -8pg = A[Ll.Xn-6pg]. (2.14) In this case, the parameter change aflects only the side-to-side shift of the attractor and not the matrix A, so A' = A. A state vector that is on the stable manifold is orthogonal to the left unstable eigenvector. Therefore the requirement that the next state vector be on the stable manifold of the unshifted fixed point can be written fl.Xn+l = 0. We must now express 6p as a function of the eigenvectors and eigen values of the system and current position Ll.Xn. Rewriting 2.14 as Ll.Xn+l = 8pg + A[Ll.Xn-8pg] and dotting both sides of this equation with fu yields 0 = fu Opg + fu A[Ll.Xn-Opg]. (2.15) We wish to solve for 8p, so to this end we vvrite A m terms of its components via the representation (see Appendix A) T T A= .\ueu fu + .\,e, f, (2.16) Substituting 2.16 into 2.15, we obtain 31

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= Dpfu g + [Aufu] [D.Xn-Opg] = 8pfu g + Aufu 6.Xn-AuDPfu g Thus {j Au fu 6.Xn p -Au 1 fu g (2.17) which gives the control perturbation we need based on the current system state as seen from the unperturbed fixed point and the precalculated quantities Au, fu and g. Let's try this scheme on the example discussed above. Substituting the values of Au, fu, 6.X and g we found earlier into 2.17, we obtain Whenever an iterate of the map enters a small box around the fixed point (whose size is limited by the extent of the local stable manifold), we change the value of the parameter by Op for one iteration. Suppose an iterate of the map lands on Xn = [ {3 , so 6.Xn = Xn X*. Then our control rule says we need to apply a perturbation 7Vl7+17 [ 4 VI?-7] [-1 2] dp = 32 VI?'-2Vl7 13' 13 A quick calculation shows that 32 -1715Vl7 104Vl7 !04Vfi -3 13

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which, when translated to the fixed point, is on the stable manifold of X' = [ ]. The control process for this example is depicted in Figure 2.3.8 below. Xn ' ' j' '' X' ' ' ' ' Figure 2.3.8. This figure illustrates the shifting of the manifolds nec essary fO'r control of the Lozi map, along with the measnred qnantities necessary to calcnlate the proper shift. An iterate of the map Xn lands near the fixed point X' (its man ifolds are the dashed lines). The correction is applied, shifting the fixed point (and its manifolds) to X'+ bpX' (solid lines). Under these dynamics the point Xn maps to Xn+l The correction is turned off, and the system state is on the stable manifold of the unshifted fixed point. The unperturbed dynamics now act on the system state under successive iterations to carry it along the stable manifold into the fixed point. We can check this by dotting our answer minus the fixed point with the unstable left eigenvector: 33

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[39J17 -17 4 -3 -4] [ 4 J17 -7] 104J17 -13'14-13. Jl?' 2J17 [ 7J17-17 1] [ 4 Jl?-.7] 104J17 13 Jl?' 2J17 = o. We can control higher periodic points similarly. Control of a single point in a periodic orbit guarantees control of the other points in the orbit. The method of control is essentially the same as one would use to control the Hcmon map, except that the control would have to be applied continually to the Hmon map to account for the non-linearity of the stable and unstable manifolds associated with the fixed points. Formulation of the OGY control rule is not difficult, even for physical systems for which no mathematical model is known. A phase space may be built from a time series (a list of successive measurements of some property of the system) by the method of delay coordinate ernbedding [7], and the fixed points and local linearized dynamics may be determined by a combination of the methods of closest approach least squares fit, and Newton's method. 34

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3. The Horseshoe 3.1. The Smale horseshoe We have seen that piecewise linear chaotic systems are easy to con trol. The analysis of the dynamics of nonlinear systems, both maps and fiows, is made simpler by observing the correspondence of their dynamics with those of an abstract chaotic structure, the Smale horse shoe. The horseshoe is a fundamental chaotic object. The dynamics of the invariant set of the Smale[S] horseshoe map mimic, in abstract, the dynamics of a generic chaotic map in the plane; that is, the horseshoe contains unstable periodic orbits and chaotic orbits and their stable and unstable sets, as does the generic chaotic map. We will investigate the dynamics of the horseshoe, and show that there are similar dynamics in the H(mon map, and that a continuous interpolation between the invariant set of the horseshoe map (along with its stable and unstable structures) and its first iterate, provides a model for the dynamics of a chaotic flow in less abstract systems. We begin with a few definitions from Devaney[9]. A system is structumlly stable if every nearby system has essentially the same dynamics. Structural stability is an important property for a chaotic system to have if we wish to control its unstable periodic orbits (UPOs). 'Ib control a chaotic system we must perturb a system parameter, and we require that the perturbed system have similar topological structure to the unperturbed system. It would be impossible to use captuTe-and-release control ( CR), for instance, if the UPO we are trying to stabilize disappeared upon a small perturbation (see section 5.3). Definition 3.1. .Let f and g be two maps. The C0 distance between f and g is given by do(!, g) =sup lf(x)-g(x)j. xEJR The cr distance is given by dr(f,g) =sup (lf(x)-g(x)j, lf'(x)-g'(x)j, ... jF(x)gr(x)j) xER. 35

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Definition 3.2. Let f : A --> A and g : B --> B be two maps. Then f and g are said to be topologically conjugate ifthere exists a homeomorphism h : A -+ B such that h f = g h. Definition 3.3. Let f : J -+ J. Then f is said to be cr structurally stable on J iF3c: > 0 such that whenever dr(f, g) < c: f'or g : J --> J, it follows thai; f is topologically conjugate to g. The classic Smale horseshoe is formed as follows. Consider the sta dium shaped region D in Figure below. Squash D in the vertical direction by a factor of 15 < and stretch it in the horizontal direction by a factor of p 2 2. Bend the region over in the shape of a horseshoe and lay the horseshoe over D in the manner shown. Now repeat this process, stretching and folding the horseshoe again by the same factors and laying it in the stadium. s Figure 3.1.1. The Smale stadium consists of a rectangular central region fitted with semi-circular endcaps. 36

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Figure 3.1. 2. The Smale stadium, sq1tashed, bent, and placed back inside itself. Call the action of the map P. There is a unique attracting fixed point p in D1 since F' is a contraction mapping, and a.s D2 is mapped to D1 all points in D1 U D2 tend top under iteration, that is, lim Fn(q) = p for all q E D1 U D2 Furthermore, any point T E S whose image is not in S for all n obeys lim pn ( r) = p. We are interested in the points s E S such that Fn(s) E S for all n. Consider the two segments of S that are mapped back inside S by the horseshoe map F. Call these H0 and H1 and their preimages Vo and Vi (see Figure 3.1.3). Since F': S-+ Sis a linear map, it preserves horizontal and vertical lines in S. The width of V0 and Vi. are p and the height of H0 and H1 are 8, and if h is any horizontal line segment in S whose image under F is also in S, then the length of F(h) is ph. Likewise if v is any vertical line segment in S whose image under F is 37

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also in S then the length of F ( v) is ov. Figure 3.1.3. The intersection of S and F(S) gives H0 and H1, and the preimages of these horizontal strips are V0 and V1 Suppose Fn(s) E S \In > 0. Then s must be in Vo U V1 F(s) E VoUV1 F2(s) E V0UVi, ... for all points not in VoUV1 map to D1 UDz. Thus, we have that s E p-n(Vo U V]) for all n > 0. The inverse image of any vertical strip of width w in Vo or V1 that extends from the bottom to the top of S is a pair of strips of width !w, one in Vi and one in V0 that extend from the bottom to the top of S. The inverse image p-l of Vo U V1 is a set of four rectangular strips of width }2 w, two in Vo and two in V1 (see Figure 3.1.5), the inverse image of p-1(V0 u V1 ) is a set of eight vertical strips of width J,w, etc. Therefore lim p-n (V0 U Vi) P n--too is the product of a Cantor set with a vertical interval. Any point s E S such that Fn(s) E S \In> 0 must be in this set which we label JL,. By the same type of reasoning, we see that if a point s E S such that y-n(s) E S \In> 0, then it must belong to a product of a Cantor set with a horizontal interval, and we label this set A_. Any point s E S 38

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such that F"(s) E S \In must be in the intersection A= A+ n A_. Figure 3.1.4. The second iterate of the Smale horseshoe map is shown above. FiguTe 3.1.5. Shown is the intersection of the horizontal and vertical strips after the first two iterates of the map. In the limit the intersection is a Cantor set. Just as we did with the tent map, we may define a symbol sequence on A. The sequence is doubly infinite and is written as (3.1) where the Bj are 0 or 1 depending on which vertical strip Vo or V1 s is in at the jth forward iterate of the map, and the s_ j are 1 or 0 depeJ,lding 39

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on which horizontal strip s is in on the jth iterate of the map's inverse. The sequence ... s-2, s_J.So, s1, s2, .. uniquely defines a point in A, and the left or right shift of the binmy point gives the backward or forward iteration of that point respectively. This shift map is a model for the dynamics of s under F' restricted to A. As F' is topologically conjugate to the shift on the symbol sequence, we can define a metric on F' by d[(s), (t)] = Is;-t;l L.r 2i (3.2) i=-oo Definition 3.4. Consider a set Q and a mapping F : Q --+ Q. Two points PI and P2 are forward asymptotic ifFn(PJ), F'n(p2) E Q V n 2: 0 and lim IF'n(PI)-P(p2)l = 0. 11-----> 0 as n--+ oo }. Consider a fixed point s = ... 111.111 ... EA. Its stable set contains not only the vertical segment l8 in which it resides, but also any segment l that maps into l8 Thus the stable set of s consists of U;,Yk(l8). 'fhe unstable set of ii E A is different in form. Let lu be the horiz,ontal 40

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segment in which !i resides. Forward iteration of the map will stretch and fold 1,, giving the structure in Figure 3.1.6. w, <'---') c Wu "'oo Figure 3.1.6. The stable and unstable sets of the point s are shown in this figure. Periodic points are dense in the horseshoe, and orbits are topologically transitive and have 8ensitive dependence on initial conditions. These conditions are the signature of chaos, and a system that can be shown to have a horseshoe, can be proven to be chaotic, at least on a subset of its at tractor, according to the definition of Devaney[9, Robert Devaney, Chaotic Dynamical Systems, Addison-Wesley, 1989]. Definition 3. 7. f : J -> J is said to be topologically transitive if for any pair of open sets U, V E J there exists k > 0 such that r(U)nV=J0. Definition 3.8. f : J -> J has sensitive dependence on initial conditions if there exists {j > 0 such that, for any x E J and any neighborhood N of x there exists a y E N and n 2': 0 such that; lr(x)-r(Y)I > o. Definition 3.9. Let V be a set. f : V -> V is said to be chaotic on V if 41

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1. f has sensitive dependence on initial conditions 2. f is topologically transitive 3. periodic points are dense in V Devaney's definition is equivalent to that of Taylor and Toohey (Def. 2.9) when V is a uniform Hausdorff space. We can see that the horseshoe is chaotic by considering the symbolic dynamics. Density of periodic points: We must exhibit an orbit that con verges to an arbitrary point s = ... s_2 s_1.s0 s1 s2 ... Let T = ... s0 ... .s,., s0 ... sn-s0 ... sn, s0 ... sn ... be the sequence that repeats the first n symbols of s. Then d[T, s] :S and r --+ s. Topological transitivity: We must exhibit a point that comes arbitrarily close to every other point in A. Consider the symbol string s* = ... 011 010 001 000,11 10 01 00, 1 0.0 1, 00 01 10 11, 000 001 010 011... formed by concatenating all possible permutations of strings of length k, k = 1, ... ,n (the commas merely delimit the groups of permutations of strings of length k). Then for some shift a* the itinerary will agree with that of any point in A to the precision we desire. Sensitive dependence on initial conditions: Let r-, s E A and have identical symbol strings in the first n places to the right of the binary point and for the first rn places to the left. Then iteration forward or backward will eventually shift out the identical strings and the orbits will diverge. To sum up the action of the horseshoe map, we note: The horseshoe mapping of the Smale stadium has an invariant set A that is the product of two Cantor sets. Dynamics on A are chaotic under the map. 42

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There are unstable periodic orbits of all periods, and periodic points have stable sets that consist of the vertical line segment in which they reside and the set of all segments that map to the vertical segment. The stable and unstable sets of points in A are orthogonal. 3.2. The Herron map and its horseshoe structure The horseshoe structure arises naturally in dissipative chaotic maps of the plane and Poincare maps of chaotic flows. Consider the Henon map Xn+l = a + byn x;, Yn+l Xn where a and bare parameters. The Hi'mon map is invertible, dissipative and has chaotic dynamics for certain values of a and b. A few thousand iterates of the Henon map are shown in Figure 3.2.7. -1 -2 FiguTe 3.2.7. Several thousand itemtes of the Henan map aTe shown joT pammeteTs a = 2, b = This figuTe and FiguTe 3.2.8 weTe pmduced by the pmgmrn Dynamics by James YoTke. 43

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This map has a fi'
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of l0 is in Lx. Under the action of the map X a and Xb will approach X' arbitrarily closely, and the length of l0 -" 0 as n -" oo. As in the horseshoe, the unstable manifold of the Hcmon at tractor A is the same for periodic points of all periods and for points in chaotic orbits. Each periodic point has its own stable manifold, and the same argument as above can be made for each of these stable manifolds, namely, any point in a segment lk E L xi, where Xj is the ,jth point J in the periodic orbit i will approach Xj arbitrarily closely as n -" oo. The set Sp of all stable manifolds for all periodic points in the Henon at tractor is dense in the basin of attraction of the at tractor. The closure of the set Sp is the entire basin, and the set that closes Sp is the set of stable manifolds Sc of the chaotic orbits in A. Definition 3.10. Let A be an invariant set for a discrete dynamical system defined by f : R" -" Rn. A hyperbolic structure for A is a continuous invariant direct sum decomposition 1AR" = EX Ell B'J.. with the property that there are constants C > 0, 0 < .\ < 1 such that: 1. ifv E B::;, then !Drn(x)v! :c; c;,n !v!; 2. if v E B;, then ID f"(x)v! :c; c.>.n !v!. The invariant set A of the Smale horseshoe has a hyperbolic structure. Essentially, a hyperbolic structure implies that all the periodic points are of saddle type. The Hcmon map may be shown to have a horseshoe structure; that is, it may be shown to contain a chaotic set and unstable points of all periods. This set is a product of Cantor sets, and lives in the unstable manifold of the Hcmon at tractor. However, we do not know whether all the points that appear to be in the attractor are part of a hyperbolic structure (there are good reasons to suppose that they are not[lO]). This means that what appears to be chaotic motion over the entire connected unstable manifold of the Henon at tractor may instead be a chaotic transient preceding asymptotic approach to any one of an infinite number of stable periodic orbits of arbitrarily high period. We take the view that although we don't know for sure whether the apparent chaotic evolution of an orbit over an attractor is 45

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really chaotic or just a chaotic transient, for the purpose of developing a control strategy there is no difference. Whether we have a true chaotic orbit or one of very long period, our goal is to stabilize low pe riod unstable orbits, and to navigate through the at tractor along these orbits. In either a simulation on a computer or in a physical experiment, chaotic orbits and orbits of very large period will appear to be the same. 3.3. The suspension of the horseshoe Our purpose in considering the horseshoe map is to use its suspension as an abstract representation of the dynamics of the flow of a chaotic dynamical system. The dynamics of a Poincare map of a flow determine the dynamics of the flow itself; that is, if the Poincare map has a periodic point of period n, the flow has a periodic orbit of period n, and if the Poincare map has a chaotic orbit, the flow has a chaotic orbit. The stable and unstable manifolds of the Poincare map of a chaotic flow act very much like the stable and unstable sets of the horseshoe, and with this in mind, we suspend the horseshoe in a way that forces the stable and unstable sets of the suspension to act like the stable and unstable manifolds of a flow. Consideration of the geometry of this suspension leads to the construction of a general framework in which control by stable subspace targeting takes place. We provide some definitions: Definition 3.11. Let


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Definition 3.12. Let y E I: and r(y) be the least positive time for which 'P7(y) (y) E I:. The Poincare map for I; is defined to be P(y) = 'PT(y)(Y), y E I:. As with the baker's map, the horseshoe can be suspended to give a three-dimensional flow. While the baker's map required some care to ensure that it was not two to one, the suspension of the horseshoe map is straightforward, and its generalization can model the actual dynamics of a physical system rather well. Consider the physical description of the iterative process that leads to the Smale horseshoe. We take a stadium D composed of a central rectangular region S and two end caps D1 and D2 stretch it linearly in the horizontal direction, while squashing it linearly in the vertical direction, fold it over and reinsert it into its original boundaries in such a way that only points in S get mapped back into S. This insures a structurally stable dynamical system consisting of the points s E A, and the mapping F restricted to A. The itinerary S(s) = ( ... s-2,s_1.so,s1,sz, ... ) is a doubly infinite sequence of symbols 1 and 0 that give a record of the travels of s under F. Take A and its stable and unstable sets W" and wu and call their union G. vVe cross G with the circle C to get G X C, and stretch and fold G X C over itself in such a way that after one circuit of the circle vertical lines in V0 and 11] have mapped into Vo 47

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and Vj. L Figure 3.3.1. We may connect the line segments in the stable set of the horseshoe ther-eby changing them to manifolds. 48

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Figure 3.3.2. The ends of the stable set8 are connected in this figure to make the stable sets into stable manifolds. When the horseshoe is stretched and folded, the manifolds unbend as the stmight segments join. The unstable set is a differentiable manifold, and we can make the stable set a differentiable manifold by including in it the paths traced by points in the right hand side of the top edge of A+ as it stretches and folds over to rE:ioin the left hand side, as well as the paths traced out by the complimentary points on the bottom of A+ (see Figures 3.3.1 and 3.3.2). We can fold other horseshoes in continuous time that have proper ties superficially similar to the apparent stretching and folding of other dynamical systems. We emphasize that just because the stretching and folding sc>ems similar, there is no guarank>e that a horseshoe structure persists through an infinite number of stretch and fold operations. We have already seen the similarity between the stable and unstable sets of the Smale horseshoe and those of the I-Icmon attractor. In section 4.1 we will look at the stretching and folding of the pendulum at.trac tor, and compare the pendulum dynamics to those of the suspended horseshoe. Depending on the values of parameters in the pendulum equation, the pendulum may have a strange at tractor corresponding to chaotic motion that includes e..xcursions over the top of the pivot, or chaotic motion for which the pendulum bob never goes over the top. In either case, there exist unstable periodic orbits whose manifolds are flip saddles, like the flip saddles of the Smale horseshoe. In the course 49

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of one system cycle, the manifolds associated with a period one orbit in either the pendulum or the suspended Smale horseshoe undergo a half twist before rejoining, forming a Mobius band. The rotation of the manifolds of the flip saddle are what make pos sible capture and release control, and a knowledge of the movement of the manifolds as an attractor goes through its cycle of stretching and folding allow us to design general control rules beyond the map based rules of OGY. 3.4. The control of the horseshoe Like the baker's map, the horseshoe map can be controlled. We imagine that we have two parameters that control the geometry of the horseshoe; b < which controls how much the stadium is flattened, and p which controls the stretching. We fix o and allow p to vary so that the position of the stable set W8 of A shifts position as p varies. The change in position of the stable set is continuous with change in p, as long as the endcaps D1 D2 do not map to the central rectangular region S. Suppose an iteration of the horseshoe map brings a system state s within c of the state s' the state we wish to control. There will be a parameter value p(s', s) for which the dynamics of the map carry s to l8 the vertical segment containing s', in one iteration. Successive iteration of the map brings any point in z,. as close to s' as we like. Figure 3.4.1 below is a sequence showing how l,. approaches s' when s' is the period one point ... 1, 1, 1.1, 1, 1.. .. The points shown are members 50

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of the invariant set of the mapping. .. . . . .. . . . . .. . .. . .. .. . . . . . . . . . . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . .. .. .. .. .. .. .. .. bl" . . . . .. .. .. .. .. .. .. .. .. .. .. .. I .. .. .. .. .. .. . . . . .. .. . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . .. .. .. .. .. .. .. .. .. .. .. .. .. .. . .. .. .. .. .. . .. .. .. .. .. .. .. .. .. .. .. Figure 3.4.1. Here, the orbit of a line l8 in the stable set of a period one point with itinerary ... 111.111... is shown under three iterations of the horseshoe rnap. In the same way we can target a preimage of ls., and again, suc cessive iteration will bring the segment as close as we like to s'. If our perturbation is limited to a small c:, we may choose a set of periodic orbits {s1,s2 ... sn}, each of which we can target by the method above. We can restrict the number of preimages of l8 we will consider in our targeting procedure, noting that the accuracy of our calculation of the 51

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position of the preimages of l8 k decreases as we go further back in time. If we choose a sufficient number of orbits, an c region around any verticalline in the stable set of one particular orbit will contain a member of the stable set of another orbit in the set {s1 s2, ... sn} Then, by threading our way through the preimages of the l8k, k = 1, ... n we can move from one orbit to another in an efficient fashion. We can define an operator that evolves local behavior continuously in the horseshoe. Consider the suspension of the horseshoe G x C, that is, the union of the stable and unstable sets of the invariant set II. of the horseshoe mapping F crossed with the circle C. Define a periodic folding operator P( T), T = t mod 21T such that P( T) ( G x C) gives the Poincare section of G x Cat timeT. Then P(O)( G X C) = P(27r)( G X C) and P(T)(G X C)= P(T 2n1r)(G x C). We noted earlier that a piece lk of the stable set II. of the horseshoe maps to lk+l after one iterate, and to lk+n after n iterates. The discrete time operator P(27r) will be designated P, and pn designates n applications of P. TheTefore P(27r) lk = Plk = lk_1 _1 and pnzk = lk+n Similarly, the unfolding operator 1 ( T), T = t mod 21T is defined so that P( T) ( G x C) is the Poincare section of G x C at time T. The opera toT P( T) takes an initial segment l and evolves it smoothly from one iterate of the horseshoe map to the next. If the initial segment lo was in the stable set of a period one point, for instance, and 10 was inside the at tractor, the map would take 10 to h, the segment one layer in and on the other side of the saddle. -when the continuous time operator is applied to lo, as T goes from 0 to 21T, P(T)lo traces out a twisting ribbon that becomes narrower each cycle as it approaches the periodic orbit. P(T)lo may even braid the ribbon. 'I'he fold operator evolves periodic orbits along their flow, and the set of periodic orbits is an infinite set of intertwined links. The orbits of a continuous time dynamical system can link and knot in exceedingly complex ways. It has recently been shown, for instance, that the set of periodic orbits of the Lorenz at tractor link in every possible way (the set of links of periodic orbits contains representatives of every tame knot isotopy class)[ll]. Let ll.(p) be the invariant set of the suspension of the horf!_eshoe, 52

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where the dependence of A on the stretching parameter p E -0 :S 0 :S 0 has been made explicit, and let A(O) be the unperturbed system. The bound 0 is such that the end-caps of the stadium never map into 8. Then A(p) is the set of interlinked periodic orbits of all periods, and P(T)A(O) is the continuous time system in which points s E A get mapped to their images P s E A at T = 2mr, n = 1, 2, .... We choose to stabilize a periodic point s'. Suppose we monitor the evolution of an initial condition s E A(O) until it is close enough to l8-, the vertical segment in which s' resides, so that a perturbation p( s, s') E -0 :S 0 :S 0 for one iteration would serve to place s in l8 We apply this perturbation for one iterate and then turn it off. The system state s is then in ls, and it approaches s' as t -+ oo. This is the discrete time control described above. We will look later at the meaning of control in continuous time from the perspective of control of the suspended horseshoe. The set A has a hyperbolic structure, so all of its periodic points under F are saddles. It is not difficult to see that a segment l E W8 is flipped over upon iteration if it is in the right side of S, and retains its orientation upon iteration if it is in the left half of S. This implies that the local stable sets of periodic orbits whose itineraries contain n 1s experience n flips, so a saddle will be a flip saddle if n is odd, and a regular saddle if n is even. Note that even though a periodic point of a Poincare section may not have a flip saddle, if its itinerary contains a 1 then the continuous time periodic orbit has a flip. Consider the n periodic orbit pn s' = P(O)s' = s' where sk = 1 for some integer k :S n. Then pk-1s' is in the right half of S, and so must flip upon iteration or during the continuous time evolution from P(27r(k1))s' to P(27r(k1) + 27r)s'. Later we will show how P twists perturbed saddles relative to each other in time, allowing us to establish a framework in which to formulate control by capture and release. 53

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4. The Dynamics of the Pendulum 4.1. Folding the horseshoe Chaos in the driven pendulum may be viewed as resulting from a stretching and folding of an invariant subset of the full phase space. This stretching and folding appears to produce an attractor with a horseshoe structure in the Poincare section. If we start with a carefully chosen set of initial conditions in the x, y phase plane and evolve them forward by wt we expect to find that they are stretched, folded and mapped inside the boundaries of the initial set. This horseshoe structure is the hallmark of chaos, and it guarantees periodic points of all periods as well as those of infinite period. The pendulum equation has not to my knowledge been shown to have a horseshoe, but its seemingly chaotic behavior and suggestive folding seem to indicate that it likely does. Although the classic Smale horseshoe preserves line segments, linearity is not a necessity for the formation of a chaotic invariant set. Recall that the horseshoe map f has an invariant Cantor set A such that (a) A contains a countable set of periodic orbits of arbitrarily long period. (b) A contains an uncountable set of unbounded non periodic motions. (c) A contains a dense orbit. Moreover, any sufficiently C1 close map j has an invariant Cantor set A with fiA topologically equivalent to JIA. We therefore proceed with confidence constructing horseshoes whose dynamics we liken to the dynamics of the vertically driven pendulum .. We establish a framework for the allowable types of horseshoes with the following definitions, lemmae and a theorem. Definition 4.1. A vertical curve x = v(y) is a curve for which 0 :S: v(y) :S: 1, lv(y!)-v(y2)l :S: fliY! -Yzl in 0 :S: Y1 :S: Y2 :S: 1 for some 0 < {.( < 1. 54

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Definition 4.2. A horizontal curve y = h(x) is a curve for which for some 0 < fl < 1. Definition 4.3. A vertical strip V is defined by V = {(x, y)jx E [v1(y), v2(y)]; y E [0, 1]} where v1 (y) < v2 (y) are non-intersecting vertical curves. Definition 4.4. A horizontal strip H is defined by H = {(x,y)jy E [h1(x),h2(x)];x E [0, 1]} where h1 ( x) < h2 ( x) are non-inl;ersecting horizontal curves. Definition 4.5. The width of a vertical or horizontal strip is defined as d(V) =max lvz(y)-VJ(Y)I, d(H) =max lhz(x)-h1(x)j yE [0,1] xE [0,1] Lemma 4.6. If V1 :J V2 :J V3 ... is a sequence of nested vertical (or horizontal) strips and if d(Vk) --. 0 as k --> oo then Vk d.;j V00 is a vertical (or horizontal) curve. Lemma 4.7. A vertical curve v(y) and a horizontal curve h(x) inter sect in precisely one point. Hypothesis 1: Let 6 be the set {1, 2, ... N} and let Hi, v; for i E 6 be disjoint horizontal and vertical strips and let f(Hi) = v;, i E 6. Hypothesis 2: f contracts vertical strips and f-1 contracts horizontal strips uniformly. Let VJ, Vz E v; be any two vertical curves bounding a vertical substrip V,' s;;; v;. Then f(Vi) n V; is a vertical strip and d(f(V,'), V;)::; vd(V,')d(V;)/d(V;) 55

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for some v E (0, 1) and 'i,j E 6. Similarly, letting h1 h2 E Hi be any two horizontal curves bounding a horizontal substrip H; <:;; Hi, then f-1(H;)nH1 is a horizontal strip and d(f 1(H;)nH,) :':: vd(H1 ) Theorem 4.8. {lO}If f is a two-dimensional homeomorphism satisfying Hypotheses 1 and 2 then f possesses an invariant set A, topologically equivalent to a shift CJ on I;, the set of bi-infinite sequences of elements of6. This theorem shows that a wide range of different nonlinear foldings can produce horseshoe structures, and if, in addition, the map f is a G'" diffeomorphism with T :2: 1, then (with a couple of additional assumptions) A is hyperbolic. Numerical evidence indicates that there are stable and unstable manifolds of periodic points in the Poincare section of the driven pendulum that act like the stable and unstable manifolds of 11.. The stretching and folding of the pendulum attractor can be mimicked by a stretching and folding of the manifolds of the suspension of a suitable horseshoe map. Let us consider how the dynamics of the vertically driven pendulum equation (see Appendix B for a physical derivation) stretch and fold a subset of the phase space with the passage of time. We examine the dynamics of the system of first order ODEs :i:=y y = -py-sinx(lAcoswt) where :i; is the angular velocity of the pendulum shaft, y the angular acceleration, p the damping factor, A the amplitude of the driving term and w the drive frequency. We will examine the dynamics in the phase plane of x and y. The first ODE :i; = y guarantees that a phase point ( J:, y) moves in the x direction in direct proportion to the value of the y component. Points in the upper half plane will move to the right and points in the lower half plane will move to the left. In the second ODE, the ,-py 56

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term forces points to move towards the x axis, and the periodic part of sin x(l-cos wt) reinforces and counteracts the -py term by causing points to move away from the x axis. Since the effect of the periodic term is modulated by sinx, points near x = and x = will be driven upward and downward with more vigor than those near f or 0. This periodic action produces a fold that is pulled from left to right on the upper half plane, and from right to left on the lower half plane. Figure 4.1.1 is a plot ofy = sinx(l-coswt). In this plot, they axis is vertical, the x axis goes from upper left to lower right, and the t axis goes from front to back. ( 4.1) Figure 4.1.1. The plot above shows the periodic ter-m responsible for the stretching dynamic of the pend11l1lm eq11ation. The sequence in Figure 4.1.2 shows the evolution of the Poincare section of the pendulum through one period of the drive cycle. The stretching and folding is clearly evident. As the at tractor stretches and folds, the stable manifolds of the unstable periodic orbits stretch and 57

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fold also. Figure 4.1.2. The sequence above shov;s 16 successive Poincare sections of the vertically driven pendulum v;hen the damping is large enough so that the bob never goes over the top. We can picture the stretching and folding in terms of horseshoes, but with an important difference. The Smale horseshoe construction maps part of the phase space out of the region of interest, resulting in an invariant set that is a product of Cantor sets. The basin of attraction of the pendulum equation is the whole phase space, and the folds at the ends of the attractor (see Figures 4.1.2 and 4.1.4) get mapped back into the central region corresponding to S in the Smale stadium. These internal folds are continually stretched out and folded deep into the interior of the attractor. Since no region is mapped out of the space, as in the Smale horseshoe, the pendulum attractor is the product of a Cantor set and a curve, rather than the product of two Cantor S(lts. If 58

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the pendulum contains a horseshoe structure, it is non-linear horseshoe, and it is most likely a product of Cantor sets imbedded in the at tractor rather than the attractor itself. We may construct pendulum-like horseshoes and make an analysis of their dynamics. In some of the gross features, the pendulum horseshoes have similar dynamics to those of the pendulum. In the horseshoe of Figure 4.1.3, a symbol sequence of three symbols would suffice to describe the dynamics on the invariant set, a Cantor set formed by the intersection of the nestings of three vertical and three horizontal strips (as the Smale horseshoe was formed of two). There is a fixed point in the center of the rectangular region, and the outer layers get mapped successively closer to the horizontal segment in the center of the rectangular region. Note the similarity to the folding of the sequence of Poincare sections of the pendulum. Furthermore, there arc two fixed points in the end caps. If the mapping included a rotation by 180 the fixed points in the end caps would be periodic of period two, as in the pendulum attractor. I. ( \; -;; (4.2) Figure 4.1.3. This particular folding resembles the folding of the pen dul-urn attmctor, and the action of the folding on the invariant set is topologically conjugate to a shift on thr'ee symbols. 59

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::r ::: "';;: ;:: ;:; :;: .. :::;:;iii :::"' ::: ::: ,;, : ::: m ::: ::: "' ::, ::: :;: ::: Figure 4.1.4. The invariant set of the pendulum horseshoe. In the mapping of Figure 4. 1.5, we can see that successive iterations will eventually map the end cap into the linear body of the at tractor. This situation is more similar to the dynamics of the pendulum and other real world attractors. There are folds within folds within folds, and the dynamics become extremely difficult, if not impossible, to an alyze using symbolic dynamics. Nevertheless, if this folding generates a horseshoe structure, then whether the remainder of the at tractor is populated with periodic orbits of extremely high period, or infinite period orbits, the horseshoe contributes chaotic dynamics on its invariant 60

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set, and long chaotic transients for orbits that pass nearby. ( "' )) ( Jl w (4.3) Figure 4.1.5. This stretching and folding is like that of the pendu lum, but because the folds are eventually mapped into the body of the attmctor, hyperbolicity cannot easily be established. The inclusion of the endcaps in the body of the attmctor essentially destroy the nested se quences of vertical strips, as the folds are neither vertical nor horizontal and could create tangencies between the stable and v.nstable manifolds. When tangencies exist, there exists the possibildy for an infinite number of stable orbits in the attmctor. When we look at a sequence of Poincare sections of the pendulum taken at successive phases of the forcing term, we see a stretching and folding of the stable and unstable manifolds that bears a striking re semblance to the stretching and folding of the stable and unstable sets of the suspended horseshoe. Specifically, line segments l whose endpoints are in the invariant set 1\.p of the pendulum are mapped to other segments with endpoints in the set 1\.p. We may imagine the continuous time evolution of a segment ls as the unfolding of a ribbon that writhes through the torus in which the pendulum solution lives, eventually find ing its way in between layers of the attractor, getting closer and closer 61

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to the periodic orbit while becoming narrower and narrower. A segment l, in the stable manifold of a periodic orbit X' eventually comes as close as we like to X', and it is this fact that we count on to make control by stable subspace targeting possible. We seek to perturb the system so that beginning with any state X0 we can find a perturbation or sequence of perturbations so that X evolves to l, E EX:., a line segment in the stable set of X'. The dynamics of the evolution of l, then assure that the orbit will approach that of X' asymptotically. The following sequence of pictures shows a line segment in the stable set of the straight down steady state of the pendulum as it evolves through one drive cycle. Notice how the segment, originally "outside" the at tractor is folded into the "inside" of the at tractor. Thereafter, its fate is to go deeper and deeper into the "interior", alternating sides as it approaches the steady state. The stable straight down state is the center manifold of a flip saddle. It is easy to see the rotation of the manifolds in this picture. The fig11res are in sequence of increasing drive phase angle from top left to bottom right, and the final frame the attract.or is in its original position, except that the stable manifold that started as a loop is now folded into the at tractor. If we viewed the entire stable manifold of the unstable steady state, we would also see the loop that replaces the one folded inside. 62

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Figure 4.1.3. These seven Poincare sections taken at equally spaced inter-vals thr-ough the drive cycle illustrate how the stable manifold is folded into the attmctor during the course of a cycle.

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The dynamics of the stable manifold above are repeated over the entire body of the attractor for all the periodic orbits, that is, pieces of stable manifolds "outside" the at tractor unfold and are compressed "inside" the at tractor. Once "inside" they migrate deeper and deeper into the fractal layers, sometimes approaching a periodic point asymp totically, and sometimes mapping back out, only to be swallowed up again. When we fold a horseshoe in the pattern of the folding of a pendulum attractor, we find that the dynamics along the stable and unstable sets of the horseshoe seem to mimic some of the dynamics of the verti cally driven pendulum if the suspension includes a half twist to make a Mobius band. I J ( ) (I Figure 4.1.4. The pendulum horseshoe above corresponds to an attrac tor that has no points surrounding the origin. We can see that points near the origin get stretched ont and there is no reinjection of the points in the fold to the central region. In a physical pendnlnrn, this condition would correspond to an attractor with minirnurn per-iod two. The at tractorconsists of a folded region that coils through the toms once in two dT"ive cycles, piercing the Poincare plane in two separate places. If we were to watch a movie of the evolution of the attractor's Poincare section with change in drive phase, we wonld see two folded regions that circled the or-igin, changing places once per cycle. Physically, this cor responds to a pendulum that always crosses the straight down position with positive velocity. 64

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Fig1lre 4.1.5. The pendulum horseshoe above corresponds to a pendulum attractorthat has points near the origin. The folds reinject an orbit that has been stretched out of the central r-egion back into the region near the origin. This situation correspond,s to a pendulum that can have zero velocity at the origin. There is then a path to the steady state along a stable manifold of the attractor-. Figure 4.1.6 shows the unstable manifold of the pendulum at tractor along with a portion of the stable manifolds of three periodic points. Notice the similarity of the loops in the stable manifold to those of the suspended horseshoe. The folding in Figure 4.1.6 corresponds to the folding of the pendulum horseshoe in Figure 4.1.5, as there is a 65

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reinjection region at 0, 0. Pigure 4.1.6. Shown are the unstable manifold of all points of all peri ods (the Z shaped r-egion) and a portion of the stable manifolds of three separate periodic points. The stable manifolds of all the nnstable pe riodic points are dense in the basin of attraction. The origin and the period two points are all .flip saddles. The rotation of the manifolds of the flip saddle are what make pos sible capture and release control, and a knowledge of the movement of the manifolds as an attractor goes through its cycle of stretching and folding allow us to design general control rules beyond the map based rules of OGY. 66

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4.2. Bifurcation and orbit structure The pendulum ODE has no analytical solutions in terms of simple func tions, but numerical integration of this equation reveals rich dynamics, both chaotic and non-chaotic. Some analytic methods are useful in establishing approximate characteristics and locations of periodic orbits, but the method of OGY is model-independent, and for the rest of the paper we will assume that we have access to a data set only. Solutions to the driven pendulum equation live naturally in a solid torus of rectangular cross section. A slice through the torus perpendicular to the minor axis reveals the familiar phase plane of the unforced pendulum. This phase plane is now a Poincare section, the map made by taking a cross section of a flow at some phase of the drive. Along the minor axis of the torus we plot the drive phase, which is 27fperiodic. The cross section is scaled in the vertical direction to fit the largest angular velocity attained on the attractor, and is 27r-periodic in the horizontal direction, corresponding to the periodic natnre of over the top rotation. time Figure 4. 2.1. The dynamics of the pendulurn live inside a torus of rectangular cross section. The left and right boundaries of the Tectangle are periodic, but the top and bottom must be unbounded. As any one of the three parameters p, w, or A is varied, and for 67

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certain sets of fixed values of the other two parameters, the motion of the pendulum goes through a series of period doubling bifurcations to chaos. We will consider only the case of variation of the damping pma meter p, as variation of the other parameters produce similar changes, except in the details. With high enough damping (and in a specific range of fixed values for w and A), the steady state () = 0 of the pendulum is straight down, even while being driven. This state is stable and its basin of attraction is the entire space, save a set of measure zero attracted to the unstable inverted state. As damping is decreased, a new stable solution appears, a period-two orbit, where the pendulum swings back and forth once every two drive cycles. There are now two basins of attraction, one for each solution. Paired with this stable sub hmmonic period-two solution is an unstable period-two solution. These solutions appear together and split from each other as the damping is decreased. Figure 4.2.2. The figure above illustrates the center, the stmight down stable state, and at the sides, the stable and unstable peTiod two oTbits. -+-+ FiguTe 4.2.3. The Btable and rmstable period two or'l;ds have separated as the damping decreases st-ill moTe. 68

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As the stable and unstable solutions separate, the basin of the stable straight down state shrinks. Finally, the basin of the straight down stable state blinks out of existence, and the only stable solution is the period-two orbit. The unstable period-two solution has merged with the stable period one solution and the basin of the period two solution is the entire space, except for a set of measure zero (the stable set of thee= 0 solution). As damping is decreased still further, the symmetric period-two solution bifurcates to two stable asymmetric period-two orbits, where swings to one side alternate between two heights. The symmetric period-two solution remains as an unstable orbit. The basins of attraction of the two asymmetric states (which are mirror images) are fract.ally entwined in each other. A section through the basin plane would reveal a Cantor structure of rather high dimension. As the stable solutions separate from each other, the dimension of this fractal decreases. Upon further decrease in damping, the stable period-two solutions bifurcate to period-four, period-four to period-eight, ete. The stable states of the pendulum keep bifurcating, but with smaller and smaller changes in the parameter until a point Pcritl is reached where an infin itesimal change in parameter produces a state where only an orbit of infinite period is stable. This is the onset of chaos. As we decrease the damping still more, this attracting infinite period orbit approaches 1T in the B direction, which corresponds to the straight up position. As the pendulum goes over the top, the at tractor collapses suddenly to two period one orbits, whirling continually over the top either clockwise or counterclockwise. This sudden collapse of a chaotic attractor is called a crisis, and it occurs whenever the basin of attraction of a chaotic orbit collides with the basin of a stable periodic orbit as a parameter is varied. More generally, a crisis occurs whenever basins collide. We continue to decrease the damping and witness another period doubling bifurcation, but now as Pcrit2 is approached we have a stable infinite period orbit that can go over the top. As damping is decreased still further, the attractor increases in size, apparently occupying more of the phase space. It is this change in the size of the at tractor, along 69

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with the attendant shift in the position of the unstable periodic orbits therein embedded that makes OGY control possible. Figure 4.2.4. The plot above is a bifurcation diagram. Damping is decr-easing fmm left to right, and angular position is along the ver-tical ax1.s. The structure of this attracting orbit is quite complex, yet embedded in this structure are the ghosts of all its previous behaviors, the unstable periodic orbits. These periodic orbits in sum determine the structure of the attracting set. When the attracting set is chaotic and has a fractal structure, it is known as a strange at tractor. As an orbit moves through phase space, it is attracted to a saddle orbit along the stable manifold, only to be driven off along the unstable manifold as it approaches the saddle. This occurs continually as the orbit moves through the varying 70

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influences of the saddles, which are dense in the at tractor. / -x --------,. . Fig'Ure 4.2.5. The Poincare section of the pend'Ul'Um in the non-over the-top mode. Fig'Ure 4.2.6. The Poincare section of the pendul"!Lm in the over-the-top mode. 71

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5. Control of Continuous Time Systems 5.1. Control by OGY Theoretically there is an infinite number of unstable saddles in the attractor of the chaotic pendulum, but in practice the lower period ones are easiest to control. The dynamics in the Poincare plane is all we need to establish control by the method of Ott, Grebogi and Yorke, and we can use the Poincare map to establish the linearized dynamics about any periodic point of interest. Now let us suppose we want to stabilize one of these periodic orbits, and all that we have is a data set of length N composed of the successive positions () and velocities iJ taken at some phase of the drive. We form the vector = [ ] which we where the dependence on the phase at which the Poincare section is taken, and the damping pis made explicit. :For this section, however, we will ignore the dependence on and p. How do we locate these unstable periodic orbits from the data alone? For the sake of demonstration, let us suppose we want the pendulum to whirl clockwise over the top, a period one orbit of the flow, or equiv alently a fixed point of the Poincare map. There is only one periodic orbit of this form embedded in the attractor, and since we arc limited to the preexistent unstable limit cycles by the nature of the algorithm, this is the orbit for which we must settle. Define some small distance E and locate all pairs of points in the data set whose Euclidean distance apart is less thanE (if we were seeking points of period n we would look for pairs of points We locate the center of mass of these points, which is a good approximation to the position of (0 the period one point we seek. Now let us compute the linearized dynamics near this point. We seek a 2 X 2 matrix A that when applied to some point (; near the periodic point gives the next iterate. One especially simple way is as follows: For the unperturbed system, write 72

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where the column vectors are two experimental points near the saddle and J+ 1 k+l are their first iterates. Solving for A we get We can average the A's obtained from several instances of close en counter, but there are two possible sources of error. Our assumption of local linearity must hold throughout the domain we choose as close encounters, otherwise our local transition matrix A will be in error. If an iterate of the Poincare map approaches the fixed point too closely, we may obtain an inaccurate estimate of the dynamics due to the error in our approximation of the location of the fixed point. Clearly, the set of points used to estimate the fixed point's location should be close to the true fixed point if we want an accurate estimate of the fixed point. Some familiarity with the dynamics is necessary to make an intel ligent choice, and in practice, I used my judgement to determine the size of the data set used to locate the fixed point, and in the choice of close encounters used to obtain A. As in the case of control of the Lozi map, we need the eigenvalues Au .\, and the eigenvectors e'u e,, of A. For convenience, we normalize the lengths of these vectors so that they are unit vectors. We also need the left, or contravariant, eigenvectors Iu and Is These vectors are orthogonal to eu and e8 and will be defined by I, eu = o Is Cs = 1. Now consider what happens as a parameter of the system (damp ing, in this case) is varied slightly. As mentioned above, the at tractor changes size with variation of damping, and the location of the fixed points moves with the attractor. Let p be the damping parameter, and 73

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8p be some small change in p. Then the location of the new fixed point upon this change will be 8p g = 8p During one drive cycle a phase point evolves to Relative to the fixed point these points may be represented by We are interested in varying the parameter p to achieve a change in the next iterate, so we write the equation to express the dynamics from the point of view of the shifted fixed point. The state vectors are now so 8pg 8pg; (5.1) where we have assumed that the local linearized dynamics of the perturbed system are near enough to those of the unperturbed system so that. the use of A for both the unperturbed and the perturbed system is justified. If the parameter p is allowed to vary, the position of the unstable limit cycles in the attractor will vary, as will the position of the attractor itself in the phase space. Our goal is to push the state variable onto the stable manifold of a desired periodic orbit. Once there, the dynamics will draw us into the fi."Xed point. The requirement that the next system state be on the stable mani fold of the unshifted fixed point may be written as fu = 0 We desire an expression equating 8p to a function of the eigenvalues and eigenvectors of A and the current position. Rewriting 5.1 as 74

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= 6pg + 6pg] and dotting by fu yields 0 = fu Opg + fu 6pg]. (5.2) We wish to solve for 6p, the perturbation of the damping that puts us on the stable manifold, so to this end we write A in terms of its components via the transformation T T A= Au.e,.Ju + A.,e8f.,. Substituting 5.3 into 5.2, we obtain so that 0 6pfu g + fu [Aueufu + Opg] 0 6pj, g + [Aufu eufu + Opg] 0 6pfu g + A.ufu Au0Pfn g], op = Au fu Au1 fu 'g which is the control law we seek. (5.3) Whenever an iterate of the Poincare map enters a small box around the fixed point (determined by either a physical restriction of an experimental parameter or by the limit to the approximation of linearity near the fixed point), we change the value of the parameter by op for one Poincare cycle. The shifted and unshifted attractors are shown in 75

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Figure 5.1.1 below .. .. ,. /.:: ,:-. . '; Figure 5.1.1. In these panels, the lobes containing the period one orbit ar-e shown. The damping in the left panel is greater than that in the right panel. As the lobes contr-act or ex:pand, the location of the fixed point changes. Local &trlx .5::f"'t)25S -.4132011 -1.&14242 hcenoo-2.62172 vcen"'-1.1653S1 f1-ei.d" FZ.:.z111 'ln. ra-,zn .oot F'l-center f,S-cru= f&-cledr f6-calc f11-lterup f12-iterdu 3-A (i""'""t ctu.rent val -2.626661 -1.13221 Figure 5.1.2. This figure shows a computer screen image of the Poincare section of the attmctor and the data extr-acted from the dynamics-. 76

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Figure 5.1.3 below shows a plot of the time series of control of a physical pendulum by the method of Ott, Grebogi and Yorke[l2] .. ... .... . . ,, ... .. ,i '; ) . .:J :. i .. .. : Iteration , ... .. I 2700 Figltre 5.1.3. The picture illustrates the control of a physical pendulum by the OGY method. Control is alternated between period one and period two. 5.2. General geometric picture of control We have seen how the stretching and folding of an attractor brings points in the stable manifold of a periodic orbit closer and closer to the orbit upon each fold. We have also seen that some manifolds twist as the attractor stretches and folds, specifically in the sequence of 16 Poincare sections of the pendulum at tractor. We can use this knowledge to gain control of a chaotic system in less than one fundamental period, that is, we can establish control before the next SOS iteration is taken. The simplest of these procedures is control by capture and release (CR), which works only for flip saddles. Flip saddles arise in period doubling bifurcations. The illustrations below show how a flip saddle is formed. 77

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Figure 5.2.1. The stable period one orbit before bifurcation. Figure 5.2.2. The stable period one orbit has become unstable, bifur-cat ing to stable period two. The first illustration shows a period one orbit and its piercing of a SOS map. A parameter is changed until the period one orbit loses stability, and a stable period two appears. One of the stable directions of the period one orbit has become unstable and is now serving also as a stable manifold of the period two orbit. Note that a half twist in the common manifold is necessary in order to form a period two or bit. Paths near the periodic orbit will follow the twisting manifolds and thus alternate sides in the surface of section map. Now suppose we move in a frame that follows the period two orbit. For this orbit to bifurcate to period four, it must undergo a half twist in this moving frame. As the orbit bifurcates, the now unstable period 78

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one and period two orbits must still be connected by their common manifold. For a new half twisted manifold to form connecting the period four orbit and its unstable progenitor, the manifold connecting the unstable period two and period one orbit must also fold, not once, but an infinite number of times. This scenario is responsible for the folded fractal structure of the strange attractor. Thus, a bifurcation to period four in a continuous time system implies an infinitely folded manifold. A cross section through this manifold will have a fractal structure. Suppose we perturb a period one flip saddle, and look at the relationship between the perturbed and unperturbed manifolds. As both the perturbed and unperturbed manifolds undergo a half twist over the course of a drive cycle, the stable manifolds (or their linearized ideal izations) must coincide at some point. Imagine the hands of two clocks side by side. One can see that at some time during the hour, the hands are colinear. Figure 5.2.3. This figure show an idealized period one flip or-bit and its manifolds. We simplify the picture somewhat if we agree to move in the refer ence frame of the unperturbed manifold. Then the perturbed manifolds 79

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will be seen to orbit around us without twisting. Compare the right hand side of Figure 5.2.4 with the sketch in Figure 5.2.5. Figure 5.2.4. The perturbed and unperturbed manifolds of a flow are shown on the left, and their Poincare section on the right. l s '\ s Figure 5.2.5. In this figure we are moving in the fmrne of the un perturbed manifolds. The perturbed manifolds will be seen to orbit us. The actual orbit will be more elliptical than circular in the case of the pendulum. 80

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Capture and release control works by perturbing the system in such a way as to place the current system state on the stable manifold of the perturbed system, and waiting until the perturbed and unperturbed manifolds are nearly co linear (we are seldom blessed with linear manifolds, so they can only be approximately co linear). At this point the perturbation is turned off, and the system evolves along the stable man ifold of the unperturbed system. No knowledge of the eigenvalues of the system is needed to implement CR control. We need only discover how long it takes for the manifolds to become colinear, but we do need a continuously adjustable control parameter. 0 no control ', ,. ' J -. .-' a+ -:---------,-------' ' .. control on control off ' I : ---Figure 5.2.6. This jigure illustrates the method of capture and release. The system is perturbed so that the curTent state is on the stable mani fold of the perturbed periodic point. The perturbation remains in effect until the perturbed and unperturbed stable manifolds are most nearly colinear. At this point we point out that the effective use of OGY control requires that the movement of the stable manifold be transverse to the direction of the unstable manifold. It is easily seen that one could 81

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choose a Poincare section at a phase where the perturbed and unper turbed stable manifolds are colinear. Should this be the case, control by OGY is impossible, as OGY requires that the perturbed and un perturbed manifolds separate so that the perturbed dynamics in the unstable direction force the system state onto the unperturbed stable manifold by cycle's end. It may be that in experiments where it was found necessary to delay the activation of control until sometime be tween SOS maps that this situation was encountered. Experimentalists sometimes have no choice as to when SOS maps are taken due to the exigencies of the experimental apparatus or type of system under study. no control p-0 .. 1. i/ . -----,-on -0 I ----'!'---I i .. I. : : :-, . . -. : .. p-180 .. ----f'_, ___ ... -.-._-_: __ ....... p-270 p-360 control off p-360 Figure 5.2. 7. Successful OGY contr-ol. The change in manifold position with perturbation is such that ther-e is a change in the unstable dir-ection. The per-turbed unstable dynamics act on the system state in such a way as to place the system state on the stable manifold of the per-iodic or-bit in one system cycle. 82

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s s s s s s s s Unsuccessful OGY control-no movement along eu Figure 5.2.8. Unwccessful OGY contr-ol. 'The change in manifold position with perturbation is such that there is no change 'in the unstable direction. The perturbed unstable dynamics act on the system state in such a way as to move the system state away from the stable manifold of the periodic orbit. From the schematic of OGY control, we can see that a little larger perturbation could drive the system state onto the stable manifold of the unperturbed system in less than one fundamental period. Figure 5.2.9 illustrates this idea. Control by time proportioned perturbation and control by capture and release will be analyzed more thoroughly and implemented in the next two sections. 83

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. ' ' ,---' ---------' ' ' no control control on "'..4 45 4>=0' 'f>=O' cont oloff tf>=? =? Figure 5.2.9. This schematic of contr-ol by time pr-opor-tioned per-tur bations shows the system being contr-olled before the next SOS map is taken. 5.3. Control by capture and release Control by capture and release (CR) works only for flip orbits. CR con trol requires knowledge of the dynamics of the system between Poincare maps, specifically, the change in fixed point with control perturbation, and the point in time at which the stable manifolds of the perturbed and unperturbed system coincide. Figure 5.3.1 is a plot of the sta ble and unstable rnanif(llds of a period-one over-the-top orbit of the 84

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pendulum at 16 Poincare phases and at three different damping levels. PiguTe 5.3.1. The figure above shows the perturbed and unperturbed manifolds of a period one orbit of the vertically driven pendulum. The phase space is periodic at the left and Tight sides, and only the upper half plane is shown. The orbit moves from left to right. In each group of three the middle manifolds are the unperturbed ones, with a damping level p0 = 0.23 in units of actual damping to small angle approximation critical damping. The left members of the trios are at an increased damping level p = 0. 25, and the right members have a damping of p = 0.21. In the 6th full group of three from the left, the unstable manifolds are nearest coincidence, and in the llth group of three the stable manifolds are nearest coincidence. In capture and release control we capture the system state at Poincare section 6, where the perturbation gives the greatest change in the unstable direction, and release the system at Poincare section 11, where the stable manifolds are almost colinear. The system then progresses to the center manifold 85

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along the stable direction. e, Figur-e 5.3.2. The per-turbed and unpertur-bed left and right stable and unstable eigenvectors of the period one orbit are shown. Let = g be the change in fixed point x0 with change in parameter p, and op be a small parameter perturbation. Our goal is to find a op that will perturb the fixed point Xo at timet, by Opg so that the system state Ll.x lies in the stable manifold of :c0 + opg. The system state Ll.x is now captured by the stable dynamics of the perturbed orbit. We leave the control on for a time tr until x lies in e, the stable manifold of the unperturbed system. The control is turned off and the system state is released into the custody of the unperturbed stable manif(Jld e8 The system will now evolve along e8 to the center manifold. Suppose we have a continuous time chaotic system that we can sample at several different Poincare phases ;, i = 1, 2, ... n over the course of the fundamental system period. Using standard techniques, we establish the location of the fixed point Xo in the Poincare map the local linearized dynamics Ll.xnn = Axn near the fixed point, the stable and unstable right and left normalized eigenvectors e8 eu, f, fu of A, and the change in fixed point with respect to parameter change g = x0 gP We also compute the time tc at which the unstable manifolds of g and x0 coincide ( e, = 0) and the time tr at which the stable 86

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manifolds of g and x0 coincide (eu = 0). We choose the Poincare phase at which we take data as close as we can to t, to maximize the effect of the parameter perturbation. The control-on time is therefore fixed at tr, the time it takes lor the manifolds to rotate from unstable coincident to stable coincident. Let bp be some small change in p. As the system parameter p varies, the attractor and the periodic orbits move in state space. We will assume that for small parameter change bp the fixed points move but the local dynamics do not change. The location of the new fixed point with change bp is bp gpx0 = 8pg. We will choose bp so that the perturbed stable manifold contains the current system state 6:I;n = Xn Xo. This condition is satisfied when fu (opg-C:.xn) = 0 or ;: fu C:.xn up= . fu 'g If we apply this correction Op the phase point will be on the perturbed stable manifold until the correction is turned off at tr. The system will then evolve along the unperturbed stable manifold into the fixed point. Near a saddle, the rate of departure of an initial condition t.x along the unstable direction eu is a function of the distance from the stable manifold. Thus, perturbing a system so that the new stable manifold contains the current system state C:.x results in a stable system so long 87

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as the perturbation remains. I i jj. [1!1 i 0 F'igure 5.2.3. This picture is a time series of the pend1tlum being con trolled by the method of capture and r-elease. The contr-ol is alternated between a period one clockwise and a period one counterclockwise rota tion. Figure 5.3.3 shows a time series of a session of control by capture and release. The control was alternately applied to period one clockwise and period one counterclockwise rotation. The B scale runs from -1r to 1r. The control parameter is velocity dependent damping, with a control off level of .23 normalized units. The control perturbation is shown in gray, and its scale runs from -.21 to .25. Circles indicate when the control was turned on and off. 5.4. Control by time proportioned perturbations The method of Ott, Grebog1 and Yorke (OGY) may be used to control a lowdimensional continuous time chaotic system when the flow may be sampled at some fundamental period to obtain a surface of section 88

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(SOS) map. Then, small parameter perturbations may suffice to stabilize the system around one of its unstable periodic orbits (UPOs). Using OGY control, we would apply a perturbation, whose size is dependent on the value of the system state and the left eigenvector of the stable manifold, for one iterate of the map. The system state will lie on the stable manifold after one fundamental period. Frequently the only parameter available in a physical system is capable of only a few discrete states, and sometimes only two; either on or off. Control using this type of pa.rameter is called bang-bang control. An air conditioner or a thermostat controlled gas heater are two types of systems that use bang-bang control. When the temperature in your house gets too low, the mercury switch in the thermostat is tripped and the furnace goes full on until the temperature rises above a preset level, at which time the furnace is turned off. If we can vary the parameter only by a fixed amount, then we must resort to schemes other than standard OGY. If we can reconstruct some of the dynamics between surface of section maps we can use time proportioned perturbation (TPP), where a fixed perturbation is applied for less than one fundamental period. We show how to implement TPP control when we have access to several SOS maps within one fundamental period. In fact, TPP needs access to several SOS maps only for the learning stage. Then, once the dynamics of the system have been reconstructed linearly in the region near the behavior of interest, we monitor only one surface of section map. The goal of time proportioned perturbation is to apply a fixed perturbation p in order to direct the system state onto the stable manifold in less than one fundamental period of the system. Usually, the stable and unstable eigenvectors e, and eu rotate about the UPO during the course of a system cycle, and the angle between them changes periodically. Furthermore, the distance between the perturbed and unperturbed orbits is periodic, and we must take these factors into account 89

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when building our control rule. Figure 5.4.1. Shown are a perturbed and an unperturbed orbit and the important features we need to build a rule for contml by time propor tioned perturbations. Suppose we have access to n surface of section maps equally spaced in time during one fundamental period T of a continuous time system for both perturbed and unperturbed states. Then we can determine the perturbed and unperturbed UPOs Xo(k6t) (k = 0, ... n-1), 6t = and Xp(k6t), respectively, the left unstable eigenvector f0(k6t) of the unperturbed orbit, and the transition matrix Bk taking a system state X(k6t) near the unperturbed periodic orbit Xo(k6t) at SOS k to its image X((k + 1)6t) at SOS k + 1. From the set of transition matrices Bk we form the set of matrices Ak where An = Eo, A1 = B1Bo, A2 = B2B1B0 etc. In other words, Ak takes an initial state X(O) and evolves it to the system state at Poincare section k. We now fit smooth periodic functions fu(t), Xo(t), Xp(t) and A(t) to the data fu(k6t), X0(k6t), Xp(k6t) and Ak Note that A(t) is not a periodic matri.'C in the sense that A(O)x(O) = x(27r) (it doesn't), but the entries of the matrix are periodic. Suppose we monitor SOS 0 and measure an initial system state X (0) near the UPO X0(0). We want X(t) to be on the stable manifold es(t) 90

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at time t. The system evolves as X(t)X0(t) = A(t)[X(O)Xo(O)j. When we turn on the perturbation p the system evolves as X(t)Xp(t) = A(t)[X(O) Xv(O)]. where we have assumed that A( t) does not differ significantly from Av(t), the dynamics of the perturbed system. Then X(t) = A(t)[X(O)Xp(O)] + Xv(t). We want, at some future timet f,(t) [X(t)X0(t)] = f,(t) (A(t)[X (0)-Xv(O)] + Xp(t)-X0(t)) = 0. Rewriting A(t)[X(O)Xv(O)] + Xv(t) X0(t) as A(t)[X(O)Xo(O) -(Xv(O)-Xo(O))]+ Xv(t)-Xo(t)
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is chaotic for the fixed parameter values a = 1. 2, w = 1.5 and for p E [0, 0.28]; we choose c0 = 0.23 as our unperturbed parameter value. We choose to operate the system with p E [0.21, 0.25], and as we are implementing control by time proportioned perturbations, we fix two possible perturbations, p1 = 0.21 and p2 = 0.25. Data sets were generated by a 4th order Runge-Kutta integrator. Data were extracted from 48 data sets consisting of 4096 values for .T andy for each of 16 equally spaced Poincare phases, and at three values of damping, p1 = 0.21, Po= 0.23, and p2 = 0.25. A PB computer program extracted the fixed points, the local linear map, its eigenvalues, and its stable and unstable left and right eigenvectors. The local transition matrices, that is, the 16 matrices that evolve a system state X k from the kth Poincare section to the k + 1 Poincare section were calculated and continuous periodic functions fitted to the sequence of entries in the i,jth position of the local transition matrices fori, j = 1, 2. These four functions a1 1(t), ar,2(t), a2,1(t) a2,2(t) form the entries in A(t) = [ ar,r (t) a1,2(t) ] a2,r(t) a2,2(t) (5.4) The functions a1 1(t), a1,2(t), a 2 ,1(t) a2,2(t) appear in the table below. In this particular case, the Poincare control plane was sampled at phase 7 ; since at this value the change in parameter moved the fixed point along the unstable manifold. Choosing this phase for control then gave the greatest controllability. Since at some drive phase cp the periodic orbit crosses the periodic boundary, the function A(t) will be discontinuous at t = cp. We avoid the problem by choosing to control in less than wt = K, hence, we actually fit A(t) to the values taken from Poincare sections 7 to 15 and 0 to 1 for a total of 11 successive Poincare sections. llJ,J = 0.291063 + 0.79il685 cos[t] -0.0905197 cos[2t] + 0.0296494 cos[3t] -0.0559318 sin[t] + 0.00286768 sin[2t] + 0.050846 sin[3t] a1,2 = 0.408048-0.220476 cos[t] 92

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-0.0905197 cos[2t] + 0.0296494 cos[:3t] -0.0559318 sin[t] + 0.00286768 sin[2t] + 0.050346 sin[3t] a2,1 = -1.29693 + 0.430001 cos[t] +0.776662 cos[2t] + 0.0230557 cos[3t] +0.786877 sin[t] 0.240939 sin[2t] 0.22432 sin[3t] a2,2 = 0.309087 + 0.400585 cos[t] +0.136693 cos[2t] + 0.0268144 cos[3t] -0.13708 sin[t] 0.0935679 sin[2t] -0.00711615 sin[3t] Shown in Figures 2a through 2d are plots of the data sets (connected piecewise linear plot) from the four entries of the matrix A and the functions used to approximate the data. !} ?!l ".' O.l:!l O.S 1 l.S ,, -(U:!l -0.5 0.' ,_, 0. ss 0. 45 1 1.5 93

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-0.? .(I.$ -0. 9 O.:l:S O.t 0.15 0.1 0.(15 0.' 1 1., Figures 5.4.2a, 5.4.2b, 5.4.2c, 5.4.2d show the fits to the data fmrn the entries of A plotted with the piecewise linear fit to the data in the entries. Fmrn top to bottom are a1,1, a1,2, a2,1, a.2,2. The unstable periodic orbits for three levels of damping were computed in a similar fashion, by fitting smooth periodic functions to the 7th through the 1st successive state vectors, forming the continuous vector function X0(t). The function X0 ( t) is shown below. x0 = 0.866553 1.97963 cos[t] -0.494012 cos[2t] + 0.0414162 cos[3t] -0.439859 sin[t] + 0.379237 sin[2t] + 0.131035 sin[3t] Yo = 1.5033 + 0.:371607 cos[t] 94

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-O.Gl76539 cos[2t] 0.0344509 cos[3t] + 0.557474 sin[t] + 0.2615 sin[2t] + 0.0335449 sin[3t] x0(t) = 0.707967-2.10011 cos[x] -0.222945 cos[2x] + 0.240454 cos[3x] +0.0253629 sin[.T] + 0.653105 sin[2x] + 0.105806 sin[3x] flo 1.49461 + 0.405109 cos[t] +0.00936983 cos[2t] 0.0279653 cos[3t] +0.553614 sin[t] + 0.275949 sin[2t] + 0.0387564 sin[3t] xt 0.772507-2.00497 cos[t] -0.47772 cos[2t] + 0.0564898 cos[3t] -0.443086 sin[t] + 0.390533 sin[2t] + 0.130429 sin[3t] flci 1.51107 + 0.333787 cos[t] -0.0459989 cos[2t] 0.0436186 cos[3t] +0.56489 sin[t] + 0.248926 sin[2t] + 0.0278085 sin[3t] Figure 5.4.3. This figure shows the fit to the x coordinate of the periodic orbit data for the three different damping levels. 95

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Figure 5.4.4. This figure shows the fit to the y coor-dinate of the periodic orbit data for the three different damping levels. The function f"(t), the left unstable eigenvector is shown below: = 0.351576 + 0.50572 cos[t] +0.0330918 cos[2t] + 0.0873002 cos[3t] 0.640942 sin[t] + 0.186759 sin[2t] 0.051183 sin[3t] f; = -0.288731-0.113669 cos[t] -0.0912112 cos[2t] + 0.122428) cos[3t] -0.819938 sin[t] + 0.193646 sin[2t]-0.158725 sin[3t] Q.H 0.' O.H 4 t5 *Q. $ *O.H Figure 5.4.5. This figure shows the data and the fit to the x coordi nate of the left unstable eigenvector of the unper-turbed orbit. 96

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4 0.95 0.9 0. 35 0.' 0.?5 0.1 F'igure 5.4. 6. This figuTe shows the data a:nd the fit to the y coordi nate of the left unstable eigenvector of the WlpeTturbed orbit. When control was implemented, we found that it was not as tight as OGY or CR control. We speculate that the accurate modeling of the system is crucial, and that the compounded error of the interpolated functions and inaccuracies in the estimates of other system characteristics conspired to confound our control. The control was, however, robust, and control was never lost over the longest run of one hour. Pictured below is a time series plot of a control session, with time on the horizontal axis and velocity on the vertical. The system was al lowed to evolve uncontrolled for a while, then TPP control was turned on. The control was then turned off and the system allowed to evolve 97

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on its own. F"------------Figure 5.4. 7. This figure shows a time series plot of the control of the damped driven pendulum equation by the method of time proportioned perturbations. 98

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6. Appendix A By definition Ae = >.e, !u e"' =Is eu = o, eu = f., es = 1, so A[eu + es] = Aueu + >.,e8 = \,eu(fueu + .f.ues) + >.,es(fses + .f.seu) Factoring, we obtain and finally 99

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7. Appendix B mg cosfJ mg VIe can derive the pendulum equation by equating forces. First, ignoring the driving and damping terms and assuming a point mass on the end of a massless rod, equate the forces along the path of the bob: 011 () mr = -mgsm Assuming velocity dependent damping, we get mrO" = -mg sin(} fO'. All motion of the pendulum will be relative to the position of the pivot. With this in mind, we note that the acceleration imparted to the bob due to the acceleration of the pivot is proportional to sin e. We differente the drive position twice to get drive acceleration, so the force felt by the pivot is -maw2 sin() cos wt. 100

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Now adding the drive term, we have mrO" = -mg sin() fl)' + amw2 sin 0 cos wt. Dividing through by mr, we get ()II 9.{) r(}f a2e = --sm -+ -w sm coswt. 1 mr r Finally we introduce a dimensionless time r such that r = and letting the double dot represent differentiation with respect to r we have .. g r a 2 () = ---sin() --{} + -w sin(! cos wt. rT2 mirT mr Letting T = j""f, A= and p = T;;'" we have, finally, e = -pfJsin(!(l-Acoswt). 101

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References [1] Zelda Gills, Christina Iwata, Rajarshi Roy, Ira Schwartz, Ioana Triandaf, Tracking Unstable Steady States: Extending the Stability Regime of a M'ult:imode Laser System, Physical Review Letters, 69, 22 (1992E). [2] Ott, C. Grebogi and J.A. Yorke, Controlling Chaos, Physical Re view Letters, 64, 1196 (1990). [3] W. L. Ditto, S. N. Rauseo, M. L. Spano, Experimental Control of Chaos, Physical Review Letters, 65, 26 (1990). [4] ,John Taylor, Pat Toohey, Chaos in Topological Spaces, preprint. [5] Rene Lozi, Un Attmcte'IJ,T Etmnger? d11, 'flype AttmcteurHenan, Journal of Physics, 39 (1978). [6] M. Henan, A Two Dimensional Mapping with a Strange Attr-actor, Communications in Mathematical Physics, 50 (1974). [7] Ute Dressler, Gregor Nitsche, Controlling Chaos Using Time Delay Coordinates, Physical Review Letters, 68, 1 (1992). [8] Steven Smale, The Mathematics of Time, Springer-Verlag (1980). [9] Robert Devaney, An Introduction to Chaotic Dynamical Systems, Benjamin/Cummings (1985). [10] John Guckenheimer, Phillip Holmes, Nonlinear Oscillations, Dy namical Systems, and Bifurcation of VectorFields, SpringerVerlag (1983). [11] Robert Ghrist, Branched Two-Manifolds Supporting all Links, Topology, 36, 2 ( 1997) [12] John Starrett, Randall Tagg, Control of a Chaotic Pammetrically Driven Pendulum, Physical Review Letters, 74, 11 (1995). 102