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Geometric control of low-dimensional chaotic systems

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Geometric control of low-dimensional chaotic systems
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Starrett, John
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209 leaves : ; 28 cm

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

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Includes bibliographical references (leaves 205-209).
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Department of Mathematical and Statistical Sciences
Statement of Responsibility:
by John D. Starrett.

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|University of Colorado Denver
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Full Text
GEOMETRIC CONTROL OF LOW-DIMENSIONAL CHAOTIC SYSTEMS
by
John D. Starrett
B.S., Metropolitan State College of Denver, 1993
M.S., University of Colorado at Denver, 1997
A thesis submitted to the
University of Colorado at Denver
in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
Applied Mathematics
2002


This thesis for the Doctor of Philosophy
degree by
John D. Starrett
has been approved
by
Alberto Sadun
Burton Simon

Date


Starrett, John D. (Ph.D., Applied Mathematics)
Geometric control of low-dimensional chaotic systems
Thesis directed by Professor William Briggs
ABSTRACT
Dissipative chaotic systems allow particularly elegant methods of control
whereby any one of an infinite number of unstable periodic orbits already present
in an attractor may be stabilized by small perturbations to some system param-
eter. The standard method of controlling chaos is to wait until the system state
comes near the target saddle orbit, then apply a perturbation such that the state
is nudged into the stable subspace of the target, whereupon the perturbation is
removed and the system is allowed to evolve to the target orbit in the stable
subspace. In this thesis, the problem of low-dimensional chaos control is formu-
lated in a geometric setting using simple diagrammatic tools to illuminate the
process. New methods are developed utilizing the relative rotation of perturbed
and unperturbed manifolds to bring the system under control, as well as new
time-optimal techniques for directing system states to the target orbit rather
than its stable subspace.
This abstract accurately represents the content of the candidates thesis. I
m


recommend its publication.
Signed
William Briggs
IV


DEDICATION
I dedicate this thesis to my family, especially my wife Marilyn and my parents
David and Grace.


CONTENTS
Figures .................................................................. xi
1. Introduction......................................................... 1
1.1 Tools and goals..................................................... 3
2. Horseshoes.......................................................... 5
2.1 Horseshoes in systems described by ODEs.......................... 15
2.1.1 Horseshoes in the pendulum......................................... 15
2.1.1.1 Horseshoes in the non-over-the-top pendulum.................... 15
2.1.1.2 Horseshoes in the over-the-top pendulum ....................... 20
3. Local models from data............................................... 30
3.1 Introduction......................................................... 30
3.2 Building local models.............................................. 31
3.3 Local models for maps and SOS maps .............................. 33
3.3.1 The target orbit ................................................ 33
3.3.2 Local dynamics for a two-dimensional map........................... 36
3.3.3 The perturbation vector.......................................... 39
3.3.4 The perturbed dynamics........................................... 39
3.4 Local models for continuous systems............................... 40
3.4.1 A local model from geometric considerations...................... 45
3.4.2 A local model from data.......................................... 51
4. Control methods for maps............................................. 61
vi


4.1 OGY control ......................................................... 61
4.1.0.1 OGY applied to the Henon map ...................................... 64
4.2 Extensions to OGY control .......................................... 68
4.2.1 Control by occasional bang-bang (OBB)........................... 68
4.2.1.1 Analysis by one-dimensional maps................................... 75
4.2.1.2 OBB with Flip saddles.............................................. 81
4.2.1.3 OBB for flip saddles with one level of perturbation................ 81
4.2.1.4 Criteria for controllability....................................... 82
4.2.1.5 OBB for flip saddles with two equal and opposite perturbations 85
4.2.1.6 Control of a pendulum by OBB....................................... 85
4.2.1.7 Summary............................................................ 86
4.2.2 Optimal map-based control.......................................... 90
4.2.3 Control by center manifold targeting (CMT)...................... 93
4.2.3.1 Chaos control and classical linear control theory.................. 93
4.2.3.2 Optimal control ................................................... 94
4.2.3.3 Time-optimality of OGY ............................................ 95
4.2.3.4 Energy-optimality of OGY .......................................... 95
4.2.3.5 Global targeting................................................... 96
4.2.3.6 Local targeting.................................................... 96
4.2.3.7 Simplifications.................................................... 97
4.2.3.8 Optimal methods.................................................... 98
4.2.3.9 CMT................................................................ 99
4.2.3.10 Control of the Lozi map by CMT.................................... 106
Vll


4.2.3.11 Control of the Henon map by CMT .......................... 109
4.2.3.12 Higher dimensional control by CMT......................... 109
4.2.3.13 Summary of map-based CMT.................................... 113
5. Control methods for continuous systems ......................... 114
5.1 The geometry of continuous control............................ 114
5.1.1 The geometry of continuous saddle orbits ..................... 114
5.1.2 Saddles in continuous systems................................. 118
5.1.3 The geometry of flip orbits .................................. 120
5.1.4 The perturbation vector and the directing vector.............. 127
5.2 Capture methods................................................. 13l
5.2.1 Capture and Release control (CR) ............................. 132
5.2.2 CR is OGY Limit .............................................. 134
5.2.3 CR control of a chaotic pendulum ............................. 136
5.2.4 Summary....................................................... 137
5.2.5 Optimal Capture Methods....................................... 138
5.2.5.1 Subspace targeting.......................................... 139
5.2.5.2 Simple proportional feedback............................... 140
5.2.6 Stable Capture Optimal Targeting (SCOT) ...................... 143
5.2.7 Direct Targeting Stable Capture (DTSC)........................ 145
5.2.8 Unstable Capture Optimal Targeting (UCOT)..................... 149
5.2.9 Direct Targeting Unstable Capture (DTUC)...................... 151
5.2.10 Summary....................................................... 153
5.3 Optimal continuous methods...................................... 156
vm


5.3.1 Variations on CMT for continuous systems....................... 156
5.3.1.1 Optimizing OGY control ..................................... 157
5.3.1.2 Optimal Anti-OGY Control.................................... 158
5.3.2 Continuous Center Manifold Targeting (CCMT).................. 159
5.3.3 Watch and Wait CMT control.................................... 163
5.3.4 Summary of optimal continuous methods......................... 164
5.4 Control by time proportioned perturbations...................... 166
5.4.0.1 TPP by simple proportional feedback ......................... 175
5.5 Simple TPP....................................................... 178
5.6 Pull TPP ........................................................ 178
5.7 Limitations of TPP............................................... 179
5.7.1 TPP controllable region........................................ 181
5.7.2 Fits for TPP models........................................... 181
5.7.2.1 Fits for the Left eigenvector fu evolution.................. 182
5.7.2.2 State evolution matrix....................................... 184
6. Summary ........................................................... 186
6.1 The big picture ................................................. 192
6.2 Routes to the stable manifold.................................... 192
6.2.1 One surface of section map, fixed perturbation, fixed on-time . 192
6.2.2 One surface of section, fixed perturbation, variable on-time .... 193
6.2.3 Capturing a system state ...................................... 194
6.3 The directing vector and time-optimal control................... 194
6.3.1 Time-optimal control for a simple saddle...................... 195
IX


6.3.2 Time-optimal control for a flip saddle........................ 195
6.4 Low dimensional optimal control ................................. 197
6.5 Definitions...................................................... 199
6.6 Notation......................................................... 201
6.7 Abbreviations.................................................... 203
References........................................................... 205
x


FIGURES
Figure
2.1 The Smale stadium on the left consists of a rectangular central re-
gion fitted with semi-circular endcaps. On the right, the stadium is
stretched, bent, and placed back inside itself.................... 7
2.2 The intersection of S and F(S) gives H0 and Hi, and the preimages
of these horizontal strips are V0 and Vi............................ 8
2.3 The second iterate of the Smale horseshoe map is shown on the left.
On the right, the intersection of the horizontal and vertical strips
after the first two iterations of the horseshoe map. In the limit, the
intersection is a Cantor set............................................ 9
2.4 The Cantor dust of the Smale horseshoe. This is a fractal set, so each
dot in the picture represents four dots which are each composed of
four dots, etc.................................................... 9
2.5 The stable and unstable sets of the point s.......................... 12
2.6 The fate of a line of points in the stable set of p under four iterates
of the map............................................................. 12
2.7 Sixteen successive Poincare sections of the periodically forced pen-
dulum in a parameter range for which it dies not go over the top. . 16
2.8 A rough guess as to the form of the pendulum horseshoe............... 17
xi


2.9 The evolution of a quadrilateral of initial conditions with corners
(-.3,1.4), (2,-.64,), (.3,-1.4,), (-2,.64) under the dynamics of the pen-
dulum equation from Poincare phase cut = 7r through 87r. The ini-
tial quadrilateral remains for comparison. The top and bottom
edges are in gray, and the left and right edges are in black. . 18
2.10 The horseshoe of the 4 times iterated Poincare map of the non-over-
the- top pendulum.................................................... 19
2.11 The evolution of a portion of the stable manifold of the periodic orbit
through the origin over a single forcing period.................... 21
2.12 An annular phase space............................................... 22
2.13 16 Poincare sections of the over-the-top pendulum in the standard
phase space.......................................................... 23
2.14 16 Poincare sections of the over-the-top pendulum in the annular
phase space........................................................ 24
2.15 The formation of the over-the-top horseshoe from times ir to 67r. . 25
2.16 The horseshoe map for the Poincare section of the over-the-top
pendulum after two forcing periods. Because the pendulum goes over
the top, the periodic boundary comes into play and we must connect
the broken numbered ends to their mates, either in a cylindrical
phase space, or an annular one....................................... 26
xii


2.17 The horseshoe map for the Poincare section of the over-the-top
pendulum after three forcing periods. Because the pendulum goes
over the top, the periodic boundary comes into play and we must
connect the broken numbered ends to their mates, either in a cylin-
drical phase space, or an annular one........................... 27
3.1 The dynamics of the logistic map with parameter r = 4 and with
an initial condition of 2/3. The web diagram is in bold, and the
progression can be followed from 2/3 to 8/9 to 32/81............ 35
3.2 The flow lines near a 2-dimensional saddle. For a map, a 2-
dimensional saddle can be either simple, with eigenvalues 0 < As <
1 < Au, flip-in with eigenvalues 1 < Xs < 0,1 < Xu, flip-
out eigenvalues Xu < 1,0 < Xs < 1, or flip with eigenvalues
Xu < 1 < Xs < 0. For a continuous 2-dimensional system, saddles
come in only simple and flip Varieties, because a flip necessitating a
Mobius band manifold for one manifold forces it for the other. ... 42
3.3 The evolution of a period-one flip orbit and its local linear stable and
unstable manifolds. Notice the scissoring of the manifolds, where the
angle between them changes continuously........................... 44
3.4 Two leaves of the foliation of the crotch of the stable and unstable
ribbons by hyperbolic sheets...................................... 44
xm


3.5 The geometry of chaos control. In each frame, the dynamics that
active are in bold while the inactive dynamics are dashed lines. In
the first frame, a system state has entered the control box, and its
path is shown under the unperturbed dynamics. If left alone the
system state would exit the controllable region along the unstable
manifold. In the second frame, the system has been perturbed, and
the path of the evolution of the state under these dynamics is in
bold, while those of the unperturbed system are dashed. In the third
frame, the system state has evolved under the perturbed dynamics
(bold) to the stable manifold of the unperturbed system (dashed).
In the fourth panel, the control parameter has been returned to its
nominal value, and the system state will now evolve along the stable
manifold of the target orbit to the target orbit itself.............. 46
3.6 Two orbits near a flip saddle; the true orbit from data, whose data
points are gray, and one from the simplified model which does not
include a correction for varying eigenvalues, whose data points are
black................................................................ 50
3.7 Another set of two orbits near a flip saddle; the true orbit from data,
whose data points are gray, and one from the simplified model which
does not include a correction for varying eigenvalues, whose data
points are black
52


3.8 The local skew coordinate system whose basis vectors are the eigen-
vectors of the local linear transformation near a periodic saddle orbit
x.................................................................. 55
3.9 The geometrical arena in which an orbit near a target orbit and
its perturbation evolves. Depending on whether the unperturbed or
perturbed system is active, the system state will evolve under these
dynamics.............................................................. 58
3.10 The local eigenvector frame of a period-one orbit in which the evolu-
tion of a local system state takes place. Notice the periodic scissoring
of the frame that results from the stretching and folding of the at-
tractor.............................................................. 59
3.11 A snapshot of a periodic orbit and its local manifolds, flanked by two
perturbed orbits and their manifolds. The roughly circular traces
intersecting the center manifolds are the paths followed by the per-
turbed orbits over a drive cycle of the pendulum relative to the un-
perturbed orbit. Units are in radians................................ 60
xv


4.1 In each panel the active dynamics are in bold while the inactive are
dashed. In the first panel we see the unperturbed manifolds (marked
with a U) in bold and the system state as well as the flow along
which it would normally evolve. In the second panel the perturbed
dynamics (marked with a P) are in effect, and the new flow along
which the system state will evolve is shown. In the third panel the
system state has evolved under the perturbed dynamics to the stable
manifold of the unperturbed system. In the fourth panel, the system
is once again governed by the unperturbed dynamics, and will evolve
along the indicated flow........................................ 65
4.2 Several thousand iterates of an initial condition inside the basin of
attraction for the Henon map. The portion of the attractor inside the
small box in the first quadrant has been magnified and superimposed
on the third quadrant to show the fractal structure of the attractor. 66
4.3 The result of OGY control applied to a period-one orbit of the Henon
map. Control is turned alternately on and off, allowing the system
to return to its chaotic state before control is reapplied. The upper
portion of the graph shows the x values, while the lower portion
shows the value of 5p while the system is being controlled...... 67
xvi


4.4 The perturbed manifold pair and their in and out boxes. The boxes
labelled inbox 1 and outbox 1 contain the controlled orbit, and are
bounded by the eigenvectors of the map and its perturbation, while
those labelled inbox 2 and outbox 2 are bounded by the reflection of
the same eigenvectors about e..................................... 72
4.5 On the left, the simple case where the eigenvalues are Xu = 7/4 and
As = 1/2 and pig = (1,1), and on the right a simple case where the
eigenvalues are \u = 7/4 and Xs = 1/2 and pig = (.15,1)......... 73
4.6 A slightly more complicated case where the eigenvalues are Xu = 1.62
and Xs = .58, the eigenvectors are e = (0.0975168, 0.995234) and
es = (-0.97948, -0.201544) and pxg = (1,1)......................... 74
4.7 A sequence of four generalized Bernoulli maps with slopes ranging
between 1 and 2. These correspond to the range of OBB controllable
systems whose unstable directions have eigenvalues between 1 and 2. 76
4.8 Bifurcation of the Bernoulli map. The parameter A is the slope of
the map............................................................ 77
4.9 The left hand map corresponds to the map that is obeyed for the
stable dynamics while the system state is in the outbox, while the
right hand map governs the stable portion of the dynamics while the
state is in the inbox..................................... 77
4.10 Three controlled orbits from three different perturbation values. . 79
xvii


4.11 An uneven division of the control box into inbox and outbox on
the left. The form of the maps for the stable direction remains un-
changed, right.................................................. 80
4.12 Two different uneven divisions of the control box into inbox and
outbox. The division with the smallest outbox keeps the controlled
orbit closest to the target orbit............................... 80
4.13 An example of the kind of attractor that results when the unstable
direction has positive eigenvalues and the stable direction has neg-
ative eigenvalues (a flip-in saddle). In this case the eigenvalues are
\u = 1.4 and \s = .75 ............................................ 82
4.14 OBB for flip saddles reduced to control on a line.................. 83
4.15 The one-dimensional control setup associated with OBB for flip sad-
dles with two equal and opposite perturbations........................ 85
4.16 On the left, the Bernoulli map associated with OBB control for flip
orbits with two equal and opposite control perturbations. On the
right, the same map with a periodic controlled orbit............ 86
4.17 The reconstructed attractor for the over-the-top mode of the para-
metrically (sinusoidal vertical) forced pendulum. The control box is
in the upper left of the fourth quadrant........................... 87
4.18 A time series of the controlled orbit of the pendulum, with the de-
layed coordinate 0{t r) on the vertical axis. The three levels of
control perturbation are shown magnified by a factor of five in the
bottom box, with the center being the nominal control value. ... 88
xviii


4.19 A close-up of the Poincare section of the controlled orbit of the pen-
dulum. The bounding box is the control box of Figure 4.17.............. 89
4.20 The phase portrait of the controlled orbit of the pendulum in natural
coordinates (9,9)................................................ 89
4.21 The perturbed (solid) and unperturbed (dashed) manifolds during
control. On the left, a system state on the stable manifold is brought
directly to the center manifold under the perturbed dynamics. This
is the special case where the perturbation moves the target orbit
along the stable manifold. On the right, a system state on the un-
stable manifold is brought directly to the center manifold under the
perturbed dynamics. This is the special case where the perturbation
moves the target orbit along the unstable manifold............... 101
4.22 Two initial conditions xi and x2 map to the target orbit under two
different perturbations 6pi and 5p2. The set of all inverse images
of the target orbit under all allowable perturbations form a linear
segment through the target orbit................................. 102
4.23 The figure shows the perturbation vector g and its inverse images
which span the first and second preimages of the target orbit. . . 104
xix


4.24 Alternating control of the Lozi map by OGY and CMT, with the x
coordinate on the vertical axis and time (in iterates) on the horizontal
axis. Note the difference in the time to control. The lower graph
shows a 35 times magnification of the controlled orbit. The first two
iterates of each controlled orbit are not shown. The scale on the x
axis just contains the attractor, and the units are non-dimensional. 108
4.25 The average accumulated control perturbation over 10 iterates for
OGY and CMT control. The Henon attractor for these parameters
just fits in a box 3.5 by 3.5 non-dimensional units, so the control box
width ranges from approximately 1/350 to 1/35 of the total attractor
width............................................................ 110
4.26 The average accumulated deviation from the target orbit over 10
iterates for OGY and CMT control. The Henon attractor for these
parameters just fits in a box 3.5 by 3.5 non-dimensional units, so the
control box width ranges from approximately 1/350 to 1/35 of the
total attractor width............................................ Ill
5.1 The dynamics near a saddle. The stable and unstable manifolds are
locally linear, and are modelled by the stable and unstable eigen-
vectors es,eu of a surface of section map SOS^t) transverse to the
center manifold (the target orbit x, which lies in the intersection
of es and eu). The hyperbolic sheets, shown in light gray, are the
surfaces along which the orbits evolve.............................. 117
xx


5.2 The geometry of chaos control at a simple saddle. In each frame, the
active dynamics are in bold while the inactive dynamics are dashed
lines. In the first frame, a system state has entered the control
box, and its path is shown under the unperturbed dynamics TJ. If
left alone the system state would exit the controllable region along
the unstable manifold. In the second frame, the system has been
perturbed, and the path of the evolution of the state under these
dynamics P is in bold, while those of the unperturbed system are
dashed. In the third frame, the system state has evolved under the
perturbed dynamics (bold) to the stable manifold of the unperturbed
system (dashed). In the fourth panel, the control parameter has been
returned to its nominal value, and the system state will now evolve
along the stable manifold of the target orbit to the target orbit itself. 118
5.3 Four different types of two-dimensional saddles. Only the simple
saddle at the upper left and the flip saddle at the lower right can
exist in continuous systems...................................... 119
5.4 The linear approximation of a flip saddle orbit. The manifolds, rep-
resented by the eigenvectors at surface of section maps transverse to
the center manifold, scissor with each other as we travel around the
meridian......................................................... 120
xxi


5.5 A snapshot of a periodic orbit and its local manifolds, flanked by two
perturbed orbits and their manifolds. The roughly circular traces
intersecting the center manifolds are the paths followed by the per-
turbed orbits over a drive cycle of the pendulum relative to the un-
perturbed orbit. Units are in radians............................. 121
5.6 Local stable and unstable manifolds of a period-one over-the-top or-
bit of the pendulum at 16 Poincare sections for 3 levels of damping.
Each group of three consists of an unperturbed orbit (center) and
two perturbations, corresponding to greater or lesser damping. . . 123
5.7 A pair of perturbed Mobius bands. The point at which the bands
are most nearly aligned is labelled tsp, which assumes these represent
the stable ribbons................................................ 124
5.8 The linear approximation of a flip saddle orbit and its perturbation
by two pairs of intersecting Mobius bands offset by a variable pertur-
bation vector g connecting the meridians. In this figure, the angle
between the manifolds changes with distance around the meridian.
Also, the perturbation vector g changes periodically in this picture,
but only in the plane of the orbit. Generally, the perturbation vector
deforms the orbit in a more complicated fashion................... 125
5.9 The linear approximation of a flip saddle orbit and its perturbation
by two pairs of intersecting Mobius bands offset by a constant vector
connecting the meridians. Here, the manifolds do not scissor, and it
is this model that we use to visualize control strategies......... 126
XXII


5.10 In the frame of reference of the unperturbed system, the perturbed
Mobius bands orbit about the unperturbed one, retaining the same
orientation. The sequence reads from left to right, top to bottom,
and the labels 07T, 27t/3, 4tt/3, and 2ir indicate the distance around
the meridian of the embedding torus.............................. 127
5.11 Local manifolds rotating about one another are equivalent to man-
ifolds that rotate about their centerlines but whose centerlines are
fixed relative to each other. The labels beneath the figures refer
to the angle around the torus that contains the attractor. This is
the representation we will use, where the manifolds associated with
the active dynamics are solid and those of the inactive dynamics are
dashed. Rotation is counterclockwise, and the manifolds are linear
and aligned with the coordinate axes............................. 128
5.12 The d-ribbon and the g-ribbon.................................... 130
5.13 The dynamics of a flip saddle orbit at a surface of section SOS^Ty 131
5.14 Control by capture and release. The system state enters the con-
trollable region, and a control perturbation is made that places the
stable manifold on the system state. The system state then evolves
in the perturbed stable manifold while the perturbed orbit structure
orbits around the unperturbed orbit. At time tsp the stable manifolds
are collinear, and the control is turned off. Thereafter, the system
evolves toward the target orbit in the unperturbed stable manifold.
xxm
135


5.15 Control of a period-one over-the-top orbit of the parametrically
forced pendulum by CR and OGY. Small modelling errors have
roughly the same effect on the nature of the controlled orbit for
both OGY and CR................................................... 137
5.16 Phase portrait of control of a period-one over-the-top orbit of the
parametrically forced pendulum by capture and release. The uncon-
trolled orbit is in gray, and the controlled orbit is superimposed in
black............................................................. 138
5.17 The manifolds are aligned with the coordinate system by changing
first the phase at which the SOS map is taken, then the time delay.
Prom the top middle to the bottom left of the plot, we see the stable
manifold come into alignment with the horizontal axis as the SOS
phase is changed, and from the bottom left to the bottom right, we
see the unstable manifold come into alignment with the vertical axis
of the system as the time delay is changed........................ 143
5.18 The first part of the SCOT control method. In the first panel a
system state has entered the control box. In the second panel the
system state is captured in the stable manifold of the perturbed
target orbit, and in the third panel the perturbed target orbit has
evolved to the stable point. Notice that the system state captured
in the stable manifold has evolved towards the perturbed target orbit. 146
xxiv


5.19 The second part of the SCOT control method. In the first panel a
system state has been placed in the stable manifold of the unper-
turbed target orbit, and a new perturbation has been applied. In
the second panel the system state has evolved to the unperturbed
target orbit, and in the third panel the perturbation has been turned
off................................................................. 146
5.20 The inverse image of the target orbit under a CR perturbation equal
to go is sufficient to determine the directing vector d............. 148
5.21 The first part of the DTSC control method. In the first panel a
system state has entered the control box. In the second panel a
perturbation is applied that is calculated to bring the system state
into the d-vector in one iterate of the map, and in the third panel
the system state has evolved into the d-vector.................... 149
5.22 The second part of the DTSC control method. In the first panel
a new perturbation has been applied to place the perturbed stable
manifold on the system state. In the second panel the system state
has evolved to the unperturbed target orbit, and in the third panel
the perturbation has been turned off.............................. 149
xxv


5.23 The first part of the UCOT control method. In the first panel a
system state has entered the control box. In the second panel the
system state is captured in the unstable manifold of the perturbed
target orbit, and in the third panel the perturbed target orbit has
evolved to the unstable point. Notice that the system state captured
in the unstable manifold has evolved away from the perturbed target
orbit.............................................................. 151
5.24 The second part of the UCOT control method. In the first panel a
system state has been placed in the unstable manifold of the unper-
turbed target orbit, and a new perturbation has been applied. In the
second panel the system state has evolved to the unperturbed target
orbit, and in the third panel the perturbation has been turned off. . 152
5.25 The inverse image of the target orbit under a UCR perturbation
equal to go is sufficient to determine the directing vector d......153
5.26 The first part of the DTUC control method. In the first panel a
system state has entered the control box. In the second panel a
perturbation is applied that is calculated to bring the system state
into the d-vector in one iterate of the map, and in the third panel
the system state has evolved into the d-vector..................... 154
xxvi


5.27 The second part of the DTUC control method. In the first panel a
new perturbation has been applied to place the perturbed unstable
manifold on the system state. In the second panel the system state
has evolved to the unperturbed target orbit, and in the third panel
the perturbation has been turned off........................... 154
5.28 The optimal OGY method. In the first panel, a system state has
entered the control box. In the second panel a perturbation has been
applied that will bring the system state to the stable unperturbed
manifold in one iterate of the map, as seen in the third panel, at
which time the control has been turned off. In the fourth panel the
system has evolved to the stable point, and in the fifth panel, at the
stable point, a new perturbation has been applied that will bring the
system to the target orbit in one iterate of the map at the stable
point, SOSsp................................................... 158
5.29 The optimal anti-OGY method. In the first panel, a system state has
entered the control box. In the second panel a perturbation has been
applied that will bring the system state to the unstable unperturbed
manifold in one iterate of the map, as seen in the third panel, at
which time the control has been turned off. In the fourth panel the
system has evolved to the unstable point, and in the fifth panel, at
the unstable point, a new perturbation has been applied that will
bring the system to the target orbit in one iterate of the map at the
unstable point, SOSup.......................................... 160
XXVll


5.30 On the left, an orthogonal eigenvector frame with a system state at
aes + beu, and on the right a skew eigenvector frame with a system
state at aes + beu................................................ 162
5.31 In each panel the active dynamics are in bold while the inactive are
dashed. In the first panel we see the unperturbed manifolds (marked
with a U) in bold and the system state as well as the flow along
which it would normally evolve. In the second panel the perturbed
dynamics (marked with a P) are in effect, and the new flow along
which the system state will evolve is shown. In the third panel the
system state has evolved under the perturbed dynamics to the stable
manifold of the unperturbed system. In the fourth panel, the system
is once again governed by the unperturbed dynamics, and will evolve
along the indicated flow......................................... 168
5.32 A wireframe torus showing 16 equally spaced Poincare sections of
the pendulum attractor in the over-the-top mode.................. 170
5.33 A wireframe torus showing 16 equally spaced Poincare sections of
a period one orbit of the pendulum in the over-the-top mode. The
point at which the orbit appears discontinuous is the point at which
the pendulum has gone over the top. The phase space has a peri-
odic boundary at that point, and the surface of section is actually a
cylinder or a torus.................................................... 171
XXVlll


5.34 The local eigenvector frame of a period-one orbit in which the evolu-
tion of a local system state takes place. Notice the periodic scissoring
of the frame that results from the stretching and folding of the at-
tractor............................................................... 172
5.35 A snapshot of a periodic orbit and its local manifolds, flanked by two
perturbed orbits and their manifolds. The roughly circular traces
intersecting the center manifolds are the paths followed by the per-
turbed orbits over a drive cycle of the pendulum relative to the un-
perturbed orbit. Units are in radians................................. 173
5.36 The lowest curve is the path of the unperturbed target orbit, and
the uppermost curve is the path of the perturbed target orbit. The
curve beginning at x(0) and moving upward is the path that the
initial system state would take if not for the control perturbation,
while the lower path is that taken under the perturbation........... 175
5.37 The gray region indicates the controllable region near the fixed point
of the SOS map........................................................ 176
5.38 The parametrically forced pendulum controlled into a period-one or-
bit by TPPSPF. Shown are the angular position, angular velocity,
and on-time. A circle appears on the on-time trace when the control
is first applied. The on-time trace is initially zero............... 177
xxix


5.39 The parametrically driven pendulum controlled into a period-one
orbit by simple TPP, where the system state is monitored continu-
ously and the control is turned off when the state is near enough to
the model of the stable manifold. Shown are the angular position,
angular velocity, and on-time...................................... 179
5.40 The parametrically driven pendulum controlled into a period-one
orbit by full TPP, where the on-time is calculated from the full model
of the system. The system state is only monitored at a single SOS.
Shown are the angular position, angular velocity, and on-time. A
circle appears on the on-time trace when the control is first applied. 180
5.41 The graph of the model of the x coordinate of the left eigenvector f
from 0 to 7T....................................................... 183
5.42 The graph of the model of the y coordinate of the left eigenvector fu
from 0 to 7T....................................................... 183
5.43 The graph of the model of the periodic portion of the state evolution
matrix from 0 to 7r.............................................. 185
xxx


1. Introduction
There is no reason why one could not use traditional linear control theory
to stabilize a chaotic system. The methods are well tested and very general,
and there is a huge literature available. Perhaps the reason that the method
of Ott, Grebogi and Yorke caught on so quickly in the nonlinear dynamics
community is that classical linear control theory was too general. Obviously a
system could be stabilized around some operating point by the application of
some perturbation whose magnitude was controlled by a feedback loop. What
caught the imagination of the chaos community was the sheer elegance of the
OGY method. Dynamicists were used to thinking about these wonderful objects
called strange attractors infinitely folded manifolds, organized by the ghosts
of departed orbits. The bizarre dynamics of a chaotic orbit as it flits from one
saddle to another in a dance of endless variation was ever present in their minds.
Upon reading the OGY paper, lightning flashed, and like the mad scientist in
the old movie who, in a moment of epiphany shouts Of course! It is so obvious!
Reverse the polarity!! there was a collective shout Of course! It is so obvious!
Nudge it onto the stable manifold!!
The paper of Ott, Grebogi and Yorke [19] opened the floodgates, and im-
mediately a torrent of new papers and methods poured out. Just a few months
later, the first physical system was controlled by Ditto et.al. [17], and at the
same time Shinbrot et.al. [47] showed how to get from one point on an attractor
to any other very quickly. This too was obvious: Use the butterfly effect!. Nor-
1


mally a chaotic orbit will visit every neighborhood of an attractor, but it may
take its time. It wanders from saddle orbit to saddle orbit like a bee from flower
to flower arriving along a trail of scent, burrowing in, then quickly departing
along another trail of scent to the next flower, and on and on until every flower
has been visited. The clever garden gnome, wishing to have his favorite flower
fertilized now, knows he cannot force the bee to come directly over, but he con-
vinces the butterfly to flap his wings and subtly waft the odor of one nearby
flower at the bee to draw him hence, then another nearby that, and another
and another, until the bee is drawn into the gnomes favorite. Shinbrot watched
the butterfly and learned his trick, coaxing an orbit just a little to this side,
then just a little to that side, until it arrived at his favorite spot on the strange
attractor.
That was fine, but not good enough for Bollt and Kostelich [8], who used
graph theory to find an optimum path through the garden to get from any one
place to any other in the least possible time. Every practitioner found some new
trick, and there were many clever tricks indeed. Ways were found to control high-
dimensional systems using several parameters [3], then using just one parameter
by toying with its timing [12]. Spatio-temporal chaos has been tamed [34], as
well as time-varying chaotic systems [37]. More recently Epureanu et.al. [21]
[20] has found an optimal method to control periodic orbits in chaotic systems
if there is a full model of the system available.
While much progress has been made, a study of the elementary geometric
and topological properties of low-dimensional control for continuous time chaotic
2


systems has been lacking. We hope this thesis fills in some of the gaps and
illuminates some of the processes. Besides developing some simple machinery to
help visualize the control process, we hope to use these tools to find the quickest
and easiest ways to stabilize unstable periodic orbits in chaotic systems.
1.1 Tools and goals
The fundamental structure that determines the evolution of a dissipative
chaotic system is an infinitely folded solution surface called a strange attractor.
The strange attractor is an invariant set that results from a continual stretching
and folding of the phase space back into itself under the dynamics of the system.
The central structure of this invariant set is called a horseshoe, after Stephen
Smales construction [49] The horseshoe is a structurally stable set of saddle
orbits whose stable, unstable, and center manifolds all have the geometric form
of Cantor sets. The Cantor set of saddles contains an infinite number of periodic
orbits of arbitrarily high period, as well as chaotic orbits. The periodic orbits
in the horseshoe organize the dynamics on the strange attractor by acting as a
skeleton over which the skin of the chaotic orbits are stretched. Chaotic orbits
in the attractor will come near each periodic orbit over and over again, only to
wander away to another periodic orbit.
Control methods for chaotic systems rely on this ergodicity to assure that
only small perturbations are necessary to achieve control. The system is allowed
to evolve in the attractor until it is close enough to a periodic orbit of interest,
the target orbit, so that a small perturbation will place the system state on
the stable manifold of that orbit. The perturbation is removed and the system
subsequently evolves on its own to the periodic orbit, or center manifold.
3


The dynamics of chaotic systems can be formulated in geometric language.
The advantage of the geometrical point of view, especially in formulating new
control strategies, is that simple dynamical diagrams can be used to simplify
extremely complex orbits and follow the system state as it evolves under the
influence of a perturbation.
The main tool of this work is a diagram of the stable and unstable mani-
folds of the target saddle orbit, along with a displaced copy of these manifolds,
which corresponds to the dynamics of the system near the target orbit and the
dynamics under the control perturbation. These diagrams are linear, and are
topologically equivalent to the (in general) nonlinear manifolds of a saddle in
the attractor of the chaotic system. The diagram, which will be referred to as
the perturbed orbit structure, is dynamic; it evolves in time in the same way that
the real perturbed and unperturbed manifolds do. Because the perturbed orbit
structure is topologically equivalent to the local nonlinear manifolds associated
with the target orbit and its perturbations, the conclusions reached by a careful
use of the diagram will be qualitatively correct.
The perturbed orbit structure will be used to design all the continuous
time control strategies in this thesis. Once the strategy is determined, the
hard work starts. The real system manifolds and their perturbations must be
modelled, and computer code must be written to simulate the control system;
alternatively, the control might be tested on a real physical system. Not all the
control strategies designed herein will be implemented; there are just too many.
Time and circumstance permitting, they will be subjects of future papers.
4


2. Horseshoes
The dynamics of chaotic systems are mainly determined by the stretching
and folding of the phase space. The primary tool for the study of stretching and
folding transformations is the Smale horseshoe [49], a mapping F : D2 D2
taking the topological unit disc D2 into itself. Iterated action of the map results
in an invariant set of Cantor dust with chaotic dynamics on it. The dynamics on
the invariant set of an appropriate Smale horseshoe map mimic the dynamics of a
particular chaotic map from which the horseshoe was derived. If we are dealing
with a chaotic flow, the Smale horseshoe is found in the Poincare sections of
that flow. Furthermore, the orbit structure of the horseshoe map, that is, the
set of periodic orbits and chaotic orbits, their stable and unstable sets, and
the structure of the trellis formed by these stable and unstable sets, remains
invariant under small perturbations. This is known as structural stability, and
it is a necessary condition for our control strategies, where we assume we can
make small perturbations to an attractor while maintaining the integrity of a
local linear saddle. We will investigate the dynamics of the horseshoe, and show
that the attractor of our main example and test system, the vertically forced
pendulum, contains horseshoes, and is therefore genuinely chaotic.
In order to make the notion of structural stability precise, we begin with a
few definitions following Devaney [15].
5


Definition 2.1 Let f and g be two maps. The C distance between f and g is
given by
d0{f,9) = sup \f(x) g(x) \.
xeM.
The Cr distance is given by
dr(f,g) = sup (|/(x) g(x)\,\f'(x) g'(x)\,...,\fr(x) -gr{x)\)
Definition 2.2 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 (i.e., the following diagram commutes).
A 4 A
h \. h 4-
B 4 B
Definition 2.3 Let f : M > M. Then f is said to be Cr structurally stable on
M if 3 e > 0 such that whenever dr(f, g) < e for g : M ¥ M, it follows that f
is topologically conjugate to g.
We now construct the Smale horseshoe. Consider the stadium shaped region
D on the left in Figure 2.1. Compress D in the vertical direction by a factor
of 5 < \ and stretch it in the horizontal direction by a factor of p > 2. Bend
the region in the shape of a horseshoe, as in the right hand side of Figure 2.1,
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.
6


Call the action of the map F. There is a unique attracting fixed point p
in Di since F is a contraction mapping, and because D2 is mapped to Tfi, all
points in Di U D2 tend to p under iteration; that is, lim Fn(q) = p for all
n^oo
q G Di U D2. Furthermore, any point r G S whose image is not in S for all n
obeys lim Fn(r) = p. We are interested in the points s G S such that Fn(s) G S
n> oo
for all n.
Consider the two segments of S that are mapped back inside S by the
horseshoe map F. Call these H0 and Hi, and their preimages Vo and Vi (see
Figure 2.2). Because 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 H0 and Hi
are S, 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 p\h\. Likewise if v is any vertical line segment
in S whose image under F is also in S then the length of F(v) is 5|u|. Suppose
Fn(s) G S' Vn > 0. Then s must be mV0U Vu F(s) eV0UVuF2(s) G VbUVi,...,
since all points not in VqUVi map to DpjD2. Thus, we have that s G F~n(V0UVi)
7



M s-m-a



V1
Figure 2.2: The intersection of S and F(S) gives H0 and Hi, and the preimages
of these horizontal strips are Vo and V\.
for all n > 0. The inverse image of any vertical strip of width w in Vo or V\ 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
of Vo U Vi is a set of four rectangular strips of width j^w, two in Vo and two in
Vi (see Figure 2.3), the inverse image of F-1(Vo U Vi) is a set of eight vertical
strips of width \w, etc. Therefore lim F~n(Vo 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+.
Likewise, 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 such that Fn(s) G S Vn must be in the
intersection A = A+ D A_. The Cantor set of the Smale horseshoe is organized
as in Figure 2.4
The dynamics of the horseshoe Cantor set are extremely complex, and we
8


Figure 2.3: The second iterate of the Smale horseshoe map is shown on the
left. On the right, the intersection of the horizontal and vertical strips after
the first two iterations of the horseshoe map. In the limit, the intersection is a
Cantor set.
Figure 2.4: The Cantor dust of the Smale horseshoe. This is a fractal set,
so each dot in the picture represents four dots which are each composed of four
dots, etc.
9


will need symbolic tools to make much headway in understanding the dynamics.
In order to understand the orbit structure of A, we construct a symbolic descrip-
tion known as a symbol sequence. The dynamics of the map in this language will
be called a symbolic dynamics. Let p E A be a point in the invariant set. Then
pn, the nth iterate of p0, is in bin 0 if it lies in Vo or H0, and that it is in bin 1 if it
lies in V\ or Hi. We can construct a doubly infinite string of symbols where the
Sj are 0 or 1 depending on which vertical strip Vo or V. s is in at the jth forward
iterate of the map, and the s_j are 1 or 0 depending 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. 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.
F is topologically conjugate to the shift on the symbol sequence, and we
can define a metric on F by
<<[(), Ml = E JfV^ (2-1)
i~oo
where (s) = ...s_2, s_i.s0, Si, s2,... and (t) = ...t_2, t-i-to, h,h,
Definition 2.4 Consider a set Q and a mapping F : Q > Q. Two points pi
and P2 are forward asymptotic if Fn(pi), Fn(p2) E Q, Vn > 0, and
lim \Fn(Pl)-Fn(p2)\ = 0.
10


Definition 2.5 Two points pi andp2 are reverse asymptotic if Fn(pi), Fn(p2) G
Q V n < 0 and lim \F~n(pi) F~n(p2)\ = 0.
n>oo
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 2.6 The stable set Ws of s is the set of points t that are forward
asymptotic to s, or Ws(s) = {t : \Fn(t) Fn(s)\ -> 0 as n -> oo} and the
unstable set Wu of s is the set of points t that are reverse asymptotic to s,
Wu(s) = {t: |F-n(£) F~n{s)\ -> 0 as n -) oo}.
Consider a fixed point s = ...111.111... G A. Its stable set contains not only
the vertical segment ls 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 segment in which s resides. Forward
iteration of the map will stretch and fold lu, giving the structure in Figure 2.5.
The intersection of the unstable set Wp of p with its stable set Wf is the
invariant set A of the Smale horseshoe. This set Ap is defined by Wf fl Wp = Ap.
Let us follow a segment of the stable set Wf as it is iterated forward. Con-
sider a segment l0 G Wp with endpoints pa,Pb G Ap and containing no other
points of Ap. Call the set of all segments so defined Lp. As only two of the points
of l0 are in Ap, the segment must map only to other segments like itself, that
is; the forward orbit of l0 is in Lp. Under the action of the map pa and pb will
approach p arbitrarily closely, and the length of l0 y 0 as n > oo. Figure 2.6
11


Figure 2.5: The stable and unstable sets of the point s.
illustrates the fate of a vertical segment p.
Definition 2.7 f : J J is said to be topologically transitive if for any pair
of open sets U,V G J there exists k > 0 such that fk(U) D V ^ 0.
Definition 2.8 / : J J has sensitive dependence on initial conditions if
there exists 5 > 0 such that, for any x G J and any neighborhood N of x there
exists ay G N and n > 0 such that \fn(x) fn(y) \ > 5.
Definition 2.9 Let f : J > J is said to be chaotic on J if
Figure 2.6: The fate of a line of points in the stable set of p under four iterates
of the map.
12


1. / has sensitive dependence on initial conditions
2. / is topologically transitive
3. periodic points are dense in J
Sometimes this alternate definition is useful:
Definition 2.10 (Taylor and Toohey)[52j A dynamical system on a topological
space is chaotic if every pair of non-void open subsets share a periodic orbit.
Devaneys definition is equivalent to that of Taylor and Toohey when J is a
uniform Hausdorff space:
Definition 2.11 (Taylor and Toohey)[52] A dynamical system on a topological
space is chaotic if every pair of non-void open subsets share a periodic orbit.
Periodic points are dense in the invariant set of the horseshoe map, and
orbits are topologically transitive and have sensitive dependence on initial con-
ditions. These conditions are the signature of chaos, and a system that can
be shown to have a horseshoe, can be proven to be chaotic, according to the
definition of Devaney [15].
We can see that the horseshoe map is chaotic by considering the symbolic
dynamics.
Density of periodic points'. We must exhibit an orbit that converges to an
arbitrary point s = ...s_2, s_i.s0> si, s2,... Let
T ...SQ...Sn,
13


be the sequence that repeats the first n symbols of s. Then d[r, s] < A-,
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* = ...Oil 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 6 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 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.
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.
14


2.1 Horseshoes in systems described by ODEs
The horseshoe structure arises naturally in dissipative chaotic maps of the
plane and Poincare maps of chaotic flows. Horseshoes have been shown to
exist in many systems, including the Henon map [28] and the Lozi map [36].
Guckenheimer and Holmes proved the existence of a horseshoe in a discrete
model of a ball bouncing on a sinusoidally driven table and the van der Pol map
of the annulus [26]. Cheng-Hsiung Hsu showed the existence of a horseshoe in
a cellular neural network [30]. Gwinn and Westervelt found horseshoes in the
attractor of the torque driven pendulum [27], and horseshoes have been found
in the Duffing oscillator.
2.1.1 Horseshoes in the pendulum
2.1.1.1 Horseshoes in the non-over-the-top pendulum
The sinusoidally forced vertical pendulum appears to be chaotic over a range
of parameters, and can be shown numerically to have a horseshoe structure in
its attractor. We consider the system described by the second order normalized
ODE
9 = p9 sin(0)(l a cos (cut)),
(2.2)
where 9 is the angular position measured from the straight down position, p is the
damping, a the forcing amplitude and ui the forcing frequency. We integrate the
equation with a fourth order Runge-Kutta integrator with step size 2-7ra;/500
and strobe the phase plane 16 times during the course of one forcing period.
When the parameters are set to p = .35, a = 1.21 and ui = 1.5 the tableaux
15


in Figure 2.7 results. For this set of parameters, the pendulum is chaotic, but
never goes over the top.
--

'll/ It "1/
%
Figure 2.7: Sixteen successive Poincare sections of the periodically forced
pendulum in a parameter range for which it dies not go over the top.
It is easy to see the stretching and folding that the attractor undergoes as
the drive cycle is traversed. One can see the fractal structure that results from
the figure in the first panel being continually bent into a sinusoid, folded over
and rotated back into itself. This action apparently forms a horseshoe structure,
though one different from the canonical Smale horseshoe we have just explored.
The pendulum seems to have a horseshoe roughly of the form of Figure 2.8
To prove a horseshoe structure exists, we must exhibit a region of the space
16


Figure 2.8: A rough guess as to the form of the pendulum horseshoe.
that is stretched, folded and mapped back into itself as in the abstract horseshoe
map.
The sequence of Poincare sections in Figure 2.1.1.1 illustrates the formation
of a horseshoe structure from a quadrilateral of initial conditions under the
dynamics of the pendulum equation. Although the pendulum is forced with a
period equal to 2ir over the forcing frequency ui, for this set of parameters the
rate of stretching is insufficient to form a horseshoe from a simple quadrilateral
after just one cycle. In fact, for this set of parameters, four cycles are required.
It is likely that there exists a set of initial conditions that would produce a
horseshoe in one period of the forcing cycle, but for our purposes, the proof by
quadrilateral is sufficient.
Upon close inspection we see that the non-over-the-top pendulum attractor
has a horseshoe of the form illustrated in Figure 2.10.
Earlier, we followed the evolution of a segment in the stable set of the Smale
horseshoe through several iterations of the map (see Figure 2.6). The horseshoe
17


: v\
\ \\
: < V N.
\. ^

: ^ V


: X
\ \
Figure 2.9: The evolution of a quadrilateral of initial conditions with corners (-
.3,1.4), (2,-.64,), (.3,-1.4,), (-2,.64) under the dynamics of the pendulum equation
from Poincare phase uit = ir through 8-k. The initial quadrilateral remains for
comparison. The top and bottom edges are in gray, and the left and
right edges are in black.
18


Figure 2.10: The horseshoe of the 4 times iterated Poincare map of the non-
over-the- top pendulum.
map of the pendulum attractor is a map from Poincare section to Poincare
section, but it is easy to see that it was formed by a continuous process of
stretching and folding. The stable set of the pendulum attractor fills the phase
space, and in fact, when the attractor is a single invariant set, the stable set
of each periodic and chaotic orbit is dense in the phase space. Points far away
from the invariant set eventually map near the set and are swallowed by the
attractor. The stable sets of the pendulum attractor are actually manifolds, for
they are continuous, differentiable two-dimensional surfaces, and thus are locally
euclidian topological spaces. The sections in the Poincare section of the stable
manifolds are also manifolds, as submanifolds of the full stable manifolds. Figure
2.1.1.1 shows the evolution of a portion of the stable manifold of the fixed point
at the origin of the pendulum. The final figure also shows the original figure for
comparison. The loop that worked its way into the interior of the attractor
is now one layer deeper and on the opposite side from the original loop. Each
segment inside the attractor works its way deeper and deeper into the attractor.


If the segment is part of a true hyperbolic (horseshoe) structure, then it is
asymptotic to the periodic orbit. This particular periodic orbit is the straight
down solution, the unstable case where the pendulum bob is unaffected by the
forcing. The horseshoe diagram of Figure 2.10 shows only the unstable manifold
of the attractor. From the sequence of Figure 2.1.1.1 we can see what a small
portion of the stable manifold looks like.
2.1.1.2 Horseshoes in the over-the-top pendulum
If the pendulum goes over the top, we have a different horseshoe structure.
In that case the periodic boundaries of the phase space come into play. Because
the right and left side of the phase space are periodic (the angular position
0 = 7r (straight up) is the same whether the top is approached clockwise or
counterclockwise) we can view the dynamics as taking place in an open annulus
(the top and bottom of the phase space are infinite-see Figure 2.12).
Figure 2.13 shows 16 Poincare sections for the over the top mode in the
standard phase plane, and Figure 2.14 shows 16 Poincare sections for the over
the top mode in the annular phase plane.
Figure 2.1.1.2 shows the horseshoe structure of the over-the-top mode of the
pendulum obtained after three forcing cycles.
The over the top pendulums horseshoe has the detailed structure shown
in Figure 2.16 after two forcing periods. There are five strips stretched across
the quadrilateral of initial conditions. Three of these came from the central
S fold that comprised the action of the non-over-the-top pendulum, and two
20


Figure 2.11: The evolution of a portion of the stable manifold of the periodic
orbit through the origin over a single forcing period.
21


Figure 2.12: An annular phase space.
22


g&
#£?£>, J # Vih^jl . J
J0^$iV P 1|1 v,sS >r /#/ fMl £ > a>/ / ,/
/ M myy f + ntc.,y Ms$0^"' ^"!i' / ?&"" ^£3~**'
Figure 2.13: 16 Poincare sections of the over-the-top pendulum in the stan-
dard phase space.
23


d 0 Cl
0 Q o
& HL j // m s
d
Figure 2.14: 16 Poincare sections of the over-the-top pendulum in the an-
nular phase space.
24


Figure 2.15: The formation of the over-the-top horseshoe from times ir to 6-7T.
25


Figure 2.16: The horseshoe map for the Poincare section of the over-the-
top pendulum after two forcing periods. Because the pendulum goes over the
top, the periodic boundary comes into play and we must connect the broken
numbered ends to their mates, either in a cylindrical phase space, or an annular
one.
strips stretched across the periodic boundary. After three forcing periods, the
horseshoe diagram appears as in Figure 2.17. The five strips fold in an S shape
as in the non-over-the-top pendulum, giving the fifteen central strips in the
figure, and in addition, the top three and bottom three strips from the two-
iterate horseshoe of Figure 2.16 stretch across the periodic boundary, adding six
more strips, for the total of twenty one strips seen in the figure.
As in the Smale horseshoe, the unstable manifold of the pendulum attractor
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 Ik G LXi, where X\ is the jth point 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 pendulum attractor is dense in the basin of attraction of
26


Figure 2.17: The horseshoe map for the Poincare section of the over-the-
top pendulum after three forcing periods. Because the pendulum goes over the
top, the periodic boundary comes into play and we must connect the broken
numbered ends to their mates, either in a cylindrical phase space, or an annular
one.
27


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 2.12 Let A be an invariant set for a discrete dynamical system de-
fined by f : Rn > Rn. A hyperbolic structure for A is a continuous invariant
direct sum decomposition T^Rn = Ef Ef with the property that there are
constants C > 0, 0 < A < 1 such that:
1. if v £ Ef, then \Df~n{x)v\ < C\n |u|;
2. if v £ E%., then \Dfn(x)v\ < CXn |u|.
The invariant set A of a generalized Smale horseshoe (one with an arbitrary
number of folds) has a hyperbolic structure. Essentially, a hyperbolic structure
is the trellis formed by the stable and unstable sets of the saddle orbits resulting
from the dynamics of the generalized Smale horseshoe. We have shown that
the pendulum attractor contains a horseshoe, but we do not know whether all
the periodic points that appear to be in the attractor are part of a hyperbolic
structure (there are good reasons to suppose that they are not [26]). This
means that what appears to be chaotic motion over the entire connected unstable
manifold of the pendulum 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. Although we dont know for sure whether the apparent
chaotic evolution of an orbit over an attractor is 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
28


is to stabilize low period unstable saddle orbits, and to navigate through the
attractor along these orbits. In either a simulation on a computer or in a physical
experiment, chaotic orbits and orbits of very large period appear to be the same.
29


3. Local models from data
3.1 Introduction
The goal of this chapter is to understand and exploit a certain dynamical
structure associated with chaos control schemes. This structure is at once geo-
metrical and dynamical: we visualize the manifolds associated with a saddle type
periodic orbit, the target orbit we wish to stabilize, and the dynamics of system
states on or near them. Near the target orbit, which can evolve in discrete or
continuous time, we set up a local coordinate system with the target orbit at
the origin, and assume linear dynamics of saddle type. Perturbation of some
system parameter p moves the target orbit, and thus shifts the origin of the local
linear dynamics, so any system state nearby will evolve under these perturbed
dynamics until the control parameter is returned to the nominal value or reset to
a new value. Thus, in addition to the current system state, we have two periodic
orbits and their manifold structures to keep track of: the target orbit and its
manifolds, and those of the perturbed target orbit. It is assumed throughout
that the perturbation is small enough so that the dynamics are essentially the
same under the perturbation, except for the relocation of the origin.
We use a geometrical interpretation because it is easy to see the consequences
of different types of perturbations of the parameter p due to the simplicity of the
structure under consideration. The nonlinear dynamics and nonlinear manifolds
can be linearized, and thus the model simplified, without loss of insight.
3.2 Building local models
30


The structure we model consists of two periodic orbits x(t, p0),x(t, p) and
their associated stable and unstable manifolds (Wpo, Wpo), (Wp, Wp). The orbit
x(t, p0) we call the target orbit, as it is the goal behavior of our control schemes.
The target orbit x(t, po) is to be considered fixed in the phase space or surface
of section (SOS) of the phase space, and the perturbed target orbit x(i, p) is to
be considered variable as a function of p.
In order to make the description and manipulation of this structure tractable,
we assume that the manifolds (Wpo, Wpo), (Wp, Wp) are nearly linear in a small
enough neighborhood of x(t, p0) and x(t, p), so that (Wpo, Wpo) (Epo,Epo) and
(Wp, Wp) (Ep, Ep), where (Epo,Epo), (Ep, Ep) are the linearized unperturbed
and perturbed sets of stable and unstable manifolds. These linear geometric
objects consisting of x(t, p0), x(t, p) and (Espo,Epo), (Esp, Ep), together with the
dynamical behavior of the points that comprise them, we call the perturbed or-
bit structure. The two periodic orbits, their locally linear manifolds, and the
dynamics on them are diffeomorphic: one can be smoothly transformed into the
other through the variation of the control parameter p.
Associated with the perturbed orbit structure are the manifolds on which
nearby orbits evolve, which are essentially nonlinear; these manifolds and the
dynamics on them are completely determined by the nature of the perturbed
orbit structure. We aim to manipulate the evolution of orbits x(i, p) near the
perturbed orbit structure in such a way as to force x(t0, p) to evolve to x(t0 +
r, po) or its stable manifold Es(po) after some time r. To that end, we model
the perturbed orbit structure and the other structures necessary for control.
31


Although we concentrate on periodically forced systems with three dimen-
sional attractors, most of the control methods developed can be adapted to
systems of other types, such as the Lorenz system or higher dimensional sys-
tems. We assume that all models are constructed entirely from data taken from
a black box: we have no a priori knowledge of the nature of the system except
that it is dissipative and chaotic, and therefore the dynamics take place on an
attractor studded with saddles. Depending on the control method under consid-
eration, we may or may not have access to data at more than one time during
the forcing period. It is common in experimental situations to have access to
only one scalar measurement of the system state, be it position, momentum, or
current.
The usual phase space model for a periodically forced system in three dimen-
sions can be constructed from a variable and its derivative. For our approach,
we require a model in phase space, so should the system under consideration
produce only a time series, we assume that a time delay reconstruction of the
phase space has been made.
If we are able to take data and turn a control parameter on or off only at
times t,t + T,t + 2T, + nT,... where T is a fundamental time of the system,
then the data defines a map and our control strategies will be limited to map
based controls, such as the method of Ott, Grebogi and Yorke. If, on the other
hand, we have access to a continuous stream of data, or to data sampled at a
rate higher than one sample per fundamental system time, then we may be able
to build a reasonably accurate continuous model that allows a greater choice of
32


control strategies.
3.3 Local models for maps and SOS maps
We begin by describing the process of building a local model from data for
a map based system.
3.3.1 The target orbit
The single most important piece of information we require is the location of
the periodic orbit we wish to stabilize, the target orbit. A standard technique for
determining the location of a period-one orbit from an SOS of a chaotic system is
to assume that several (at least two) successive piercings of the SOS within e of each other indicates the presence of a period-one orbit. For periods higher
than one, say, period m, let xn,xre+1, ...,x+m be m + 1 successive piercings of
the SOS. Then if
l-X-ra -^n+m|> l^n+1 '^n+m+l|) > |^n+m1 ^n+2m1| ^
we assume that there is a period m orbit. Suppose our data shows
l^ra -^ra-l-11 > |^7i+l -^n+2 [; |xn-)_2 ^71+31 > |^n+3 ^7x+41 ^
By our criterion this indicates the presence of a period-one orbit. Ideally, we
should have several such data sets to choose from.
There are several possible approaches to approximating the location of the
target orbit x using these subsets of data. We could use least squares to find the
center of gravity of each set of successive points within e of each other, then
use least squares again on the resultant points. As Baker [4] and other authors
33


[46] [48] suggest, we could also make a least squares fit to the whole ensemble.
The problem with these approaches is that they ignore the dynamical component
that determines the placement of these points. The situation for a simple saddle
is subject to the greatest error, since points are moving exponentially away from
the fixed point at each iterate of the map, and for any particular set of points
within e of each other, there is a systematic error in the center of these points
away from the true fixed point. A least squares fit to an ensemble of sets of such
measurements could correct this error, but again, the dynamics of evolution on
the attractor will usually confound the method, because approach to a fixed
point in a map is generically biased to an approach from a particular direction
or directions. Flip-in and flip-out saddles are more forgiving, because along at
least one direction the system state bounces back and forth about the fixed point
as the map iterates forward. Full-flip saddles are the most forgiving of all, as
points bounce back and forth across both the stable and unstable directions.
Even for a flip saddle, however, ignoring the effect of the dynamics can be
deleterious. As a simple and easily visualized one-dimensional example, consider
the logistic map
xn+1 = rxn(l xn). (3.1)
For r = 4 the period-one points are x 0, x = 3/4, and the derivative at x = 3/4
is 2, so near x = 3/4 the eigenvalue is 2, and the map will act locally about
the point 3/4 as the unstable manifold of a flip saddle. Points near x = 3/4
approximately double, their distance from x with each iterate, hopping from side
to side in the process. Figure 3.1 illustrates the dynamics of the logistic map.
34


If we took a single sample of data consisting of three data points near x, say
Figure 3.1: The dynamics of the logistic map with parameter r = 4 and with
an initial condition of 2/3. The web diagram is in bold, arid the progression can
be followed from 2/3 to 8/9 to 32/81.
2/3,8/9,32/81, then a least squares fit, which amounts to an average in the
one-dimensional case, gives a fixed point x = 158/243, which misses the mark
by approximately 0.0997942.
In order to build a local linear model from the data, we postulate a simple
linear map near our computed fixed point x = 158/243. The map will have the
form xn+x x = m(xn x). Solving for m, with xn+i = 8/9,xn+2 = 32/81,
we obtain m = 31/29, which misses the mark (the true slope is 2) by about
0.931034 (using xn = 2/3,xn+i = 8/9 is much worse: it gives a slope of 29/2).
35


This is an extreme example, as the initial data point 2/3 is rather distant
from the true fixed point 3/4. Nonetheless, if we include the assumption about
the form of the map in our calculation, we are led to solve the system
xn+i x = m{xn x)
Ui+ 2 3?)
We call this the dynamics adjusted method. The result is that for xn =
2/3,xn+x = 8/9,xn+i = 8/9,xn+2 = 32/81, we get x = 64/87, m = -20/9
which have errors of around 0.0143678 and 0.2222222 respectively. Using many
more points, we can lower the least squares error in both x and m, but at the
expense of needing much more extra data. For points closer than in the above
example to the true fixed point, we can do much better using the dynamics ad-
justed method, while the least squares method increases in accuracy very slowly.
For this simple one-dimensional example, it is clear that there is an advantage
to incorporating the procedure for finding the fixed point into that for finding
the eigenvalue. Clearly, the closer the data points are to the fixed point, the
better the outcome, as the map is more nearly linear in this region.
3.3.2 Local dynamics for a two-dimensional map
For a two-dimensional map, we can also use the dynamics adjusted proce-
dure. The advantage is that, given a relatively noiseless system with approxi-
mately linear saddles, we can make one calculation using the data set from the
closest approach rather than a least squares fit to the data sets of all close ap-
proaches. The result of using the closest approach will be that the true dynamics
36


will be closest to linear of all the close approach sets, and a linear approximation
will be very accurate.
Suppose that we have built a surface of section map from data and that x;
is the ith piercing of the SOS, and that our data set contains four successive
piercings of the SOS. Then, assuming that the dynamics of the four successive
x
piercings is determined by a linear map centered at x =
, the origin for the
linear map at the Poincare section, the local system state is Xj = x* x, and
the local linear dynamics are determined by
a b Xi
c d Vi
Xi+l = -DX;
The x coordinates of the four successive piercings are labelled x\,x2, x3, x4
and the y coordinates are 2/i, 2/2? 2/3,2/4- Then we can solve
"
a b X\ X £3 X x2 X X4 X
c d yi-y fa-y fa y fa- y
a b Xi X X2 X X2 X X3 X
c d fa-y fa-y fa y fa y
a b X2 X £3 X X3 X X4 X
c d fa-y fa- y fa y fa- y
for x, y, a, b, c, d, yielding
X2X4yiyiX^+X2X3y2+y2X^XlX^y2 X2X4V2-yjX%+X\X3yz-X2X3V3 +XlXjy3+yix\-XlXjyi
X2yi-2x3yi+X4yixiy2+3x3y22x4y2+2xiy3-3x2y3+X4y3-xiy4+2x2y4-X3y4
r- y2y4Xi-xiy%+y2y3X2+x2y3-yiy4X2-y2y4X2-x3y%+yiy3X3-y2y3X3+yiy4X3+x4y%-yiy3X4
* y2Xi-2y3Xi+y4Xi-y1X2+3y3X2-2yiX2+2yiX3-3y2X3+y4X3-yiX4+2y2X4-y3X4
37


(3.2)
a Xjyz-Xjyx ^ xrx4-x?.xz _ y2V3~yiy4 ^ xiy4-X3V2
x\y3xzyx x\yz-xzy\ xiy^-x^yx xiy^-x^yx
Thus, we have the period-one orbit x =
xx
%2
and the state evolution
matrix D =
a b
for a two-dimensional system that can be modelled as a
c d
two-dimensional map. If the system is noisy, it may be more prudent to average
the above method over several data sets.
A similar procedure can be used for periodic points of higher period. A
periodic point of period m will follow the dynamics
xo /(xq) > /2(x0) >,..., > fm 1(xo) Xq,
so we look for close approaches to each of /l(x0), 0 < i < m 1 and proceed as
before, building a local model about each computed point in the orbit.
For higher dimensional systems, say dimension 3, the procedure is roughly
the same, except that now we need to solve a system of 12 equations in 12
unknowns for each point in a periodic orbit. This requires five successive data
points within e of each other: four is not enough, but we can arrange five data
points to get the equations
7
a b c X\ X X2 X X3 X X2 x Xs X X4 X
efg Vl ~ V V2 ~ V V3 X = V2 y h V Vi x
h i j Z\ z z2 z z3 z Z2 z Z3 Z Z4 z
38


- -
a b c X2 X fa X f 4 X £3 X X4 X £5 X
efg fa y fa y £4 % fa-y fa-y fa- x
h % j fa Z fa Z Z4 z fa z fa z fa z
a b c Xi X X2 x X4 X X2 X X3 X fa X
efg fa-y fa- y fa- x fa-y fa-y fa- x
h i j fa z fa z fa z fa z fa z fa z
a b c X1 X X 3 X X 4 X x2 X £4 x x$ X
e f 9 fa-y fa-y fa- x fa-y fa-y fa- x
h i j fa z fa z fa z fa z fa z fa z
which can be solved for x, y, z, a, b, c, d, e, /, g, h, i. As the dimension of the sys-
tem increases, the size of e and the size of the data set become more important
as limiting factors in the success of the method.
3.3.3 The perturbation vector
In order to make a useful model for the purpose of controlling the system,
we will need to include the effect of a perturbation of the control parameter p on
the position of the target orbit x. Depending on the level of precision required,
and the number of possible parameter values, we can take data at the nominal
value of p = p0 and one other value pi or at several other values pi,...,pm and
locate the position of the target orbit. A curve can be fit to these points to
derive an expression for the vector g =
3.3.4 The perturbed dynamics
Near x(p0) the evolution of the system can be described by xra+i = Axn.
For systems with a continuously variable control parameter p, we can perturb
39


the system by 5p so that the perturbed target orbit is now x(p) = x(p0) + Spg,
where g = In this frame of reference, the evolution of a local system state
x is described by xre+1 5pg = A{~x.n 5pg) where we have assumed that
A{p) = A(po) for Sp small.
We have built a local map-based model near a target orbit that is sufficient to
implement two-dimensional OGY control, and for two-dimensional Occasional
Bang-Bang (OBB) (and its variants) and Center Manifold Targeting (CMT),
which will be introduced later in this paper.
3.4 Local models for continuous systems
In order to implement the continuous control schemes to be introduced later,
we require at least some knowledge of the dynamics between surfaces of section.
We confine ourselves to periodically forced systems with three-dimensional at-
tractors. These systems come equipped with a natural fundamental period, the
period of the forcing function. The attractor of such a system lives naturally
in a torus of rectangular cross section, where a cross section of the torus is the
phase plane (more exactly, the Poincare plane), with time parameterizing the
meridian. Depending on the nature of the system, the phase plane could be
a simple rectangular region, a cylinder or annulus for a system with one peri-
odic variable, or a torus for a system with two periodic variables. The Duffing
oscillator, for instance would naturally evolve in a simple planar phase space,
while our perennial example the pendulum would evolve in a cylinder or annulus
for natural phase plane variables 9, 9 and in a torus for time delayed variables
9(t), 9(t + r).
40


We can picture a portion of an orbit x(t) of a chaotic system F(x, i, p) in
three dimensions as a particle following the flow lines near a nonlinear saddle.
The nonlinear saddle is composed of two two-dimensional surfaces, the stable
and unstable sets Ws(t),Wu(t) of the center manifold. The independent vari-
able t merely indicates the evolution of these manifolds in time when viewed
in the phase plane. They do not evolve in time when the attractor is viewed
as a whole. Depending on the system, these manifolds may be infinite in ex-
tent, or confined to a bounded basin of attraction. Although each orbit in the
attractor comes equipped with its own stable and unstable manifolds, and lies
in their intersection, for the purpose of modelling, we ignore all the manifold
structures but those of the target orbit and its perturbations. In this paper, we
further simplify this picture by making a linear approximation Es(t), Eu(t) to
Ws(t), Wu(t) locally, so that when we visualize an orbit nearby a target orbit,
we imagine a particle following hyperbolic flow lines as illustrated in Figure 3.2.
The linear approximations ES,EU to Ws. Ww near the center manifold can
be made explicit by considering the linear map describing the evolution of the
system at a surface of section SOSk transverse to the flow. The subscript k
indicates that the SOS is the (k + l)st of n, 0 < k < n 1 surfaces of section,
where it is assumed that there is some fundamental period T, either from a
periodic forcing function (t) = (t + T) or from a parameterization of the
orbit x(i) = x(f + T). A continuous orbit pierces the surface of section each
fundamental period, and if x(t + nT) = x(t) for some n, we say that x is a
41


Figure 3.2: The flow lines near a 2-dimensional saddle. For a map, a 2-
dimensional saddle can be either simple, with eigenvalues 0 < As < 1 < Xu,
flip-in with eigenvalues 1 \? < 1, or flip with eigenvalues Xu < 1 < As < 0. For a continuous 2-
dimensional system, saddles come in only simple and flip varieties, because a
flip necessitating a Mobius band manifold for one manifold forces it for the other.
periodic orbit of period n.
In what follows we assume that the n surfaces of section are evenly spaced in
time. Let x be a period-one orbit, i.e. x(t) = x(t + T). At SOS0 there is a map
D0 that evolves some system state x near x forward in time by T, the period
of the orbit x. Let us consider the linearization about x and the linear map D0
whose action is approximately D0 : xn+i = D0xn. The eigenvectors eu,es of
Dq are tangent to the stable and unstable manifolds Ws, Wu at x and collinear
with ES,EU at SO So- At each SOSk there is a linear map Dk approximating
the action of Dk, and whose eigenvectors are tangent to the stable and unstable
manifolds at x. If the system is scaled properly, the normalized stable and
unstable eigenvectors eu, es of Dk for each k are appropriate local linear models
42


for the mappings at each SOSk Figure 3.3 shows a caricature of the local linear
manifolds Es, Eu about a period-one flip orbit x excised from the attractor.
Generically, the orbit would not be circular, but otherwise, the caricature is a
good one. Notice that the manifolds periodically scissor with each other, that
is, the angle between the stable and unstable linear subspaces changes as time
advances as we move around the meridian.
If n > oo, the surfaces of section SOSk densely foliate the torus, and
the set of stable eigenvectors from each Dk form a continuous ribbon, a two-
dimensional manifold with boundary whose cross section is strictly linear at
each surface of section. Likewise, the unstable eigenvectors form a continuous
ribbon as n v oo, and the intersection of these ribbons is the periodic orbit
x. Call the ribbon formed by the stable eigenvectors the stable ribbon and that
formed by the unstable eigenvectors the unstable ribbon. These ribbons are a
geometric model for the frame in which the continuous local linear dynamics
near x evolve. System states in the stable and unstable ribbons flow towards
the center manifold and away from the center manifold, respectively. A system
state in this linear frame cannot pass through either of the ribbons, for that
would violate uniqueness of orbits.
An orbit in this local frame evolves along the hyperbolic flow lines as in
the SOS map, but now we picture the hyperbolae as curved two-dimensional
sheets nestled in a stack inside the stable and unstable ribbons. The hyperbolae
along which the orbits evolve foliate the crotch of the intersection of the rib-
bons. Figure 3.4 illustrates the geometrical situation showing two leaves of the
43


Figure 3.3: The evolution of a period-one flip orbit and its local linear stable
and unstable manifolds. Notice the scissoring of the manifolds, where the angle
between them changes continuously.
foliation.
center manifold
(target orbit)
Figure 3.4: Two leaves of the foliation of the crotch of the stable and unstable
ribbons by hyperbolic sheets.
Our geometrical picture is almost complete. Our goal is to use a perturba-
tion or series of perturbations calculated to place the system state on or near


either the target orbit or its stable manifold. Geometrically, we have the situ-
ation illustrated in Figure 3.5, where a perturbation is applied for some time r
sufficient to bring the system state to the stable manifold of the target orbit.
In order to achieve our goal, we must have a continuous model of the change in
target orbit with the perturbation The model for is built by interpolating
the position of x(t, p) for continuous values of p G {pmin, Pmax) from a limited
set of measurements (pmin,
Although there is no compelling mathematical reason to choose a linear
model for |^, we choose to do so because of physical constraints in experimen-
tal situations. It may be difficult to take data at several different parameter
settings: for instance, the experimental forced pendulum used by Starrett and
Tagg [51] used a magnet mounted on a small audio speaker to induce eddy cur-
rent damping in a shaft mounted copper disc as a control actuator. Because the
speaker coil would heat up dangerously when a fixed perturbation was applied
for the duration of a data taking run, it was necessary to make many painstak-
ing mechanical adjustments and measurements in order to take data at different
damping levels. Therefore, in that study, data was taken at only three levels of
damping and a piecewise linear model was built using only two data points per
piece.
3.4.1 A local model from geometric considerations
We are now ready to build a model for the dynamics near a period-one orbit
in a continuous, periodically forced system. A suitable model consists of the
periodic orbit itself, its associated local linear stable and unstable manifolds, a
45


Figure 3.5: The geometry of chaos control. In each frame, the dynamics
that active are in bold while the inactive dynamics are dashed lines. In the
first frame, a system state has entered the control box, and its path is shown
under the unperturbed dynamics. If left alone the system state would exit
the controllable region along the unstable manifold. In the second frame, the
system has been perturbed, and the path of the evolution of the state under
these dynamics is in bold, while those of the unperturbed system are dashed.
In the third frame, the system state has evolved under the perturbed dynamics
(bold) to the stable manifold of the unperturbed system (dashed). In the fourth
panel, the control parameter has been returned to its nominal value, and the
system state will now evolve along the stable manifold of the target orbit to the
target orbit itself.
manifold orthogonal to the stable manifold, and a state evolution matrix whose
entries vary continuously with the phase of the system. The manifold orthogonal
to the stable manifold is used as an indication of when the system state is in
the stable manifold, since the dot product of the local state vector with a vector
in that manifold will be zero. With this model, we may evolve a system state
near the periodic orbit forward in time using the state evolution matrix while
keeping track of the eigenvector frame in which it evolves.
46


In detail, suppose we have a periodically forced dynamical system whose
evolution takes place on a strange attractor, and that we have a set of data
taken at equally spaced intervals so that a set of surface of section maps may be
formed. Then, periodic orbits, as well as the local dynamics near the periodic
orbits, may be extracted by standard methods.
We may use the formulae (3.2) to compute the fixed points and local linear
maps at each SOS. We can then smoothly interpolate between the fixed points
to get a model of the target orbit. The matrices Dj of the local linear maps
that take a system state in any particular SOS to the next iterate in the same
SOS will provide what we need to build the continuous state evolution matrix.
In the attractor, the system state x near the target orbit x evolves in a frame
of the local stable and unstable manifolds of x, thus, we build our local linear
model from the stable and unstable eigenvectors es and eu of the Dj, which
are tangent to the local stable and unstable manifolds Es, Eu of x. Our model
system therefore acts like the solution x = (t)x0 to a linear differential equation
x = F(£)x. We appeal to Floquets theorem, which deals with the form of the
solutions to such an ODE:
Theorem 3.1 (Floquets Theorem) Suppose x = F(t)x with Fnxn contin-
uous and T-periodic. There are n linearly independent solutions f>k that can
be arranged as columns in a matrix $(t), known as a fundamental matrix.
Then each fundamental matrix $(t) o/x = F(t)x can be written as the product
$(t) = P(t)ect, where Pnxn is T-periodic and Cnxn is a constant matrix.
47


We aim to build a solution in the form of a Floquet fundamental matrix.
First, let us reason geometrically. As the dynamics near x cannot allow the
system state x to cross a local manifold, we expect that x will evolve along the
hyperbolic sheets foliating the crotch of the saddle. The evolution along these
sheets will be a combination of (possibly time-varying) exponential evolution
along the stable and unstable eigenvectors by an amount determined by the
stable and unstable eigenvalues As(t), Au(t), at least in the linear approximation.
The matrix of eigenvalues will be diagonal if the system state is written in a
basis of the eigenvectors, so we choose an initial surface of section SOSq, whose
evolution matrix is D0, and let So have as its columns the eigenvectors of D0. We
write the initial condition x0 in a basis of the eigenvectors of D0: x0 = S^Xq.
Letting So
si(0) s2(0)
, we see that the components of x0 determine the
si(0)
+ £2(0)
s2(0)
as well as its
length of the vector x0 = S(i)x = £i(0)
position in the local eigenvector frame of So; that is, its angle in relation to the
angle between the column vectors of So- Now consider the expression S(t)£(t)x,
where S (t) is a continuous time matrix whose columns are vectors co-linear with
the stable and unstable ribbons of x. S(t) defines the local frame in which the
system state x evolves near x, and when the initial condition is written as x and
£(i) =
\K\tlT 0
0 \K\,it
, then y =
3/2 (*)
= E{t) =
whose components change exponentially. Writing S(t) =
the local linear system f(t) = S(t)y = yi(t)
si (*)
+ 3/2 (*)
|As|i/Ta;i
|A|*/t:e2
Si(t) S2(t)
s2(i)
is a vector
we have
48


Thus these local dynamics can be described as a linear combination of two
independently rotating vectors, each of whose coefficients changes exponentially.
Because the coefficients change exponentially, they cannot change sign, and the
system state must evolve solely in the quadrant in which it began initially,
that is, between the linear stable and unstable manifolds in which the initial
condition began.
The picture is still not complete. The above model was built on the assump-
tion that the eigenvalues associated with the eigenvectors of the local linear sys-
tem were constant. For a map, there is no continuous time, so the eigenvalues
must be constant, and for a continuous system the calculation of eigenvectors
at any particular surface of section SOSk will yield approximately the same
eigenvalues, as the calculated eigenvalues are averaged over the same period.
However, when data are compared to a simplified model x(i) = S(t)E(t)SL there
is an obvious difference. Figure 3.6 shows a single cycle near a period-one flip
orbit taken from data and the evolution of the same initial system state under
the constant eigenvalue model. The data points in gray are the true system
states and the points in black are the model system states taken at 16 points
during the cycle. The lines indicate the local linearized stable manifolds Es(k),
1 < k < 16, at the sixteen surface of sections SOSk Because the system states
for both the model and the real data are so close to the stable manifold, the
dynamics are dominated by the stable eigenvalue As(t). The stable eigenvalue
is obviously varying in time, so we need a term in our model to correct for
the variation about the constant average eigenvalue over a cycle. The unstable
49


Figure 3.6: Two orbits near a flip saddle; the true orbit from data, whose
data points are gray, and one from the simplified model which does not include
a correction for varying eigenvalues, whose data points are black.
50


eigenvalue Au(t) is also a function of time, as can be seen by Figure 3.7. In
this figure the lines represent the unstable manifolds, and the other features are
as in Figure 3.6. The deviation of the simple model from data is even more
pronounced in this case than it was for when the data were near the stable
manifolds. Again, because the data and model states are so close to the un-
stable manifolds, the unstable eigenvalue dominates the dynamics, and we are
confident in stating that the unstable eigenvalue Xu is a varying function of
time. Let us define a diagonal matrix P(t) such that the diagonal entries act
|AS|*/T 0
as corrections to the constant eigenvalues XS,XU in E(t) =
0 \XU\T
Ps(t) 0
. The correct local model is therefore
0 pu(t)
x = S(t)P(t)E(t)-xo, which is in Floquet form x = S(t)E(t)x0 if P(t) is ab-
sorbed by S(t) as S(t) = S(t)P(t), (because P(t) is also T periodic). This is
exactly the geometric model we want, and it is in the form of a Floquet solution.
Thus, we write P(t) =
3.4.2 A local model from data
Now let us construct a continuous time model from data. Suppose there
are m surface of section maps SOSq, SOSi, ...SOSm-1 evenly spaced around the
torus. Let x^j be the ith iterate of an orbit nearby the periodic orbit x, at SOSj,
and let Aj be the matrix that evolves a system state xy from SOSj to SOSj+1,
i.e. AjX,j = Xjj+i. Define Dj so that DjX,j = J([ Aj x* j = Xj+ij, where the
product over the indices is cyclic on the set {0,1,..., m 1}; that is,
Dj 11 Aj AjiAj2AoAm\Am2---AjjriAj
3
51


Figure 3.7: Another set of two orbits near a flip saddle; the true orbit from
data, whose data points are gray, and one from the simplified model which does
not include a correction for varying eigenvalues, whose data points are black.
52


The set of all D* would therefore be
Do AmiAm2---AiA.Q
D\ AQAm-\...A2Ai
D2 A1A0...A3A2
Let Sj be the matrix of eigenvectors of Dj. We call Sj a scissoring matrix
because of the way the angle between the eigenvectors occupying its columns
changes with j. The scissoring matrix is of fundamental importance to time pro-
portioned perturbations (TPP) because it contains a description of the moving
target we are trying to hit, the stable eigenvector es(t). Now RjSj Sj+i for
some matrix Rj\ that is, Rj takes the eigenvectors of Dj into those of Dj+1, so
Rj = Sj+iSj1-
We model the evolution of x^- to xij+1 by a small exponentiation along
the phase path between the eigenvectors followed by a small rotation. To
begin our computations in the local eigenvector frame, we write in a basis
of the eigenvectors of Dj\ = S'1Xjj. Then, we evolve Xy in the frame
of the eigenvectors to where the diagonal matrix Hj is the nonro-
tational component of Aj. Now we rewrite HjS~lx.ij in the Euclidean basis:
Sj{HjSjlyiitj). Note that this matrix SjHjSj1 is just the diagonal form of the
local evolution matrix Aj that takes a state in SOSj to its image in SOSj+1.
We then rotate the eigenvectors of Dj into the eigenvectors of Dj+1 by
Rj (Sj Hj j) (each separate eigenvector is rotated to its image, but because
53


the eigenvectors scissor, the term rotate is a misnomer). Now we have
-l
R^S-
*i,3
= X
i,j+1-
We can further decompose Hj by Hj = PjE, where
M1/m o glnlA,!1/ 0 ln|As| e m 0
i ii O 1 0 eln|Au|1/m ln|A| 0 e m
is a purely exponential matrix, with XS,XU the eigenvalues of Dj, and Pj is a
diagonal matrix
Pj 0
Pj =
0 q3
that acts as a periodic perturbation to the exponential matrix. We can think of
A,,1/ and A1//m as the average eigenvalues of the A,-, and the Pj as corrections
to these average eigenvalues to account for the differences between the Aj.
We now have an expression that evolves the system state Xjj in the jth
Poincare section to the system state x^+i in the j + lst Poincare section, written
in terms of a matrix of eigenvectors Sj of the jth SOS, the matrix of eigenvalues
E of the full transition matrix Dj, a matrix Rj that rotates the eigenvectors
of the jth SOS to those of the (j + l)st SOS, and a matrix Pj that accounts for
the non-rotational and non-constant exponential evolution:
Pj^jPjESj Xjj Xjj+i.
This perturbative matrix Pj accounts for the modelling errors resulting from the
linearization of the nonlinear system, as well as the variation in the eigenvalues
54


Figure 3.8: The local skew coordinate system whose basis vectors are the.
eigenvectors of the local linear transformation near a periodic saddle orbit x.
over a cycle and the change in length experienced by a vector due to its being
written in a basis of scissoring eigenvectors (see Figure 3.8).
To evolve the system state x^- in SOSj to the system state XjJ+m in
SOSj+m, we use
as the E and Pj are diagonal and therefore commute.
We would like to have an expression for the exponential part of the evolution
in the limit as the number of surfaces of section goes to infinity. For time t and
forcing period T, with me N the number of surfaces of section, take t/T £ [0,1),
n = G [0, m), and let n be the integer nearest n. Then lim = lim = 4
Because Rj = Sj+\Sj \ this expression reduces to
moo
m>oo
. Therefore
glnlAsly, q
lim
in|A|
0 e rn
Q glnlAul'j,
55


The continuous time matrix that models the constant exponential evolution is
therefore
m
Inf |
e t
* 0
ln| Au| ^
0 e~r~
Similarly, in the limit of an infinite number of surfaces of section, the discrete
models for the scissoring matrix S and the periodic exponential perturbation
matrix P become continuous models S(t) and P(t) for these aspects of the local
linear model.
We may numerically model the local linear dynamics as a continuous system
then, by fitting periodic functions to the entries of the periodic matrices S, P
and E. Therefore the evolution equation for the local linear dynamics near an
experimentally measured orbit is
x(i) = S(t)P(t)E(t)S0 x0.
Thus our model is of the form of a fundamental matrix of Floquet, where
S(t)P(t) is the periodic matrix decomposed into the a rotating eigenvector frame
S(t) and a periodic perturbation P(t), and an exponential matrix E(t).
The local solution about the periodic orbit is therefore described by the
evolution of an initial state within a moving frame of the stable and unstable
manifolds as linearly approximated by the stable and unstable eigenvectors,
which scissor periodically and rotate about the unstable periodic orbit. Co-
evolving with this unperturbed system is a family of perturbed systems modelled
in the same way, and parameterized by the perturbation vector Spg. Figure 3.9
shows a caricature of a flip orbit and its perturbation. Except for the fact
56


that the figure shows circular orbits, and that the perturbation has a constant
displacement, the figure is fairly accurate in terms of what we expect from the
geometry of a flip orbit and its perturbation.
Figure 3.10 shows the evolution of a frame of local eigenvectors for a period-
one full flip orbit taken from data from a simulation of the equation for a ver-
tically forced pendulum. The periodic orbit and local eigenvector frames were
computed as suggested in this paper. Except for the actual path of the orbit,
which for the pendulum is an over-the-top orbit and thus passes through the
periodic boundary, the geometry is very similar to the caricature of Figure 3.3.
Although our model contains all the essential dynamics, our diagram of an
idealized perturbed orbit structure, as seen in Figure 3.9 lacks another feature
common to real perturbed orbit structures: the vector g from x(p0) to x(p) is
shown as varying only in the plane or the orbit, whereas g in a real system varies
continually over the course of a forcing cycle. For comparison, refer to Figure
3.11 which shows the effect of a constant positive and negative perturbation of
the damping parameter on a vertically forced pendulum on the perturbation
vector g relative to a period-one orbit.
57


Figure 3.9: The geometrical arena in which an orbit near a target orbit and
its perturbation evolves. Depending on whether the unperturbed or perturbed
system is active, the system state will evolve under these dynamics.
58


Figure 3.10: The local eigenvector frame of a period-one orbit in which the
evolution of a local system state takes place. Notice the periodic scissoring of
the frame that results from the stretching and folding of the attractor.
59


-0.16 w 0.16
Figure 3.11: A snapshot of a periodic orbit and its local manifolds, flanked
by two perturbed orbits and their, manifolds. The roughly circular traces inter-
secting the center manifolds are the paths followed by the perturbed orbits over
a drive cycle of the pendulum relative to the unperturbed orbit. Units are in
radians.
60


4. Control methods for maps
4.1 OGY control
The paper of Ott, Grebogi and Yorke (OGY) [19] on a new method for con-
trolling chaotic dynamical systems appeared in 1990, and was the first in a flood
of papers on methods explicitly designed to control chaotic dynamical systems.
Although OGY control was later shown to be a specialization of classical control
theory [56] [43], it inspired many variations because of the elegance with which
it addressed a particulary interesting problem. The application of classical con-
trol techniques generically results in a system whose operating point in state
space is stabilized but with a non-zero control cost. Simply put, in most cases
it takes energy to change the characteristic behavior of a system. The beauty of
the OGY method was that it targeted a system that already contained an infi-
nite number of unstable periodic orbits, and that it chose to stabilize only this
restricted class of operating points. Furthermore, by the nature of the method
and the special nature of the operating points it acts on, the resulting control
had, at least theoretically, a vanishing control cost. In this chapter, we explain
the OGY method, and offer several improvements and extensions.
Let -F(x, t, p) be a continuous time dissipative chaotic system with one acces-
sible scalar parameter p and a periodic orbit x(t, p) of saddle type, and designate
by x(t, p) some orbit near x(t, p). In order to model the dynamics near the tar-
get orbit x(t, p), we set x(i, p) = x(t, p) x(t, p), and take a surface of section
transverse to x (from now on we suppress the independent variables). Then
61


near x the system can be modelled as a linear map xn+1 = Axn where A is an
nxn matrix assumed to have no p dependency for 5p small. Similarly, we may
make local models for true maps, and what follows will apply equally well. If
we perturb the system by changing the value of p to p + 5p, the attractor, and
hence the periodic point x will shift in phase space by x. + 5pg where g = 0. A
chaotic system is subject to bifurcations upon large changes in parameters, so
we must restrict the values of 5p to those for which the orbit structure remains
intact.
Chaotic systems are ergodic, so we can count on the system state to eventu-
ally come as near as we like to the target orbit x. When an orbit xn comes near x
we aim to perturb the system for one iterate in such a way that the maps action
xn+1 5pg = A(xn 5pg) forces xn+i to lie in the stable subspace of x. This
is not always possible for systems whose dimension is greater than two, because
the surface of section map may have more than one unstable direction. When
the system has a two-dimensional map, however, it is generically controllable
by this procedure (the exceptional case arises when the control parameter shifts
the fixed point solely along the stable direction).
Let us consider a system whose dynamics near the target orbit can be well
characterized by a two-dimensional map. For this map, the eigenvectors es, eu
of A are tangent to the stable and unstable manifolds Ws and Wu of the target
orbit x(p). Form the pair of vectors (fu,f,) with the properties
fis es 1 fs ^u 6
0 * gu 1
(4.1)
62


The system state x+1 is in the (linear) stable subspace es when xn+1-fu = 0.
To determine the OGY control perturbation bp necessary to achieve this, we
solve fu (xn+i Spg) = iu A(x bpg) = 0 for bp. Because A describes the
evolution near a saddle orbit, it has a complete set of eigenvectors, and can be
decomposed as
[eseu]_1
A = [eseu]
As 0
0 A,.
Now
by (4.1), so we may write
10
01
A [ese
1 Cn o 1 1
1 3 o 1 f T LU
or
A [Gsew]
0 AfT
The condition for control is thus
W 0
[Asesfs -t- Aueufu ]
' ([Asesfs + Xueufu ](xn bpg) + Spg)
[Asfu 6afs Y Afu 6ufu ] (p^n Spg) A bpiu g
= Aufu xn bpXJu g + bpfu g = 0.
Solving for bp gives
bp
Xu fu '
(4.2)
Xu 1 f g
Thus, bp is the control perturbation that puts xn+1 in the (linear) stable man-
ifold. Once there, the natural dynamics of the system will evolve the system
63


state to the target orbit x. Figure 4.1 shows schematically how OGY and other
Stable Subspace Targeting (SST) methods work.
4.1.0.1 OGY applied to the Henon map
We will demonstrate the OGY method on the Henon map, an affine map
of the plane that is chaotic for certain sets of parameters. Figure 4.2 shows the
attractor of the Henon map
xn+1 = (a + p) x\ + byn
Dn+1 %n
where a = 6/5, b = 1/5 and p is the control parameter.
We choose to stabilize the fixed point Vu-2y con^roi js applied
each time the system state enters a control box, a controllable region surrounding
the target orbit that can be determined by, among several other criteria, the
maximum control perturbation for which the target orbit retains its character.
With the control activated, each time the system state enters the control box,
the perturbation is calculated and applied for one iterate of the map. The system
state will then lie near the stable manifold, its proximity depending on the local
linearity of the stable and unstable manifolds. The system state is then measured
and control reapplied as needed. In this experiment, perturbations were applied
each iterate no matter how close the system state came to the target orbit.
While the system was controlled, the magnitude of 5p would converge to values
of order 10~7. The control was turned alternately turned on and off, each time
allowing the system to wander out of the controllable region before control is
64


Figure 4.1: In each panel the active dynamics are in bold while the inactive are
dashed. In the first panel we see the unperturbed manifolds (marked with a U)
in bold and the system state as well as the flow along which it would normally
evolve. In the second panel the perturbed dynamics (marked with a P) are in
effect, and the new flow along which the system state will evolve is shown. In
the third panel the system state has evolved under the perturbed dynamics to
the stable manifold of the unperturbed system. In the fourth panel, the system
is once again governed by the unperturbed dynamics, and will evolve along the
indicated flow.
65


Figure 4.2: Several thousand iterates of an initial condition inside the basin
of attraction for the Henon map. The portion of the attractor inside the small
box in the first quadrant has been magnified and superimposed on the third
quadrant to show the fractal structure of the attractor.
66


P !--------- ;------ i------- f
time
Figure 4.3: The result of OGY control applied to a period-one orbit of the
Henon map. Control is turned alternately on and off, allowing the system to
return to its chaotic state before control is reapplied. The upper portion of the
graph shows the x values, while the lower portion shows the value of Sp while
the system is being controlled.
67


reapplied. Figure 4.3 shows a time series of a typical OGY control session using
the Henon map.
4.2 Extensions to OGY control
In this section we extend the applicability of OGY control to systems with
some restrictions on the type of perturbation that can be applied. We assume
that either the system is a map, or that the only information available about
the current state is that obtained from a surface of section map.
4.2.1 Control by occasional bang-bang (OBB)
Bang-Bang control is a method whereby control is achieved by the applica-
tion of a fixed, or several different fixed perturbations, rather than a continuous
range of perturbations, and generally requires a flexible scheduling. If we have
available only fixed perturbation levels, and can apply control only at regular
intervals and for fixed durations, standard OGY will not work. We demonstrate
a method that will control maps and continuous systems at a surface of section,
albeit imprecisely, with a single fixed perturbation of fixed duration. We call
this Occasional Bang-Bang (OBB).
Much work has been done on so-called bang-bang control in the classical
control theory literature [23] [11] [22], but very little has been written from the
viewpoint of chaos control. Starrett and Tagg [51] showed that a parametrically
forced pendulum could be controlled by the application of a fixed perturbation
for an amount of time proportional to the deviation of the current state from
the target orbit. This type of bang-bang control is known as time proportioned
perturbations (TPP), and uses a controller whose on-time is variable. Epureanu
68


and Dowell [21] considered a one-dimensional OGY problem in the course of
investigation of higher dimensional OGY control and found that a bang-bang
solution was optimal for that case. We propose and demonstrate a controller
that is effective in controlling an orbit nearby a target orbit when the controller
can take on only one or two perturbed values and only one value of on-time.
Specifically, we consider systems represented by maps and surface of section
maps, whose on-time is equal to a single iterate of the map, and whose parameter
values can take on only two: (off, p+) or three values (p_, off, p+).
Suppose we have a chaotic map which has an orbit x we wish to stabilize,
and our control parameter can take on only two values, p0, the nominal value,
and the perturbation pi. Because we are working with a map, we do not have
the option to control for part of a cycle as we do for bang-bang in a continuous
system. We settle instead for stabilizing another orbit, or set of orbits, in a
small region near the target orbit. The orbit we create will be generically a
chaotic or a very high period orbit, but it will be confined to a region near
the periodic orbit of interest. Control is achieved by applying the perturbation
irregularly, scheduled so as to guarantee that the system state will be alternately
driven towards the stable manifold of the unperturbed target orbit and allowed
to move away a little before control is applied again.
As a simple example, choose a two-dimensional chaotic map / with state
variable x whose linearization near a saddle orbit x is x,:+i = Axj, where A is
a 2 x 2 matrix with eigenvalues 0 < \s < 1 < \u and associated eigenvectors
es,eu, with x = x x. The system has a parameter p whose nominal value
69


is set to po, and we allow for one level of perturbation 5p = pi po, where
Pi is the perturbed parameter value. Under the perturbation the dynamics
will be Xj+1 5pg = A(xj 5pg) Denote by es,eu,x the perturbed stable and
unstable eigenvectors, and the perturbed target orbit, respectively. Then the
perturbed target orbit x is 6pg, where g = So that control will be possible,
we assume that g does not lie along the stable direction. The dynamics under
the perturbation will be x 5pig = yl(x 5pig) where, by a slight abuse of
notation, we allow
5pi = 0 for p = po
8 Pi = Pi Po for p = pi
When the system enters a control box delineated by the region bound by
the eigenvectors es,eu,es,eu, the controller decides to turn on or off based on
whether the system state is nearer the perturbed es or the unperturbed es stable
manifolds. The controllable region actually extends past eu towards the outside
of the box (see Figure 4.4) but its boundary in this direction is not easy to char-
acterize, since the extent of the region depends largely on the linearity of the
map [53]. If we make the assumption that the deviation from linearity is roughly
the same on both sides of eu, we can extend the region by reflection of the box
bounded by es,eu,es,e about eu. Points entering the reflected box can be
controlled, but will be asymptotic to orbits in the box bounded by es, eu, es, eu.
We therefore subdivide the parallelogram enclosed by the perturbed and unper-
turbed manifold sets into two regions of equal area, one bounded in the unstable
direction by the dividing line and the unperturbed stable manifold, which we
call the outbox, and one bounded in the unstable direction by the dividing line
70