NONLINEAR SCIENCE
WORLD SCIENTIFIC SERIES ON
Series A
Vol. 66
Series Editor: Leon O. Chua
DIFFERENTIAL GEOMETRY APPLIED
TO DYNAMICAL SYSTEMS
Jean-Marc Ginoux
Université du Sud, France
World Scientific
NEW JERSEY
•
LONDON
•
SINGAPORE
•
BEIJING
•
SHANGHAI
•
HONG KONG
•
TA I P E I
•
CHENNAI
Contents
I Dynamical Systems
18
1 Introduction
1.1 Galileo’s pendulum . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 D’Alembert transformation . . . . . . . . . . . . . . . . . . . . .
1.3 From differential equations to dynamical systems . . . . . . . . .
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21
22
2 Dynamical Systems
2.1 State space – phase space . . . . . . . . . . . .
2.2 Definition . . . . . . . . . . . . . . . . . . . . .
2.3 Existence and uniqueness . . . . . . . . . . . . .
2.4 Flow, fixed points and null-clines . . . . . . . . .
2.5 Stability theorems . . . . . . . . . . . . . . . . .
2.5.1 Linearized system . . . . . . . . . . . .
2.5.2 Hartman-Grobman linearization theorem
2.5.3 Liapounoff stability theorem . . . . . . .
2.6 Phase portraits of dynamical systems . . . . . . .
2.6.1 Two-dimensional systems . . . . . . . .
2.6.2 Three-dimensional systems . . . . . . . .
2.7 Various types of dynamical systems . . . . . . .
2.7.1 Linear and nonlinear dynamical systems .
2.7.2 Homogeneous dynamical systems . . . .
2.7.3 Polynomial dynamical systems . . . . . .
2.7.4 Singularly perturbed systems . . . . . . .
2.7.5 Slow-Fast dynamical systems . . . . . .
2.8 Two-dimensional dynamical systems . . . . . . .
2.8.1 Poincaré index . . . . . . . . . . . . . .
2.8.2 Poincaré contact theory . . . . . . . . . .
2.8.3 Poincaré limit cycle . . . . . . . . . . . .
2.8.4 Poincaré-Bendixson Theorem . . . . . .
2.9 High-dimensional dynamical systems . . . . . .
2.9.1 Attractors . . . . . . . . . . . . . . . . .
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CONTENTS
2
2.9.2 Strange attractors . . . . . . . .
2.9.3 First integrals and Lie derivative
2.10 Hamiltonian and integrable systems . .
2.10.1 Hamiltonian dynamical systems
2.10.2 Integrable system . . . . . . . .
2.10.3 K.A.M. Theorem . . . . . . . .
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6 Integrability
6.1 Integrability conditions, integrating factor and multiplier . . . . .
6.1.1 Two-dimensional dynamical systems . . . . . . . . . . .
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93
3 Invariant Sets
3.1 Manifold . . . . . . . .
3.1.1 Definition . . . .
3.1.2 Existence . . . .
3.2 Invariant sets . . . . . .
3.2.1 Global invariance
3.2.2 Local invariance
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4 Local Bifurcations
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . .
4.2 Center Manifold Theorem . . . . . . . . . . . . . .
4.2.1 Center manifold theorem for flows . . . . . .
4.2.2 Center manifold approximation . . . . . . .
4.2.3 Center manifold depending upon a parameter
4.3 Normal Form Theorem . . . . . . . . . . . . . . . .
4.4 Local Bifurcations of Codimension 1 . . . . . . . . .
4.4.1 Saddle-node bifurcation . . . . . . . . . . .
4.4.2 Transcritical bifurcation . . . . . . . . . . .
4.4.3 Pitchfork bifurcation . . . . . . . . . . . . .
4.4.4 Hopf bifurcation . . . . . . . . . . . . . . .
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5 Slow-Fast Dynamical Systems
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Geometric Singular Perturbation Theory . . . . . . . . . . .
5.2.1 Assumptions . . . . . . . . . . . . . . . . . . . . .
5.2.2 Invariance . . . . . . . . . . . . . . . . . . . . . . .
5.2.3 Slow invariant manifold . . . . . . . . . . . . . . .
5.3 Slow-fast dynamical systems – Singularly perturbed systems
5.3.1 Singularly perturbed systems . . . . . . . . . . . . .
5.3.2 Slow-fast autonomous dynamical systems . . . . . .
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CONTENTS
6.2
6.3
3
6.1.2 Three-dimensional dynamical systems . . . . . . . . . . . 96
First integrals – Jacobi’s last multiplier theorem . . . . . . . . . . 101
6.2.1 Jacobi’s last multiplier theorem . . . . . . . . . . . . . . 102
Darboux theory of integrability . . . . . . . . . . . . . . . . . . . 103
6.3.1 Algebraic particular integral – General integral . . . . . . 103
6.3.2 General integral . . . . . . . . . . . . . . . . . . . . . . . 105
6.3.3 Multiplier . . . . . . . . . . . . . . . . . . . . . . . . . . 107
6.3.4 Algebraic particular integral and fixed points . . . . . . . 108
6.3.5 Homogeneous polynomial dynamical system of degree m 109
6.3.6 Homogeneous polynomial dynamical system of degree two 115
6.3.7 Planar polynomial dynamical systems . . . . . . . . . . . 121
II Differential Geometry
126
7 Differential Geometry
7.1 Concept of curves – Kinematics vector functions . . . . .
7.1.1 Trajectory curve . . . . . . . . . . . . . . . . . .
7.1.2 Instantaneous velocity vector . . . . . . . . . . . .
7.1.3 Instantaneous acceleration vector . . . . . . . . .
7.2 Gram-Schmidt process – Generalized Frénet moving frame
7.2.1 Gram-Schmidt process. . . . . . . . . . . . . . .
7.2.2 Generalized Frénet moving frame. . . . . . . . . .
7.3 Curvatures of trajectory curves – Osculating planes . . . .
7.4 Curvatures and osculating plane of space curves . . . . . .
7.4.1 Frénet trihedron – Serret-Frénet formulae . . . . .
7.4.2 Osculating plane . . . . . . . . . . . . . . . . . .
7.4.3 Curvatures of space curves . . . . . . . . . . . . .
7.5 Flow curvature method . . . . . . . . . . . . . . . . . . .
7.5.1 Flow curvature manifold . . . . . . . . . . . . . .
7.5.2 Flow curvature method . . . . . . . . . . . . . . .
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8 Dynamical Systems
138
8.1 Phase portraits of dynamical systems . . . . . . . . . . . . . . . . 138
8.1.1 Fixed points . . . . . . . . . . . . . . . . . . . . . . . . . 138
8.1.2 Stability theorems . . . . . . . . . . . . . . . . . . . . . 140
9 Invariant Sets
148
9.1 Invariant manifolds . . . . . . . . . . . . . . . . . . . . . . . . . 148
9.1.1 Global invariance . . . . . . . . . . . . . . . . . . . . . . 149
9.1.2 Local invariance . . . . . . . . . . . . . . . . . . . . . . 149
CONTENTS
9.2
9.3
4
Linear invariant manifolds . . . . . . . . . . . . . . . . . . . . . 151
Nonlinear invariant manifolds . . . . . . . . . . . . . . . . . . . 157
10 Local Bifurcations
10.1 Center Manifold . . . . . . . . . . . . . . . . . . . .
10.1.1 Center manifold approximation . . . . . . .
10.1.2 Center manifold depending upon a parameter
10.2 Normal Form Theorem . . . . . . . . . . . . . . . .
10.3 Local bifurcations of codimension 1 . . . . . . . . .
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11 Slow-Fast Dynamical Systems
11.1 Slow manifold of n-dimensional slow-fast dynamical systems .
11.2 Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.3 Flow Curvature Method – Singular Perturbation Method . . .
11.3.1 Darboux invariance – Fenichel’s invariance . . . . . .
11.3.2 Slow invariant manifold . . . . . . . . . . . . . . . .
11.4 Non-singularly perturbed systems . . . . . . . . . . . . . . .
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12 Integrability
199
12.1 First integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
12.1.1 Global first integral . . . . . . . . . . . . . . . . . . . . . 199
12.1.2 Local first integral . . . . . . . . . . . . . . . . . . . . . 200
12.2 Linear invariant manifolds as first integral . . . . . . . . . . . . . 201
12.3 Darboux theory of integrability . . . . . . . . . . . . . . . . . . . 205
12.3.1 General integral – Multiplier . . . . . . . . . . . . . . . . 205
12.3.2 Homogeneous polynomial dynamical system of degree two 207
12.3.3 Planar polynomial dynamical systems . . . . . . . . . . . 208
13 Inverse Problem
13.1 Flow curvature manifold of polynomial dynamical systems
13.1.1 Two-dimensional polynomial dynamical systems .
13.1.2 Three-dimensional polynomial dynamical systems
13.2 Inverse problem for polynomial dynamical systems . . . .
13.2.1 Two-dimensional polynomial dynamical systems .
13.2.2 Three-dimensional polynomial dynamical systems
III Applications
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210
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14 Dynamical Systems
218
14.1 FitzHugh-Nagumo model . . . . . . . . . . . . . . . . . . . . . . 218
14.2 Pikovskii-Rabinovich-Trakhtengerts model . . . . . . . . . . . . 219
CONTENTS
5
15 Invariant sets - Integrability
15.1 Pikovskii-Rabinovich-Trakhtengerts model
15.2 Rikitake model . . . . . . . . . . . . . . .
15.3 Chua’s model . . . . . . . . . . . . . . . .
15.4 Lorenz model . . . . . . . . . . . . . . . .
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220
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16 Local bifurcations
227
16.1 Chua’s model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
16.2 Lorenz model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
17 Slow-Fast Dynamical Systems
17.1 Piecewise Linear Models 2D & 3D . . . . . . . . .
17.1.1 Van der Pol piecewise linear model . . . .
17.1.2 Chua’s piecewise linear model . . . . . . .
17.2 Singularly Perturbed Systems 2D & 3D . . . . . .
17.2.1 FitzHugh-Nagumo model . . . . . . . . .
17.2.2 Chua’s model . . . . . . . . . . . . . . . .
17.3 Slow Fast Dynamical Systems 2D & 3D . . . . . .
17.3.1 Brusselator model . . . . . . . . . . . . .
17.3.2 Pikovskii-Rabinovich-Trakhtengerts model
17.3.3 Rikitake model . . . . . . . . . . . . . . .
17.4 Piecewise Linear Models 4D & 5D . . . . . . . . .
17.4.1 Chua’s fourth-order piecewise linear model
17.4.2 Chua’s fifth-order piecewise linear model .
17.5 Singularly Perturbed Systems 4D & 5D . . . . . .
17.5.1 Chua’s fourth-order cubic model . . . . . .
17.5.2 Chua’s fifth-order cubic model . . . . . . .
17.6 Slow Fast Dynamical Systems 4D & 5D . . . . . .
17.6.1 Homopolar dynamo model . . . . . . . . .
17.6.2 Mofatt model . . . . . . . . . . . . . . . .
17.6.3 Magnetoconvection model . . . . . . . . .
17.7 Slow manifold gallery . . . . . . . . . . . . . . . .
17.8 Forced Van der Pol model . . . . . . . . . . . . . .
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Discussion
18 Appendix
18.1 Lie derivative . .
18.2 Hessian . . . . .
18.3 Jordan form . . .
18.4 Connected region
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257
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258
259
CONTENTS
18.5 Fractal dimension . . . . . . . . . . . . . . . . . . . . . . . . . .
18.5.1 Kolmogorov or capacity dimension . . . . . . . . . . . .
18.5.2 Liapounoff exponents – Wolf, Swinney, Vastano algorithm
18.5.3 Liapounoff dimension and Kaplan-Yorke conjecture . . .
18.5.4 Liapounoff dimension and Chlouverakis-Sprott conjecture
18.6 Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18.6.1 Concept of curves . . . . . . . . . . . . . . . . . . . . .
18.6.2 Gram-Schmidt process and Frénet moving frame . . . . .
18.6.3 Frénet trihedron and curvatures of space curves . . . . . .
18.6.4 First identity . . . . . . . . . . . . . . . . . . . . . . . .
18.6.5 Second identity . . . . . . . . . . . . . . . . . . . . . . .
18.6.6 Third identity . . . . . . . . . . . . . . . . . . . . . . . .
18.7 Homeomorphism and diffeomorphism . . . . . . . . . . . . . . .
18.7.1 Homeomorphism . . . . . . . . . . . . . . . . . . . . . .
18.7.2 Diffeomorphism . . . . . . . . . . . . . . . . . . . . . .
18.8 Differential equations . . . . . . . . . . . . . . . . . . . . . . .
18.8.1 Two-dimensional dynamical systems . . . . . . . . . . .
18.8.2 Three-dimensional dynamical systems . . . . . . . . . . .
18.9 Generalized Tangent Linear System Approximation . . . . . . . .
18.9.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . .
18.9.2 Corollaries . . . . . . . . . . . . . . . . . . . . . . . . .
6
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267
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270
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271
271
271
272
272
273
Mathematica Files
277
Bibliography
282
Index
297
List of Figures
1
Synopsis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
1.1
Galileo’s pendulum . . . . . . . . . . . . . . . . . . . . . . . . .
20
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
2.10
2.11
2.12
Free fall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Volterra-Lotka predator-prey model . . . . . . . . . . . . . . . .
Phase plane stability diagram . . . . . . . . . . . . . . . . . . . .
Inverted pendulum . . . . . . . . . . . . . . . . . . . . . . . . .
stability diagram . . . . . . . . . . . . . . . . . . . . . . . . . .
Saddle-focus . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Poincaré limit cycle . . . . . . . . . . . . . . . . . . . . . . . . .
Duffing oscillator . . . . . . . . . . . . . . . . . . . . . . . . . .
Lorenz butterfly . . . . . . . . . . . . . . . . . . . . . . . . . . .
Spherical pendulum . . . . . . . . . . . . . . . . . . . . . . . . .
Hénon-Heiles Hamiltonian . . . . . . . . . . . . . . . . . . . . .
Transversal Poincaré section (p2 , q2 ) of Hénon-Heiles Hamiltonian
26
27
32
33
36
37
43
45
48
51
53
53
3.1
Stable W S and unstable W U manifolds . . . . . . . . . . . . . .
58
4.1
Part of the center manifold in green . . . . . . . . . . . . . . . .
61
6.1
General integral . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
7.1
Osculating plane . . . . . . . . . . . . . . . . . . . . . . . . . . 134
8.1
Duffing oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . 144
9.1
Local invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
10.1 Center manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
11.1 Van der Pol slow manifold . . . . . . . . . . . . . . . . . . . . . 192
11.2 Chua’s slow invariant manifold in (xz)-plane . . . . . . . . . . . 196
11.3 Lorenz slow manifold . . . . . . . . . . . . . . . . . . . . . . . . 198
LIST OF FIGURES
8
12.1 Local first integral of Van der Pol model . . . . . . . . . . . . . . 201
12.2 Volterra-Lotka’s first integral . . . . . . . . . . . . . . . . . . . . 204
12.3 First integral of quadratic system . . . . . . . . . . . . . . . . . . 206
17.1 Van der Pol piecewise linear model slow invariant manifold . .
17.2 Chua’s piecewise linear model slow invariant manifold . . . .
17.3 FitzHugh-Nagumo model slow invariant manifold . . . . . . .
17.4 Chua’s cubic model slow invariant manifold . . . . . . . . . .
17.5 Brusselator’s model slow invariant manifold . . . . . . . . . .
17.6 (PRT) model slow invariant manifold . . . . . . . . . . . . .
17.7 Rikitake model slow invariant manifold . . . . . . . . . . . .
17.8 Chua’s model invariant hyperplanes in (x1 , x2 , x3 ) phase space
17.9 Chua’s model invariant hyperplanes in (x1 , x2 , x3 ) phase space
17.10 Chua’s model slow invariant manifold . . . . . . . . . . . . .
17.11 Chua’s slow invariant manifold . . . . . . . . . . . . . . . .
17.12 Dynamo model slow invariant manifold . . . . . . . . . . . .
17.13 Mofatt model slow invariant manifold . . . . . . . . . . . . .
17.14 Magnetoconvection slow invariant manifold . . . . . . . . . .
17.15 Chemical kinetics model - Neuronal bursting model . . . . .
17.16 Forced Van der Pol model slow invariant manifold . . . . . .
17.17 Chua’s cubic model attractor structure . . . . . . . . . . . . .
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233
234
235
236
237
238
239
242
243
245
246
247
248
250
251
253
256
List of Tables
15.1 Invariant manifolds of the (PRT) model . . . . . . . . . . . . . . 222
15.2 Invariant manifolds of the Rikitake model . . . . . . . . . . . . . 223
15.3 Invariant manifolds of the Lorenz model . . . . . . . . . . . . . . 226
Index
attractor, 46, 47, 251, 255, 260, 262
187, 199, 204, 205, 207–209, 221,
attractor structure, 256
255
autonomous, 17, 23, 24, 28, 59, 249, 252, Darboux invariance theorem, 15, 16, 103,
255
104, 148, 149, 181, 183, 187, 232,
234–239, 241, 243, 244, 246, 247,
Bendixson, 44–46
249, 250, 252, 254, 255
bifurcation, 59, 60, 64, 68, 72, 74, 76, 77,
diffeomorphism, 29, 270
127, 137, 160, 174, 179, 229, 230,
differential equation, 13, 19–24, 27, 41, 42,
278–280
44, 46, 59, 62, 64, 70, 92, 93, 95–
Brusselator, 16, 219, 237, 281
101, 127, 138, 160, 167, 181, 219,
223, 224, 240, 242, 244, 245, 271,
Cauchy, 13, 24, 25, 40, 46, 59, 89, 93, 96,
272
99, 127, 128, 138, 160, 167, 181,
differential geometry, 13, 14, 17, 127, 131,
210, 212
154, 181, 254, 263, 278
centre manifold, 15, 16, 54, 58, 60–65, 68,
70–72, 82, 127, 137, 160, 161, 163, dissipative, 46–48, 101, 260, 262
164, 166–171, 174, 179, 227, 229, divergence, 47, 93, 97
Duffing, 45, 144, 169, 170, 277–279
230, 254, 255, 278, 279
dynamical system, 13–16, 18, 22–30, 33,
chaotic attractor, 238
34, 37–47, 49, 50, 54–66, 68–70,
Chua, 14, 16, 38, 80, 88, 89, 195, 219, 224,
72–74, 76–78, 89–92, 95–97, 99–
225, 227, 233, 234, 236, 240–246,
101, 103–115, 118, 119, 127, 128,
256, 278–281
131–138, 140–142, 144–150, 155–
Chua’s invariant hyperplanes, 242, 243
157, 159–161, 163, 164, 166–169,
Chua’s slow invariant manifold, 196, 236
171, 174, 175, 178, 180–183, 185,
codimension, 64, 70, 137, 160, 179, 230
188, 192, 193, 197, 199–202, 205,
complex dynamics, 80
207, 208, 210, 212–216, 228, 229,
conservative, 47, 50, 101, 262
254, 255, 257, 264, 266, 272–275,
curvature, 13, 14, 127, 129, 131–136, 154–
277, 278, 280
156, 181, 192, 197, 202, 254, 255,
263, 264, 266–268
Fenichel, 15, 78, 79, 81, 82, 85, 88, 91, 180,
curvature of the flow, 13
185, 187, 254
curve, 13, 27, 40–42, 44, 72, 93, 99, 101,
first integral, 14–16, 49–51, 100–102, 104–
102, 128, 131–136, 139, 154, 263,
106, 108, 110, 111, 114–119, 122–
264, 266, 267, 271
125, 148, 149, 156, 199–209, 221,
224, 225, 254, 255, 278, 279
Darboux, 14, 16, 94, 99, 101, 103–118, 121,
123, 124, 148, 149, 159, 183, 184, FitzHugh-Nagumo, 16, 218, 235, 280, 281
INDEX
300
non-autonomous, 231, 252
148, 180, 181, 186, 188, 189, 193,
non-singularly perturbed systems, 16, 90,
194, 197, 200, 231, 235, 236, 238,
91, 197, 255, 279
239, 244, 247–249, 254, 255, 281
nonlinear invariant manifold, 157, 221, 254 slow invariant manifold, 14–17, 78, 82, 83,
normal forms, 15, 65, 66, 68–70, 127, 137,
85, 88, 90, 91, 137, 180, 182, 185–
160, 174, 175, 178, 179, 220, 254,
189, 191, 193, 194, 196, 220, 232,
258
233, 236–240, 243–251, 253–255
null-clines, 25, 26, 28
slow-fast dynamical systems, 14, 39, 78, 89,
90, 180, 181, 184, 185, 192, 231,
orbit, 29, 45, 51, 55, 78, 180, 261
237, 255, 278–280
osculating plane, 14, 131–136, 140–144, 146, spherical pendulum, 50, 51, 200, 277
202, 203, 234
stability, 29, 30, 32, 36, 59, 73–75, 127,
137, 138, 140, 142, 144, 147, 148,
pendulum, 19, 20, 154
180, 218
phase, 23, 24, 26, 27, 30, 32, 33, 44, 45,
strange attractor, 14, 47, 50, 260–263
47, 49, 59, 90, 101, 138, 232, 234,
242, 243, 245–248, 250, 253, 256, tangent linear system approximation, 17, 181–
260
183, 272, 274–276
piecewise linear model, 16, 80, 231–234, torsion, 131, 134–136, 154, 181, 197, 266–
240–243, 280, 281
268
Pikovskii, 16, 158, 219, 220, 238, 280, 281 trajectory curve, 13, 14, 24–26, 42, 44, 46,
pitchfork bifurcation, 14, 64, 72, 75, 76,
47, 49–53, 101, 127–132, 136, 137,
137, 160, 174, 179, 229, 230
155, 156, 181, 201, 202, 254–256,
Poincaré, 15, 43, 44, 46, 92, 128, 144
260, 261, 263, 264, 266–268
Poincaré index, 14
transcritical, 137, 160, 170, 179
Poincaré section, 53, 260
transcritical bifurcation, 72–74
quadratic system, 206
relaxation oscillations, 27, 39, 231
relaxation oscillator, 79
Rikitake, 16, 223, 239, 246, 255, 280, 281
saddle, 31, 33, 35, 37, 40, 45, 73, 137, 142,
144, 145, 160, 179
saddle points, 52
saddle-focus, 145
saddle-node, 145
saddle-node bifurcation, 72, 73
singular approximation, 39, 85, 88, 90, 190,
192, 195, 200, 201, 235–237
Singular Perturbation Method, 81, 185, 192,
196, 236, 278
singularly perturbed systems, 14–16, 28, 39,
58, 78–80, 82–84, 87, 89–91, 137,
Van der Pol, 14, 16, 27, 39, 41, 43, 44, 190,
192, 200, 201, 211, 215, 218, 231,
232, 235, 252, 253, 278–280
Volterra, 27, 28, 37, 118, 153, 156, 204, 279
INDEX
fixed point, 15, 16, 25, 26, 28–31, 33–35,
37, 40–46, 55–57, 60, 61, 70, 72–
77, 108, 114, 127, 128, 132, 134,
137–147, 185, 214, 216, 218, 219,
234, 241, 243, 256
fixed point stability, 15, 254
flow curvature manifold, 13, 15, 16, 127,
136–144, 146, 148, 151–154, 156,
157, 159–161, 163, 164, 166–168,
170, 171, 174, 175, 179, 182, 184,
186, 188, 191–193, 196, 197, 201,
202, 204, 206, 207, 209–216, 218–
221, 224, 225, 227, 232–241, 243,
244, 246–250, 252, 254–256
flow curvature method, 13, 15, 17, 90, 109,
127, 136, 137, 159, 185, 186, 190,
192, 195–197, 200, 252, 254–256,
276, 279
Forced Van der Pol, 17, 252, 255, 281
Galois, 147, 181
Geometric Singular Perturbation Theory, 14,
15, 58, 78, 79, 81, 82, 90, 91, 180,
185, 186, 188, 190–193, 195–197,
201, 254, 255, 278
Grobman, 29, 58
Groebner, 215, 216
Hénon-Heiles Hamiltonian, 52, 53, 277
Hamiltonian, 14, 49, 50, 52, 53, 199, 262
harmonic oscillator, 49
Hartman, 29, 58
homeomorphism, 29, 270
homopolar dynamo, 16, 246, 247
Hopf, 77, 137, 160
Hopf bifurcation, 76, 77
hyperbolic, 29, 40, 56, 57, 78, 81, 180
hyperbolic points, 52
299
127, 137, 148–152, 154–159, 180,
184, 199, 201, 224–226
invariant tori, 52
inverse problem, 16, 137, 210, 214, 255,
280
inverted pendulum, 33, 37
Jacobian, 29, 30, 33, 34, 37–39, 41, 54–57,
60, 61, 70, 71, 73–75, 77, 90, 91,
95, 129, 140–142, 145, 146, 161,
180–186, 234, 273–276
Jordan fom, 258, 259
Jordan form, 31, 35, 41, 258
K.A.M. theorem, 14, 52
K.A.M. tori, 52, 53
Kapteyn-Bautin, 125, 209, 278
LaSalle, 54
Liapounoff, 14, 29, 30
Liapounoff dimension, 260, 262, 263
Liapounoff exponents, 261–263
Lie derivative, 14, 15, 42, 49, 94, 119, 140–
142, 149, 150, 153, 154, 157, 158,
184, 187, 192, 197, 200, 201, 221,
226, 232, 234, 237–239, 241, 244,
246, 247, 249, 250, 252, 255, 257
limit cycle, 14, 42–44, 46, 77, 277
linear, 29, 37, 38, 56, 65, 67, 134, 137,
148, 151, 152, 154–156, 159, 175,
180–183, 201, 233, 240–242, 244,
245, 258, 272–276
linear invariant manifold, 15, 153, 156, 202–
205, 207, 208, 220, 254
Liouville, 47, 92, 199, 262
local bifurcations, 254
Lorenz, 16, 48, 90, 91, 147, 197, 198, 212,
216, 225, 226, 229, 230, 249, 255,
263, 278–280
Lorenz butterfly, 48, 277
Lorenz slow manifold, 198
implicit function theorem, 55, 82, 84, 85,
87, 88, 186
integrability, 54, 92–103, 127, 148, 159, 199,
205, 254, 255, 278–280
magnetoconvection, 16, 249, 250, 281
invariant manifold, 54, 56–58, 60, 71, 78, Mofatt, 16, 246, 248, 249, 281
104, 105, 107, 108, 110, 112–115,