Differential geometry applied to dynamical systems / Jean-Marc Ginoux.

Format:

Book

Author/Creator:

Ginoux, Jean-Marc.

Contributor:

Anne and Joseph Trachtman Memorial Book Fund.

Series:

World Scientific series on nonlinear science. Monographs and treatises ; Series A, v. 66.

World scientific series on nonlinear science. Series A. ; vol. 66

Language:

English

Subjects (All):

Dynamics.

Geometry, Differential.

Physical Description:

xxvii, 312 pages : illustrations (some color) ; 24 cm + 1 CD-ROM (4 3/4 in.)

Place of Publication:

Hackensack, N. J. : World Scientific, 2009.

Summary:

Accompanying CD-ROM contains ... "the Mathematica files 'MF XX' with which [the examples in the book] have been elaborated."--P. xii.

Contents:

Dynamical Systems 1

1 Differential Equations 3

1.1 Galileo's pendulum 3

1.2 D'Alembert transformation 5

1.3 From differential equations to dynamical systems 6

2 Dynamical Systems 7

2.1 State space - phase space 8

2.2 Definition 8

2.3 Existence and uniqueness 8

2.4 Flow, fixed points and null-clines 9

2.5 Stability theorems 13

2.5.1 Linearized system 13

2.5.2 Hartman-Grobman linearization theorem 13

2.5.3 Liapounoff stability theorem 13

2.6 Phase portraits of dynamical systems 14

2.6.1 Two-dimensional systems 14

2.6.2 Three-dimensional systems 18

2.7 Various types of dynamical systems 22

2.7.1 Linear and nonlinear dynamical systems 22

2.7.2 Homogeneous dynamical systems 22

2.7.3 Polynomial dynamical systems 22

2.7.4 Singularly perturbed systems 23

2.7.5 Slow-Fast dynamical systems 24

2.8 Two-dimensional dynamical systems 24

2.8.1 Poincaré index 24

2.8.2 Poincaré contact theory 26

2.8.3 Poincaré limit cycle 27

2.8.4 Poincaré-Bendixson Theorem 29

2.9 High-dimensional dynamical systems 31

2.9.1 Attractors 31

2.9.2 Strange attractors 32

2.9.3 First integrals and Lie derivative 34

2.10 Hamiltonian and integrable systems 34

2.10.1 Hamiltonian dynamical systems 34

2.10.2 Integrable system 35

2.10.3 K.A.M. Theorem 37

3 Invariant Sets 41

3.1 Manifold 41

3.1.1 Definition 41

3.1.2 Existence 42

3.2 Invariant sets 42

3.2.1 Global invariance 42

3.2.2 Local invariance 44

4 Local Bifurcations 47

4.1 Center Manifold Theorem 47

4.1.1 Center manifold theorem for flows 48

4.1.2 Center manifold approximation 49

4.1.3 Center manifold depending upon a parameter 53

4.2 Normal Form Theorem 54

4.3 Local Bifurcations of Codimension 1 60

4.3.1 Saddle-node bifurcation 62

4.3.2 Transcritical bifurcation 63

4.3.3 Pitchfork bifurcation 64

4.3.4 Hopf bifurcation 66

5 Slow-Fast Dynamical Systems 69

5.1 Introduction 69

5.2 Geometric Singular Perturbation Theory 72

5.2.1 Assumptions 72

5.2.2 Invariance 73

5.2.3 Slow invariant manifold 74

5.3 Slow-fast dynamical systems - Singularly perturbed systems 81

5.3.1 Singularly perturbed systems 81

5.3.2 Slow-fast autonomous dynamical systems 81

6 Integrability 85

6.1 Integrability conditions, integrating factor, multiplier 85

6.1.1 Two-dimensional dynamical systems 86

6.1.2 Three-dimensional dynamical systems 89

6.2 First integrals - Jacobi's last multiplier theorem 94

6.2.1 First integrals 94

6.2.2 Jacobi's last multiplier theorem 95

6.3 Darboux theory of integrability 96

6.3.1 Algebraic particular integral - General integral 96

6.3.2 General integral 98

6.3.3 Multiplier 100

6.3.4 Algebraic particular integral and fixed points 102

6.3.5 Homogeneous polynomial dynamical systems of degree m 102

6.3.6 Homogeneous polynomial dynamical systems of degree two 108

6.3.7 Planar polynomial dynamical systems 114

Differential Geometry 121

7 Differential Geometry 123

7.1 Concept of curves - Kinematics vector functions 124

7.1.1 Trajectory curve 124

7.1.2 Instantaneous velocity vector 124

7.1.3 Instantaneous acceleration vector 125

7.2 Gram-Schmidt process - Generalized Frénet moving frame 125

7.2.1 Gram-Schmidt process 126

7.2.2 Generalized Frénet moving frame 126

7.3 Curvatures of trajectory curves - Osculating planes 127

7.4 Curvatures and osculating plane of space curves 129

7.4.1 Frénet trihedron - Serret-Frénet formulate 129

7.4.2 Osculating plane 130

7.4.3 Curvatures of space curves 131

7.5 Flow curvature method 133

7.5.1 Flow curvature manifold 133

7.5.2 Flow curvature method 133

8 Dynamical Systems 135

8.1 Phase portraits of dynamical systems 135

8.1.1 Fixed points 135

8.1.2 Stability theorems 137

9 Invariant Sets 145

9.1 Invariant manifolds 145

9.1.1 Global invariance 146

9.1.2 Local invariance 147

9.2 Linear invariant manifolds 148

9.3 Nonlinear invariant manifolds 155

10 Local Bifurcations 159

10.1 Center Manifold 159

10.1.1 Center manifold approximation 159

10.1.2 Center manifold depending upon a parameter 167

10.2 Normal Form Theorem 175

10.3 Local bifurcations of codimension 1 181

11 Slow-Fast Dynamical Systems 183

11.1 Slow manifold of n-dimensional slow-fast dynamical systems 184

11.2 Invariance 187

11.3 Flow Curvature Method-Singular Perturbation Method 188

11.3.1 Darboux invariance-Fenichel's invariance 190

11.3.2 Slow invariant manifold 191

11.4 Non-singularly perturbed systems 200

12 Integrability 203

12.1 First integral 203

12.1.1 Global first integral 203

12.1.2 Local first integral 204

12.2 Linear invariant manifolds as first integral 206

12.3 Darboux theory of integrability 209

12.3.1 General integral - Multiplier 209

12.3.2 Darboux homogeneous polynomial dynamical systems of degree two 211

12.3.3 Planar polynomial dynamical systems 212

13 Inverse Problem 215

13.1 Flow curvature manifold of polynomial dynamical systems 215

13.1.1 Two-dimensional polynomial dynamical systems 215

13.1.2 Three-dimensional polynomial dynamical systems 217

13.2 Flow curvature manifold symmetry (parity) 218

13.2.1 Two-dimensional polynomial dynamical systems 219

13.2.2 n-dimensional polynomial dynamical systems 220

13.3 Inverse problem for polynomial dynamical systems 222

13.3.1 Two-dimensional polynomial dynamical systems 222

13.3.2 Three-dimensional polynomial dynamical systems 223

Applications 225

14 Dynamical Systems 227

14.1 FitzHugh-Nagumo model 227

14.2 Pikovskii-Rabinovich-Trakhtengerts model 228

15 Invariant Sets-Integrability 229

15.1 Pikovskii-Rabinovich-Trakhtengerts model 229

15.2 Rikitake model 231

15.3 Chua's model 232

15.4 Lorenz model 234

16 Local Bifurcations 237

16.1 Chua's model 237

16.2 Lorenz model 239

17 Slow-Fast Dynamical Systems - Singularly Perturbed Systems 241

17.1 Piecewise Linear Models 2D & 3D 241

17.1.1 Van der Pol piecewise linear model 241

17.1.2 Chua's piecewise linear model 243

17.2 Singularly Perturbed Systems 2D & 3D 245

17.2.1 FitzHugh-Nagumo model 245

17.2.2 Chua's model 247

17.3 Slow Fast Dynamical Systems 2D & 3D 248

17.3.1 Brusselator model 248

17.3.2 Pikovskii-Rabinovich-Trakhtengerts model 249

17.3.3 Rikitake model 250

17.4 Piecewise Linear Models 4D & 5D 251

17.4.1 Chua's fourth-order piecewise linear model 251

17.4.2 Chua's fifth-order piecewise linear model 253

17.5 Singularly Perturbed Systems 4D & 5D 255

17.5.1 Chua's fourth-order cubic model 255

17.5.2 Chua's fifth-order cubic model 257

17.6 Slow Fast Dynamical Systems 4D & 5D 258

17.6.1 Homopolar dynamo model 258

17.6.2 Mofatt model 260

17.6.3 Magnetoconvection model 261

17.7 Slow manifold gallery 263

17.8 Forced Van der Pol model 263

Discussion 265

Appendix A 269

A.1 Lie derivative 269

A.2 Hessian 270

A.3 Jordan form 270

A.4 Connected region 271

A.5 Fractal dimension 272

A.5.1 Kolmogorov or capacity dimension 273

A.5.2 Liapounoff exponents-Wolf, Swinney, Vastano algorithm 273

A.5.3 Liapounoff dimension and Kaplan-Yorke conjecture 274

A.5.4 Liapounoff dimension and Chlouverakis-Sprott conjecture 275

A.6 Identities 276

A.6.1 Concept of curves 276

A.6.2 Gram-Schmidt process and Frénet moving frame 277

A.6.3 Frénet trihedron and curvatures of space curves 279

A.6.4 First identity 280

A.6.5 Second identity 281

A.6.6 Third identity 282

A.7 Homeomorphism and diffeomorphism 283

A.7.1 Homeomorphism 283

A.7.2 Diffeomorphism 283

A.8 Differential equations 283

A.8.1 Two-dimensional dynamical systems 283

A.8.2 Three-dimensional dynamical systems 284

A.9 Generalized Tangent

Linear System Approximation 285

A.9.1 Assumptions 285

A.9.2 Corollaries 285.

Notes:

Includes bibliographical references (pages 297-307) and index.

Local Notes:

Acquired for the Penn Libraries with assistance from the Anne and Joseph Trachtman Memorial Book Fund.

ISBN:

9789814277143

9814277142

OCLC:

311763235