This book aims to present a new approach called Flow Curvature Method that applies Differential Geometry to Dynamical Systems. Hence, for a trajectory curve, an integral of any n-dimensional dynamical system as a curve in Euclidean n-space, the curvature of the trajectory -- or the flow -- may be analytically computed. Then, the location of the points where the curvature of the flow vanishes defines a manifold called flow curvature manifold. Such a manifold being defined from the time derivatives of the velocity vector field, contains information about the dynamics of the system, hence identifying the main features of the system such as fixed points and their stability, local bifurcations of codimension one, center manifold equation, normal forms, linear invariant manifolds (straight lines, planes, hyperplanes).
In the case of singularly perturbed systems or slow-fast dynamical systems, the flow curvature manifold directly provides the slow invariant manifold analytical equation associated with such systems. Also, starting from the flow curvature manifold, it will be demonstrated how to find again the corresponding dynamical system, thus solving the inverse problem.
Author(s): Jean-marc Ginoux
Series: WSSNS0066
Edition: Har/Cdr
Publisher: World Scientific Publishing Company
Year: 2009
Language: English
Pages: 341
Tags: Математика;Топология;Дифференциальная геометрия и топология;
Contents......Page 16
Preface......Page 8
Acknowledgments......Page 14
List of Figures......Page 24
List of Examples......Page 26
Dynamical Systems......Page 30
1.1 Galileo’s pendulum......Page 32
1.2 D’Alembert transformation......Page 34
1.3 From differential equations to dynamical systems......Page 35
2. Dynamical Systems......Page 36
2.3 Existence and uniqueness......Page 37
2.4 Flow, fixed points and null-clines......Page 38
2.5.3 Liapouno. stability theorem......Page 42
2.6.1 Two-dimensional systems......Page 43
2.6.2 Three-dimensional systems......Page 47
2.7.3 Polynomial dynamical systems......Page 51
2.7.4 Singularly perturbed systems......Page 52
2.8.1 Poincare index......Page 53
2.8.2 Poincare contact theory......Page 55
2.8.3 Poincare limit cycle......Page 56
2.8.4 Poincare-Bendixson Theorem......Page 58
2.9.1 Attractors......Page 60
2.9.2 Strange attractors......Page 61
2.10.1 Hamiltonian dynamical systems......Page 63
2.10.2 Integrable system......Page 64
2.10.3 K.A.M. Theorem......Page 66
3.1.1 Definition......Page 70
3.2.1 Global invariance......Page 71
3.2.2 Local invariance......Page 73
4.1 CenterManifold Theorem......Page 76
4.1.1 Center manifold theorem for flows......Page 77
4.1.2 Center manifold approximation......Page 78
4.1.3 Center manifold depending upon a parameter......Page 82
4.2 Normal FormTheorem.......Page 83
4.3 Local Bifurcations of Codimension 1......Page 89
4.3.1 Saddle-node bifurcation......Page 91
4.3.2 Transcritical bifurcation......Page 92
4.3.3 Pitchfork bifurcation......Page 93
4.3.4 Hopf bifurcation......Page 95
5.1 Introduction......Page 98
5.2.1 Assumptions......Page 101
5.2.2 Invariance......Page 102
5.2.3 Slow invariant manifold......Page 103
5.3.2 Slow-fast autonomous dynamical systems......Page 110
6.1 Integrability conditions, integrating factor, multiplier......Page 114
6.1.1 Two-dimensional dynamical systems......Page 115
6.1.2 Three-dimensional dynamical systems......Page 118
6.2.1 First integrals......Page 123
6.2.2 Jacobi’s last multiplier theorem......Page 124
6.3.1 Algebraic particular integral – General integral......Page 125
6.3.2 General integral......Page 127
6.3.3 Multiplier......Page 129
6.3.5 Homogeneous polynomial dynamical systems of degree m......Page 131
6.3.6 Homogeneous polynomial dynamical systems of degree two......Page 137
6.3.7 Planar polynomial dynamical systems......Page 143
Differential Geometry......Page 150
7. Differential Geometry......Page 152
7.1.2 Instantaneous velocity vector......Page 153
7.2 Gram-Schmidt process – Generalized Fr´enet moving frame......Page 154
7.2.2 Generalized Fr´enetmoving frame......Page 155
7.3 Curvatures of trajectory curves – Osculating planes......Page 156
7.4.1 Frenet trihedron – Serret-Frenet formulae......Page 158
7.4.2 Osculating plane......Page 159
7.4.3 Curvatures of space curves......Page 160
7.5.2 Flow curvaturemethod......Page 162
8 .1.1 Fixed points......Page 164
8.1.2 Stability theorems......Page 166
9.1 Invariantmanifolds......Page 174
9.1.1 Global invariance......Page 175
9.1.2 Local invariance......Page 176
9.2 Linear invariantmanifolds......Page 177
9.3 Nonlinear invariantmanifolds......Page 184
10.1.1 Center manifold approximation......Page 188
10.1.2 Center manifold depending upon a parameter......Page 196
10.2 Normal Form Theorem.......Page 204
10.3 Local bifurcations of codimension 1......Page 210
11. Slow-Fast Dynamical Systems......Page 212
11.1 Slow manifold of n-dimensional slow-fast dynamical systems......Page 213
11.2 Invariance......Page 216
11.3 Flow Curvature Method – Singular Perturbation Method......Page 217
11.3.1 Darboux invariance – Fenichel’s invariance......Page 219
11.3.2 Slow invariant manifold......Page 220
11.4 Non-singularly perturbed systems......Page 229
12.1.1 Global first integral......Page 232
12.1.2 Local first integral......Page 233
12.2 Linear invariant manifolds as first integral......Page 235
12.3.1 General integral – Multiplier......Page 238
12.3.2 Darboux homogeneous polynomial dynamical systems of degree two......Page 240
12.3.3 Planar polynomial dynamical systems......Page 241
13.1.1 Two-dimensional polynomial dynamical systems......Page 244
13.1.2 Three-dimensional polynomial dynamical systems......Page 246
13.2 Flow curvature manifold symmetry (parity)......Page 247
13.2.1 Two-dimensional polynomial dynamical systems......Page 248
13.2.2 n-dimensional polynomial dynamical systems......Page 249
13.3.1 Two-dimensional polynomial dynamical systems......Page 251
13.3.2 Three-dimensional polynomial dynamical systems......Page 252
Applications......Page 254
14.1 FitzHugh-Nagumomodel......Page 256
14.2 Pikovskii-Rabinovich-Trakhtengerts model......Page 257
15.1 Pikovskii-Rabinovich-Trakhtengerts model......Page 258
15.2 Rikitakemodel......Page 260
15.3 Chua’smodel......Page 261
15.4 Lorenzmodel......Page 263
16.1 Chua’smodel......Page 266
16.2 Lorenz model......Page 268
17.1.1 Van der Pol piecewise linear model......Page 270
17.1.2 Chua’s piecewise linear model......Page 272
17.2.1 FitzHugh-Nagumo model......Page 274
17.2.2 Chua’s model......Page 276
17.3.1 Brusselator model......Page 277
17.3.2 Pikovskii-Rabinovich-Trakhtengerts model......Page 278
17.3.3 Rikitake model......Page 279
17.4.1 Chua’s fourth-order piecewise linear model......Page 280
17.4.2 Chua’s .fth-order piecewise linear model......Page 282
17.5.1 Chua’s fourth-order cubic model......Page 284
17.5.2 Chua’s fifth-order cubic model......Page 286
17.6.1 Homopolar dynamo model......Page 287
17.6.2 Mofatt model......Page 289
17.6.3 Magnetoconvection model......Page 290
17.8 Forced Van der Polmodel......Page 292
Discussion......Page 294
A.1 Lie derivative......Page 298
A.3 Jordan form......Page 299
A.4 Connected region......Page 300
A.5 Fractal dimension......Page 301
A.5.2 Liapounoff exponents – Wolf, Swinney, Vastano algorithm......Page 302
A.5.3 Liapounoff dimension and Kaplan-Yorke conjecture......Page 303
A.5.4 Liapounoff dimension and Chlouverakis-Sprott conjecture......Page 304
A.6.1 Concept of curves......Page 305
A.6.2 Gram-Schmidt process and Fr´enet moving frame......Page 306
A.6.3 Fr´enet trihedron and curvatures of space curves......Page 308
A.6.4 First identity......Page 309
A.6.5 Second identity......Page 310
A.6.6 Third identity......Page 311
A.8.1 Two-dimensional dynamical systems......Page 312
A.8.2 Three-dimensional dynamical systems......Page 313
A.9.2 Corollaries......Page 314
Mathematica Files......Page 320
Bibliography......Page 326
Index......Page 338
