Preface ....................................................... vii
Acknowledgments .............................................. xiii
List of Figures ............................................. xxiii
List of Examples .............................................. xxv
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 Frenet moving
frame ................................................. 125
7.2.1 Gram-Schmidt process ........................... 126
7.2.2 Generalized Frenet 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 Frenet trihedron - Serret-Frenet formulae ...... 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 pieccwise 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.l 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 Liaponnoff dimension and Kaplan-Yorkc
conjecture ..................................... 274
A.5.4 Liaponnoff dimension and Chlouverakis-Sprott
conjecture ..................................... 275
A.6 Identities ............................................ 276
A.6.1 Concept of curves .............................. 276
A.6.2 Gram-Schmidt process and Frenet moving frame ... 277
A.6.3 Frenet 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
Mathematica Files ............................................. 291
Bibliography .................................................. 297
Index ......................................................... 309
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