Handbook of dynamical systems.

Preface......Page 3
List of Contributors......Page 4
Contents......Page 5
Preliminaries of Dynamical Systems Theory......Page 6
General definition of a dynamical system......Page 8
The state space......Page 9
Time set  R......Page 10
Time set  Z+......Page 11
Volume preserving systems......Page 12
Reversible systems......Page 13
Shifts......Page 14
Gradient systems and (non-)recurrence......Page 15
Poincaré map......Page 16
Taking the time  t  map......Page 17
The notion of genericity......Page 18
Sard\'s theorem and Thom\'s tranversality lemma......Page 19
Jet extensions and transversality......Page 21
Transversality with respect to (semi-)algebraic sets......Page 22
The Kupka-Smale theorem......Page 24
Generic properties of symplectic, Hamiltonian and volume preserving systems......Page 27
Generic properties which are not based on transversality: the Closing Lemma......Page 31
Generic local bifurcations of functions (Catatrophe Theory)......Page 33
Centre manifolds and generic local bifurcations of differential equations......Page 35
Generic bifurcations in centre manifolds......Page 37
Generic bifurcations in centre manifolds......Page 39
Concluding remarks......Page 42
Structural stability and moduli......Page 43
References......Page 45
Prevalence......Page 48
Introduction......Page 50
Examples where generic properties do not hold almost everwhere......Page 51
Examples of properties that hold generically and almost everywhere......Page 54
Linear prevalence......Page 56
Properties of prevalence......Page 60
Proving prevalent results......Page 64
Regularity of functions......Page 65
Projections and embeddings......Page 66
Symmetric attractors for generic or prevalent smooth  Γ-invariant dynamical systems......Page 68
Non-uniform hyperbolicity and reducibility of one-dimensional quasiperiodic cocycles in  S L (2, R)......Page 69
Nondegeneracy of double-periodic solutions of nonlinear Cauchy--Riemann equations......Page 70
Operators with singular continuous spectrum......Page 71
Prevalence and genericity of rapid mixing......Page 72
Nonlinear prevalence......Page 73
Nonlinear prevalence without an underlying linear structure......Page 74
Nonlinear prevalence with an underlying linear structure......Page 75
Prevalence of fat solenoidal attractors......Page 77
Kupka--Smale theorem for diffeomorphisms and vector fields......Page 78
Genericity of ergodicity, mixing, weak mixing......Page 79
Genericity of non-Lipschitz Anosov foliations......Page 80
Newhouse phenomena......Page 81
Stable intersection of two Cantor sets on a line......Page 82
Non-Abelian prevalence......Page 83
Relative prevalence and applications in mathematical economics......Page 84
Stronger forms of genericity and prevalence......Page 85
References......Page 86
Local Invariant Manifolds and Normal Forms......Page 94
Construction of invariant manifolds: the graph transform......Page 96
(Locally) invariant manifolds of fixed points......Page 107
Invariant foliations......Page 109
Linearizations and partial linearizations......Page 113
Normal forms......Page 117
Normal forms in bifurcation theory......Page 122
Liapunov-Schmidt reduction......Page 125
References......Page 128
Complex Exponential Dynamics......Page 130
Preliminaries from dynamics......Page 132
Types of fixed or periodic points......Page 134
Preliminaries from complex analysis......Page 137
Role of the singular values......Page 139
The Julia set......Page 141
The Fatou set......Page 143
The filled Julia set......Page 145
The Mandelbrot set......Page 146
External rays......Page 148
Some folk theorems......Page 149
Landing points of external rays......Page 151
Rays landing on the  p / q  bulb......Page 153
The size of limbs and the Farey tree......Page 156
Further remarks......Page 157
Appendix: Angle doubling......Page 158
Computing the Julia set......Page 160
Explosions......Page 161
Misiurewicz points......Page 162
Hyperbolic components of period 1--3......Page 163
Hyperbolic components with higher periods......Page 165
Cantor bouquets......Page 168
The idea of the construction......Page 169
Cantor N-bouquets......Page 171
Straight brushes......Page 175
Connectedness properties of Cantor bouquets......Page 179
Uniformization of the attracting basin......Page 180
Indecomposable continua......Page 181
Topological preliminaries......Page 182
Construction of  Λ......Page 183
Dynamics on  Λ......Page 185
Final comments and questions......Page 187
Structural instability......Page 188
Other near-real parameters......Page 192
Hairs in the parameter plane......Page 193
Questions and problems......Page 194
Untangling hairs......Page 195
The period doubling bifurcation......Page 196
Fingers......Page 198
The kneading sequence......Page 200
Augmented itineraries......Page 202
Untangling the hairs......Page 204
Back to the parameter plane......Page 207
Back to polynomials......Page 208
The polynomial family......Page 209
External rays......Page 212
Other families of maps......Page 213
Maps with polynomial Schwarzian derivative......Page 214
The tangent family......Page 218
Asymptotic values that are poles......Page 220
Bifurcation to an entire function......Page 223
Cantor bouquets and cantor sets......Page 224
References......Page 226
Some Applications of Moser\'s Twist Theorem......Page 230
Examples of action-angle variables......Page 232
Convex Hamiltonians with  n = 1  degree of freedom......Page 233
Action-angle variables for particles in  Rn......Page 234
A generating function construction of the action-angle variables......Page 235
A mechanical interpretation of generating functions......Page 236
The twist condition......Page 238
Basic statements of KAM theory......Page 240
Moser\'s twist theorem......Page 241
A variational approach to Moser\'s twist theorem......Page 242
Reduction to a difference equation......Page 243
Applications......Page 245
References......Page 250
KAM Theory: Quasi-periodicity in Dynamical Systems......Page 254
The `classical\' KAM theorem......Page 256
Related developments: outline......Page 257
Discussion......Page 258
Complex linearization......Page 259
Measure and category......Page 260
Discussion......Page 261
Circle maps......Page 262
Small divisors again......Page 263
A KAM theorem for circle maps......Page 264
Discussion......Page 266
Area preserving annulus maps......Page 268
Moser\'s Twist Mapping Theorem......Page 269
Discussion......Page 270
Introduction......Page 271
Affine structure......Page 272
The perturbation problem......Page 273
Formulation of the normally hyperbolic KAM theorem......Page 275
KAM Theory for Lagrangean tori in Hamiltonian systems......Page 277
Formulation of the Lagrangean KAM theorem......Page 278
Discussion......Page 279
Applications in Classical, Quantum, and Statistical Mechanics......Page 282
Discussion......Page 284
Unicity of KAM tori......Page 285
Paley--Wiener estimates and Diophantine frequencies......Page 286
Parametrized KAM Theory......Page 288
The parametrized dissipative KAM theorem......Page 289
Direct consequences of the parametrized approach......Page 292
Quasi-periodic bifurcations: dissipative setting......Page 293
Quasi-periodic Hopf bifurcation......Page 295
Fattening the parameter domain of invariant  n -tori......Page 296
The parameter domain of invariant  (n+ 1) -tori......Page 298
Non-parallel dynamics......Page 299
Final remarks......Page 300
Quasi-periodic bifurcation theory in other settings......Page 301
Hamiltonian cases......Page 302
Exponential condensation......Page 304
Destruction of resonant tori......Page 306
Lower dimensional isotropic invariant tori......Page 308
The parametrized Hamiltonian KAM theorem......Page 309
Lower dimensional tori in individual Hamiltonian systems......Page 313
Historical remarks......Page 318
Excitation of elliptic normal modes......Page 319
Higher dimensional coisotropic invariant tori......Page 320
Atropic invariant tori......Page 322
Motivation......Page 323
Formulation of the global KAM theorem......Page 324
Example: the spherical pendulum......Page 326
Monodromy in the nearly integrable case......Page 327
Discussion......Page 328
Conclusion......Page 329
References......Page 330
Reconstruction Theory and Nonlinear Time Series Analysis......Page 350
An experimental example: the dripping tap......Page 352
The reconstruction theorem......Page 353
The reconstruction theorem and nonlinear time series analysis: discrimination between deterministic and random time series......Page 357
Box counting dimension and its numerical estimation......Page 359
Stationarity and reconstruction measures......Page 362
Definition of stationarity and reconstruction measures......Page 363
Examples......Page 364
Correlation dimensions and entropies......Page 365
Numerical estimation of correlation integrals and the corresponding dimensions and entropies......Page 368
Classical time series analysis......Page 371
Determinism and auto covariances......Page 373
Predictability and correlation integrals......Page 375
Miscellaneous subjects......Page 377
Liapunov exponents......Page 378
The Kantz--Diks test -- discriminating between time series and testing for reversibility......Page 380
References......Page 381
Homoclinic and Heteroclinic Bifurcations in Vector Fields......Page 384
Introduction......Page 386
Homoclinic orbits to hyperbolic equilibria......Page 388
Homoclinic orbits to nonhyperbolic equilibria......Page 394
Heteroclinic cycles with hyperbolic equilibria......Page 396
Analytical and geometric approaches......Page 398
Normal forms and linearizability......Page 399
Shil\'nikov variables......Page 401
Lin\'s method......Page 404
Homoclinic centre manifolds......Page 405
Stable foliations......Page 406
Shil\'nikov variables......Page 408
Homoclinic centre manifolds......Page 410
N-pulses and N-periodic orbits......Page 411
Robust singular dynamics......Page 412
Singular horseshoes......Page 415
The boundary of Morse--Smale flows......Page 419
Homoclinic-doubling cascades......Page 421
Intermittency......Page 423
Creation of 1-periodic orbits......Page 425
Shil\'nikov\'s saddle-focus homoclinic orbits......Page 427
Bi-focus homoclinic orbits......Page 431
Belyakov transitions......Page 432
Resonant homoclinic orbits......Page 433
Inclination-flips......Page 436
Orbit-flips......Page 438
Coexisting homoclinic orbits......Page 439
Degenerate homoclinic orbits......Page 441
Homoclinic orbits to nonhyperbolic equilibria......Page 442
Heteroclinic cycles with saddles of identical Morse index......Page 445
T-points: Heteroclinic cycles with saddles of different index......Page 450
Heteroclinic cycles with nonhyperbolic equilibria......Page 453
Heteroclinic cycles containing periodic orbits......Page 455
Introduction and hypotheses......Page 456
Belyakov--Devaney transition......Page 458
Reversible systems with SO(2)-symmetry......Page 460
Reversible and Hamiltonian flip bifurcations......Page 462
Coexisting homoclinic orbits......Page 463
Degenerate homoclinic orbits......Page 464
Saddle-centre homoclinic orbits......Page 466
Homoclinic orbits to nonhyperbolic equilibria......Page 469
Heteroclinic cycles and snaking......Page 470
Nilpotent singularities......Page 472
Hopf/saddle-node bifurcations in generic and reversible systems......Page 474
1:1 resonances in reversible systems......Page 478
 02+ i ω resonances in reversible systems......Page 479
Equivariant systems......Page 481
Robust heteroclinic cycles......Page 482
Asymptotic stability of heteroclinic networks......Page 490
Bifurcations from heteroclinic cycles......Page 494
Forced symmetry breaking......Page 498
Homoclinic orbits in systems with  Z2 -symmetry......Page 500
Related topics......Page 502
Topological indices......Page 503
Moduli......Page 505
Existence results......Page 509
Numerical techniques......Page 510
Singularly perturbed systems......Page 511
Infinite-dimensional systems......Page 512
References......Page 514
Author Index......Page 530
Subject Index......Page 544