Global Analysis of Dynamical Systems

Contents......Page 6
Preface......Page 12
1 Forced oscillations and bifurcations......Page 16
2 Historical behaviour in smooth dynamical systems......Page 78
References......Page 80
On Floris as a friend......Page 82
3.1 On homoclinic bifurcations......Page 84
3.2 Implicit formalism for affine-like maps......Page 85
3.3 Parabolic composition......Page 92
References......Page 102
4 Strong resonances and Takens\'s Utrecht preprint......Page 104
4.1 Setting and notation......Page 106
4.2 Z[sub(q)]-equivariant normal forms......Page 107
4.3 Weak resonance......Page 109
4.4 Strong resonances......Page 110
4.5 Strong resonance for q = 4......Page 111
4.6 From the normal form to the full dynamics......Page 122
References......Page 123
5 Semi-local analysis of the k : 1 and k : 2 resonances in quasi-periodically forced systems......Page 128
5.1 Preliminaries......Page 130
5.2 Normal form analysis......Page 131
5.3 Semi-local bifurcation analysis of the k : 1 resonance......Page 133
5.4 Semi-local bifurcation analysis of the k : 2 resonance......Page 136
5.5 Conclusions......Page 139
References......Page 143
6.1 General setting and some history of the subject......Page 146
6.2 Specific setting and presentation of the results......Page 147
6.3 Local bifurcations......Page 155
6.4 Rescalings......Page 159
6.5 Bifurcations of heteroclinic saddle connections......Page 166
6.6 Study of X[sub(λ,δ)] + O(δ²) as a perturbation of a family of Hamiltonian systems......Page 170
6.7 Uniformity of local bifurcation diagrams with respect to δ......Page 179
References......Page 180
7 Exponential confinement of chaos in the bifurcation sets of real analytic diffeomorphisms......Page 182
7.1 Motivation: the Hopf—Takens bifurcation......Page 183
7.2 Setting of the problem and main result......Page 184
7.3 Applications......Page 187
7.4 Averaging......Page 192
7.5 Linearization near a hyperbolic orbit......Page 199
7.6 Exponential confinement: proof of the main theorem 7.1......Page 211
7.7 Concluding remarks......Page 215
References......Page 223
8 Takens—Bogdanov bifurcations without parameters and oscillatory shock profiles......Page 226
8.1 Examples......Page 227
8.2 Bifurcations from lines of equilibria......Page 230
8.3 Bifurcations from planes of equilibria......Page 235
8.4 Normal forms......Page 239
8.5 Scaling, alias blow-up......Page 242
8.6 Complete integrability to scaling order zero......Page 243
8.7 Slow flow of first integrals to order ε......Page 245
8.8 Elliptic integrals and Weierstrass functions......Page 246
8.9 Poincaré flows of first integrals......Page 249
8.10 Higher order: Poincaré flow, averaging, Melnikov......Page 253
8.11 Geometry of Poincaré maps......Page 256
8.12 Stiff hyperbolic balance laws......Page 261
Appendix. Derivation of the normal form......Page 264
References......Page 272
9 Global bifurcations of periodic orbits in the forced Van der Pol equation......Page 276
9.1 The slow flow and its bifurcations......Page 278
9.2 Symmetry and return maps......Page 282
9.3 Degenerate decompositions and fixed points of H......Page 285
9.4 Concluding remarks......Page 288
References......Page 290
10 An unfolding theory approach to bursting in fast—slow systems......Page 292
10.1 A framework for classifying bursters......Page 293
10.2 The codimension-one burster......Page 299
10.3 Codimension-two bursters......Page 301
10.4 Codimension-three bursters......Page 312
Acknowledgements......Page 319
References......Page 320
11 The intermittency route to chaotic dynamics......Page 324
11.1 Saddle—nodes of diffeomorphisms......Page 325
11.2 Transition maps......Page 328
11.3 Global aspects: ghost dynamics......Page 331
11.4 Prevalence of local and global strange attractors......Page 335
11.5 Persistence of tangencies......Page 337
References......Page 341
12.1 Baby Mandelbrot sets......Page 344
12.2 The complex saddle—node......Page 345
12.3 Dynamics of P₀......Page 347
12.4 Homoclinic points......Page 348
12.5 Ecalle cylinders for P[sub(c)]......Page 349
12.6 Polynomial-like maps......Page 350
12.7 Proof of theorem 12.1......Page 351
References......Page 352
13 Excitation of elliptic normal modes of invariant tori in volume preserving flows......Page 354
13.1 Background on elliptic normal modes......Page 355
13.2 Aim and notation......Page 357
13.3 Persistence theorem for tori of codimension greater than one......Page 358
13.4 Persistence theorem for tori of codimension one......Page 361
13.5 Excitation of the elliptic normal mode......Page 362
References......Page 365
14.1 Setting of the problem......Page 368
14.2 Normalization of the Hamiltonian......Page 373
14.3 Dynamics of the normal forms......Page 377
14.4 Implications for the original system......Page 382
Acknowledgements......Page 385
References......Page 386
15 Global dynamics and fast indicators......Page 388
15.1 A model problem in 1½ degrees of freedom......Page 389
15.2 Bounding the region of interest......Page 390
15.3 Estimating the fraction of integrability......Page 393
15.4 A model problem in 2½ degrees of freedom......Page 398
15.5 Computation of the local exponential growth of the distance......Page 399
15.6 Conclusion......Page 401
Appendix. Taylor series methods of integration......Page 402
References......Page 403
16.1 Granger causality......Page 406
16.2 Information theoretic test statistic......Page 408
16.3 Bootstrap procedures......Page 411
16.4 Monte Carlo simulations......Page 412
16.5 Summary and discussion......Page 416
References......Page 418
17 Birkhoff averages and bifurcations......Page 420
17.1 General assumptions and notations......Page 421
17.2 Likely rotation numbers......Page 422
17.3 Discontinuity of averages......Page 424
17.4 Local embedding flows for the saddle—node......Page 426
17.5 The Mather invariant and return maps......Page 428
17.6 Intermittency near the saddle—node bifurcation......Page 429
17.7 Boundary crisis bifurcations......Page 431
References......Page 432
18.1 Main result......Page 434
18.2 Applications and connections to probability and number theory......Page 440
Appendix. Equilibrium measures and thermodynamic formalism......Page 444
References......Page 445
19 Existence of absolutely continuous invariant probability measures for multimodal maps......Page 448
19.1 Setting of the problem......Page 449
19.2 The multimodal summability condition......Page 450
19.3 Proof of theorem 19.1, part (a)......Page 452
19.4 Proof of theorem 19.1, part (b)......Page 457
References......Page 462
20 On the dynamics of the renormalization operator......Page 464
20.1 Formulation of the main result......Page 465
20.2 Decompositions......Page 467
Appendix. Behavior of the composition operator......Page 472
Acknowledgements......Page 474
References......Page 475
F......Page 476
L......Page 477
W......Page 478
Y......Page 479