2-D Quadratic Maps and 3-D ODE Systems. A Rigorous Approach

Contents......Page 10
Preface......Page 8
Acknowledgements......Page 14
1.1 Introduction......Page 16
1.2 A chain of rigorous proof of chaos......Page 18
1.3.1 Characteristic multiplier......Page 22
1.3.2 The generalized Poincar´e map......Page 23
1.3.3.1 Existence of periodic orbits......Page 25
1.3.3.2 Interval arithmetic......Page 26
1.3.4 Mean value form......Page 28
1.4 The method of fixed point index......Page 29
1.4.1 Periodic points of the TS-map......Page 31
1.4.2 Existence of semiconjugacy......Page 32
1.5 Smale’s horseshoe map......Page 34
1.5.1 Some basic properties of Smale’s horseshoe map......Page 35
1.5.2 Dynamics of the horseshoe map......Page 37
1.5.3 Symbolic dynamics......Page 38
1.6.1 Sil’nikov criterion for smooth systems......Page 41
1.6.2 Sil’nikov criterion for continuous piecewise linear systems......Page 42
1.7 The Marotto theorem......Page 43
1.8.1 The checking routine algorithm......Page 45
1.8.2 Efficacy of the checking routine algorithm......Page 46
1.9 Shadowing lemma......Page 48
1.9.1 Shadowing lemmas for ODE systems and discrete mappings......Page 50
1.9.2 Homoclinic orbit shadowing......Page 51
1.10 Method based on the second-derivative test and bounds for Lyapunov exponents......Page 53
1.11.1 Algorithm based on the Wiener model......Page 54
1.11.2 Algorithm based on the Hammerstein model......Page 57
1.12 Methods based on time series analysis......Page 58
1.13 A new chaos detector......Page 61
1.14 Exercises......Page 62
2.1 Introduction......Page 64
2.2 Equivalences in the general 2-D quadratic maps......Page 65
2.3 Invertibility of the map......Page 74
2.4 The H´enon map......Page 78
2.5.1 Finding Smale’s horseshoe maps......Page 79
2.5.2 Topological entropy......Page 80
2.5.3 The verified optimization technique......Page 83
2.5.4 The Wiener and Hammerstein cascade models......Page 84
2.5.5 Methods based on time series analysis......Page 85
2.5.6 The validated shadowing......Page 86
2.5.8 A new chaos detector......Page 87
2.6.1 Existence and bifurcations of periodic orbits......Page 88
2.6.2 Recent bifurcation phenomena......Page 89
2.6.3 Existence of transversal homoclinic points......Page 91
2.6.4 Classification of homoclinic bifurcations......Page 109
2.6.5 Basins of attraction......Page 114
2.6.6 Structure of the parameter space......Page 115
2.7 Exercises......Page 118
3.1.1 Existence of unbounded orbits......Page 120
3.1.2 Existence of bounded orbits......Page 122
3.2 A zone of possible chaotic orbits......Page 124
3.2.1 Zones of stable fixed points......Page 126
3.3 Boundary between different attractors......Page 127
3.4 Finding chaotic and nonchaotic attractors......Page 138
3.5 Finding hyperchaotic attractors......Page 146
3.6 Some criteria for finding chaotic orbits......Page 154
3.7 2-D quadratic maps with one nonlinearity......Page 155
3.8 2-D quadratic maps with two nonlinearities......Page 163
3.9 2-D quadratic maps with three nonlinearities......Page 164
3.10 2-D quadratic maps with four nonlinearities......Page 166
3.12 2-D quadratic maps with six nonlinearities......Page 168
3.13 Numerical analysis......Page 169
3.13.1 Some observed catastrophic solutions in the dynamics of the map......Page 170
4.1 Introduction......Page 174
4.2.1 Geometry of a piecewise linear vector field in R3......Page 179
4.2.2 Straight line tangency property......Page 181
4.2.3 The real Jordan form......Page 183
4.2.4 Canonical piecewise linear normal form......Page 186
4.2.5 Poincar´e and half-return maps......Page 190
4.3 The dynamics of an orbit in the double-scroll......Page 191
4.3.1 The half-return map π0......Page 192
4.3.2 Half-return map π1......Page 200
4.3.3 Connection map......Page 207
4.4 Poincar´e map π......Page 209
4.4.1 V1 portrait of V0......Page 210
4.4.2 Spiral image property......Page 211
4.5.1 Homoclinic orbits......Page 212
4.5.2 Examination of the loci of points......Page 217
4.5.3 Heteroclinic orbits......Page 225
4.5.4 Geometrical explanation......Page 229
4.5.5 Dynamics near homoclinic and heteroclinic orbits......Page 230
4.6 Subfamilies of the double-scroll family......Page 234
4.7 The geometric model......Page 235
4.8 Method 2: The computer-assisted proof......Page 244
4.8.1 Estimating topological entropy......Page 245
4.8.2 Formula for the topological entropy in terms of the Poincar´e map......Page 251
4.9 Exercises......Page 253
5.1 Introduction......Page 254
5.2 Asymptotic stability of equilibria......Page 255
5.3 Types of chaotic attractors in the double-scroll......Page 259
5.4 Method 1: Rigorous mathematical analysis......Page 260
5.4.1 The pull-up map......Page 261
5.4.2 Construction of the trapping region for the doublescroll......Page 262
5.4.3 Finding trapping regions using confinors theory......Page 267
5.4.4 Construction of the trapping region for the R¨ossler-type attractor......Page 272
5.4.5 Macroscopic structure of an attractor for the double-scroll system......Page 280
5.4.6.1 Birth of the double-scroll......Page 283
5.4.6.2 Death of the double-scroll......Page 284
5.4.6.3 The hole-filling double-scroll......Page 291
5.4.7 Bifurcation diagram......Page 294
5.4.7.1 Comparison of numerical and analytical bifurcation dia- grams......Page 295
5.5.2 Construction of the 1-D Poincar´e map......Page 296
5.5.3 Properties of the 1-D Poincar´e map π.......Page 304
5.5.3.2 Bimodality and self-similarity (vector scaling)......Page 305
5.5.4.3 1-D Poincar´e map π. for a hole-filling orbit......Page 306
5.5.5 Periodic points of the 1-D Poincar´e map π.......Page 307
5.5.5.2 Period-2 points of π.......Page 308
5.5.5.3 Period-n point of the map π.......Page 310
5.5.5.4 Localization of limit cycles......Page 314
5.5.5.5 Structure and order of the appearance of periodic orbits......Page 318
5.5.6 Bifurcation diagrams using confinors theory......Page 322
5.6 Exercises......Page 327
Bibliography......Page 330
Index......Page 352