Introduction to Chaos: Physics and Mathematics of Chaotic Phenomena

Cover Page......Page 1
Title: Introduction to Chaos: Physics and Mathematics of Chaotic Phenomena......Page 2
ISBN 075030507X......Page 3
4 Chaos in realistic systems......Page 4
References......Page 5
Preface to the first (Japanese) edition......Page 6
Preface to the English edition......Page 7
1.1 What is chaos?......Page 8
1.2 Characteristics of chaos......Page 12
1.3 Chaos in Nature......Page 16
2.1 Li–Yorke theorem and Sharkovski's theorem......Page 20
2.2 Periodic orbits......Page 23
2.3 Li–Yorke theorem (continued)......Page 29
2.4 Scrambled set and observability of Li–Yorke chaos......Page 30
2.5 Topological entropy......Page 32
2.6 Denseness of orbits......Page 40
2.7 Invariant measure......Page 45
2.8 Lyapunov number......Page 47
2.9 Summary......Page 48
3.1 Pitchfork bifurcation and Feigenbaum route......Page 49
3.2 Condition for pitchfork bifurcation......Page 58
3.3 Windows......Page 64
3.4 Intermittent chaos......Page 71
4.1 Conservative system and dissipative system......Page 78
4.2 Attractor and Poincaré section......Page 84
4.3 Lyapunov numbers and change of volume......Page 89
4.4 Construction of attractor......Page 92
4.5 Hausdorff dimension, generalized dimension and fractal......Page 95
4.6 Evaluation of correlation dimension......Page 102
4.7 Evaluation of Lyapunov number......Page 108
4.8 Global spectrum—the f(α) method......Page 113
1A Periodic solutions of the logistic map......Page 123
2A Möbius function and inversion formula......Page 124
2B Countable set and uncountable set......Page 125
2C Upper limit and lower limit......Page 127
2D Lebesgue measure......Page 128
2E Normal numbers......Page 129
2F Periodic orbits with finite fraction initial value......Page 130
2G The delta function......Page 133
3A Where does the period 3 window begin in the logistic map?......Page 134
3B Newton method......Page 136
3C How to evaluate topological entropy......Page 138
4A Generalized dimension D[sub(q)] is monotonically decreasing in q......Page 140
4B Saddle point method......Page 146
4C Chaos in double pendulum......Page 147
4D Singular points and limit cycle of van der Pol equation......Page 150
4E Singular points of the Rössler model......Page 154
References......Page 156
Solutions......Page 157
K......Page 174
W......Page 175
Back Page......Page 176