Introduction to chaos and coherence Book

	Preface
1	Introduction	1
2	Fractals	3
2.1	A Cantor set	4
2.2	The Koch triadic island	5
2.3	Fractal dimensions	7
3	The logistic map	9
3.1	The linear map	9
3.2	Definition of the logistic map. Scaling and translation transformations	10
3.3	The fixed points and their stability	11
3.4	Period two	14
3.5	The period doubling route to chaos. Feigenbaum's constants	15
3.6	Chaos and strange attractors	16
3.7	The critical point and its iterates	17
3.8	Self-similarity, scaling and universality	19
3.9	Reversed bifurcations. Crisis	21
3.10	Lyapunov exponents	23
3.11	Statistical properties of chaotic orbits	26
3.12	Dimensions of attractors	27
3.13	Tangent bifurcations and intermittency	29
3.14	Exact results at [lambda] = 1	31
3.15	Predicted power spectra. Critical exponents. Effect of noise	33
3.16	Experiments relevant to the logistic map	34
3.17	Poincare maps and return maps	35
3.18	Closing remarks on the logistic map	37
4	The circle map	38
4.1	The fixed points	38
4.2	Circle maps near K = 0. Arnol'd tongues	39
4.3	The critical value K = 1	42
4.4	Period two, bimodality, superstability and swallowtails	42
4.5	Where can there be chaos?	45
5	Higher dimensional maps	49
5.1	Linear maps in higher dimensions	49
5.2	Manifolds. Homoclinic and heteroclinic points	52
5.3	Lyapunov exponents in higher dimensional maps	54
5.4	The Kaplan-Yorke conjecture	56
5.5	The Hopf bifurcation	57
6	Dissipative maps in higher dimensions	58
6.1	The Henon map	58
6.2	The complex logistic map	62
6.3	Two-dimensional coupled logistic map	65
7	Conservative maps	80
7.1	The twist map	80
7.2	The KAM theorem	83
7.3	The rings of Saturn	83
8	Cellular automata	87
9	Ordinary differential equations	92
9.1	Fixed points. Linear stability analysis	94
9.2	Homoclinic and heteroclinic orbits	95
9.3	Lyapunov exponents for flows	97
9.4	Hopf bifurcations for flows	98
10	The Lorenz model	101
11	Time series analysis	107
11.1	Fractal dimension from a time series	107
11.2	Autoregressive models	109
11.3	Rescaled range analysis	113
11.4	The global temperature: an example	115
	App. A1 Period three in the logistic map	119
	App. A2 Lyapunov exponents algorithm	122
A2.1	Lyapunov exponents for maps	122
A2.2	Lyapunov exponents for flows	123
A2.3	Practical hints	124
	Further Reading	126
	Index	128