Dynamical systems : examples of complex behaviour

Cover
......Page 1
Series: Universitext
......Page 2
Title: Dynamical Systems. Examples of Complex Behaviour
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Copyright
......Page 5
Preface......Page 6
Table of Contents......Page 8
1 Introduction......Page 10
2.1 Some general notions......Page 16
2.2 Autonomous systems of ODEs......Page 18
A. Time continuous systems......Page 24
B. Time discrete systems......Page 27
2.4 Chaos in differential and difference equations. Theconcept of an attractor......Page 31
2.5 Interaction, or the interplay between concentrationor reaction and diffusion......Page 45
2.6 Discrete and continuous systems. The Poincaré return map
......Page 50
2.7 Stability and bifurcations; generic properties......Page 51
2.8 The Hopf bifurcation......Page 55
2.9 Lotka-Volterra equations......Page 57
2.10 Stable, unstable, and center manifolds......Page 61
3.1 The topology of graphs......Page 70
3.2 Floer homology......Page 71
3.3 Conley theory: examples and results......Page 79
3.4 Cohomological Conley index......Page 86
3.5 Homotopical invariants......Page 89
3.6 Continuation properties of the Conley index......Page 101
4.1 The entropy of a process as an asymptotic quantity......Page 108
4.2 Positive entropy and chaos......Page 112
4.3 Symbolic dynamics......Page 117
5.1 The metric approach to topological entropy
......Page 120
5.2 Complexity and intrinsic scales......Page 123
6.1 Probability spaces and measure preserving maps......Page 128
6.2 Ergodicity......Page 130
6.3 Entropy and information......Page 135
6.4 Invariant measures......Page 147
6.5 Stochastic processes......Page 151
6.6 Stochastic bifurcations......Page 158
7.1 Lyapunov exponents......Page 162
7.2 Hyperbolicity......Page 165
7.3 Information loss......Page 174
8.1 Cellular automata......Page 178
8.2 Boolean networks......Page 184
References......Page 190
Index......Page 194