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1 Introduction 1

2 Stability of dynamical systems,bifurcations,and generic properties 7

2.1 Some general notions 7

2.2 Autonomous systems of ODEs 9

2.3 Examples:Bifurcation depending on a parameter λ∈R 15

2.4 Chaos in difierential and difierence equations.The concept of an attractor 22

2.5 Interaction.or the interplay between concentration or reaction and diffusion 36

2.6 Discrete and continuous systems.The Poincaré return map 41

2.7 Stability and bifurcations;generic properties 42

2.8 The Hopf bifurcation 46

2.9 Lotka-Volterra equations 48

2.10 Stable,unstable,and center manifolds 52

3 Discrete invariants of dynamical systems 61

3.1 The topology of graphs 61

3.2 Floer homology 62

3.3 Conley theory:examples and results 70

3.4 Cohomological Conley index 77

3.5 Homotopical invariants 80

3.6 Continuation properties of the Conley index 92

3.7 The discrete Conley index 93

4 Entropy and topological aspects of dynamical systems 99

4.1 The entropy of a process as an asymptotic quantity 99

4.2 Positive entropy and chaos 103

4.3 Symbolic dynamics 108

5 Entropy and metric aspects of dynamical systems 111

5.1 The metric approach to topological entropy 111

5.2 Complexity and intrinsic scales 114

6 Entropy and measure theoretic aspects of dynamical systems 119

6.1 Probability spaces and measure preserving maps 119

6.2 Ergodicity 121

6.3 Entropy and information 126

6.4 Invariant measures 138

6.5 Stochastic proeesses 142

6.6 Stochastic bifurcations 149

7 Smooth dynamical systems 153

7.1 Lyapunov exponents 153

7.2 Hyperbolicity 156

7.3 Information loss 165

8 Cellular automata and Boolean networks as examples of discrete dynamical systems 169

8.1 Cellular automata 169

8.2 Boolean networks 175

References 181

Index 185