Jürgen Jost
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Max-Planck-Institut für Mathematik in den Naturwissenschaften

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Dynamical Systems. Examples of Complex Behaviour

Springer, 2005

Table of Contents

1. Introduction

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

2.1 Some general notions
2.2 Autonomous systems of ODEs
2.3 Examples: Bifurcation depending on a parameter λ ∈ R
2.4 Chaos in differential and difference equations. The concept of an attractor
2.5 Interaction, or the interplay between concentration or reaction and diffusion
2.6 Discrete and continuous systems. The Poincar'e return map
2.7 Stability and bifurcations; generic properties
2.8 The Hopf bifurcation
2.9 Lotka-Volterra equations
2.10 Stable, unstable, and center manifolds

3. Discrete invariants of dynamical systems

3.1 The topology of graphs
3.2 Floer homology
3.3 Conley theory: examples and results
3.4 Cohomological Conley index
3.5 Homotopical invariants
3.6 Continuation properties of the Conley index
3.7 The discrete Conley index

4. Entropy and topological aspects of dynamical systems

4.1 The entropy of a process as an asymptotic quantity
4.2 Positive entropy and chaos
4.3 Symbolic dynamics

5. Entropy and metric aspects of dynamical systems

5.1 The metric approach to topological entropy
5.2 Complexity and intrinsic scales

6. Entropy and measure theoretic aspects of dynamical systems

6.1 Probability spaces and measure preserving maps
6.2 Ergodicity
6.3 Entropy and information
6.4 Invariant measures
6.5 Stochastic processes
6.6 Stochastic bifurcations

7. Smooth dynamical systems

7.1 Lyapunov exponents
7.2 Hyperbolicity
7.3 Information loss

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

8.1 Cellular automata
8.2 Boolean networks

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