Lecture: Nonlinear Dynamics I/II

Wintersemester 2004/2005

Jürgen Kurths und Marco Thiel (V), Udo Schwarz (Ü + Comp-Sem)

1. Introduction

2. Simple and basic examples

   1. Tent map

   2. Logistic map and discrete population models

   3. Lorenz model for atmospheric convection and for lasers

3. Local bifurcations

   1. Qualitative properties

   2. Saddle-node bifurcation

   3. Transcritical bifurcation

   4. Pitchfork bifurcation

   5. Hopf bifurcation

4. Basic terms for dissipative systems

   1. Flows in phase space

   2. Attractors

   3. Measures

5. Quantitative description of nonlinear dynamics

   1. Deterministic chaos und Lyapunov-exponents

   2. Self-similarity and fractal dimensions

   3. Limited predictability and entropies

   4. Recurrence

   5. Routes to chaos

6. Synchronization phenomena of coupled chaotic systems

   1. Complete synchronization

   2. Phase synchronization

7. Nonlinear time series analysis

   1. Embedding

   2. Recurrence analysis

   3. Fractal dimensions

   4. Lyapunov exponents

8. Complex Networks

8.1 Random networks

8.2 Small-world networks

8.3 Scale-free networks

9. Constructive influences of noisy fluctuations

   1. Noise-induced phase transitions

   2. Stochastic resonance

10. Distributed nonlinear systems

10.1 Dynamics in lattices of coupled maps

10.2 Spatio-temporal chaos

11. Spatio-temporal pattern formation

11.1. Symmetry breaking

11.2. Turing instabilities in reaction diffusion equations

11.3. Bifurcations in the Be´nard problem

11.4. Bifurcation analysis of the Ginzburg-Landau equation

Literatur

• Alligood, K.T., Sauer, T., Yorke, J.: Chaos, Springer, 1996.

• Anishchenko, V., Astakhov, V., Neiman, A., Vadivasova, T., Schimansky-Geier, L.: Nonlinear Dynamics of Chaotic and Stochastic Systems, Springer, 2002.

• Leven, R., Pompe, B., Koch, B.: Chaos in dissipativen Systemen, Akademie-Verlag, VCH Verlagsges., 1989

• Murray, J. D.: Mathematical Biology, Springer, 2002

• Nicolis, G.: Introduction to Nonlinear Science, Cambridge University Press, 1995.

• Ott, E.: Chaos in Dynamical Systems, Cambridge University Press, 1993.

• Pikovsky, A., Rosenblum, M., Kurths, J.: Synchronization - A universal concept in nonlinear sciences, Cambridge University Press, 2001.

• Strogatz, S.H.: Nonlinear Dynamics and Chaos, Addison-Wesley Publ., 1996.

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