Content

The following points will be illustrated by means of concrete examples from different disciplines.
1) Introduction
From linear to non-linear. Historical overview and examples.
2) Dynamic systems
* Study of one-dimensional differential systems:
- with straight line as state space: balance points and stability
- with the circle as state space: uniform and non-uniform oscillators, synchronization
* Study of two-dimensional differential systems
- Phase plane and phase portraits:
balance points, characterizing the nature of balance points through linearization and its conditions.
Special characteristics of phase portraits in conservative and reversible systems.
- Limit cycles and conditions for their existence.
Gradient systems. Lyapounov functions.
Poincaré-Bendixon theorem.
- Strange attractors (chaotic behaviour)
- Liénard systems and relaxation oscillators. Weak non-linear oscillators and pertubation theory.
3) Bifurcation analysis
- Concepts from bifurcation theory for parameter-dependent non-linear systems: bifurcations and their normal forms (including Hopf-bifurcation, homoclinic bifurcation), catastrophe theory
- Coupled oscillators: synchronization, quasi periodicity
- Poincaré pictures
4) Chaos theory
- Deterministic chaos, chaos in continuous time systems and in discrete time systems
- Lorenz' equation, discrete Lorenz' picture
- Logistic picture
- Roads to chaos: Lyapounov exponent, fractal dimension
5) Spatial pattern formation
- Pattern formation in cellular automatics
- Turing structures in reaction-diffusion problems
- Pattern formation in hydrodynamic problems
6) Numerical methods for continuation and bifurcation analysis
- Numerical continuation of solution branches
- Numerical methods for the calculation of periodical solutions and their stability (Monodromy matrix), calculating the Lyapounov exponents.
- Functionality and use of software packages for the analysis of dynamic systems and bifurcation analysis.

Course material

Study cost: 51-75 euros (The information about the study costs as stated here gives an indication and only represents the costs for purchasing new materials. There might be some electronic or second-hand copies available as well. You can use LIMO to check whether the textbook is available in the library. Any potential printing costs and optional course material are not included in this price.)

Handbook/articles and literature/toledo.

Content

Exercises and practical sessions with the lecture: Non-linear systems.

Course material

Handbook/articles and literature/toledo.