Chaos and Non-Linear Phenomena

Aim

The aim of the course is that the students should get an orientation in the special properties of non-linear systems of theoretical and practical interest, and also an orientation in some topical areas of research. After finishing the course the student knows how to:

• explain basic concepts in non-linear physics such as chaos, fractal dimensions, integrability, solitons and quantum chaos, and exemplify important applications of these concepts in different scientific disciplines.
• model simple non-linear dynamical systems mathematically, and demonstrate some of their characteristic properties by analyzing bifurcations and calculating Lyapunov-exponents and fractal dimensions for their attractors.
• explain mechanisms for appearance of chaos in conservative systems and properties of the corresponding quantum mechanical systems, and solve problems by applying the theory for Hamiltonian systems on simple models.
• derive soliton solutions in special mathematical models for wave motion in non-linear media, and exemplify their physical significance.

Content

Introduction. Experiments and simple models. Vibrations in mechanical systems and electrical circuits. Piecewise linear maps and deterministic chaos. Universal behavior of quadratic maps. Bifurcations. Poincaré maps. Lyapunov exponents. Fractal dimensions. Period-doubling. Homoclinic and heteroclinic orbits. The intermittency route to chaos. Strange attractors in dissipative dynamical systems. The transition from quasiperiodicity to chaos. Regular and irregular motion of conservative systems. Integrable and non-integrable systems. Solitons and breathers with applications. Chaos in quantum systems.