Tuesdays 14-16 and Thursdays 14-16. Room S4, Department of Mathematics. First lecture: January 20, 2004.
The course will be lectured by Alain Schenkel (room 421). The course assistant is Kalle Kytölä (room 420).
The exercise sessions are Thursdays 16-18 (after the lecture) in room S3, Department of Mathematics. If you find the time inconvenient, you can leave written solutions to my mailbox (with label Kytölä) on the 5th floor by the time of the exercise class.
The theory of dynamical systems is one of the major achievements of the previous century in Mathematical Physics. While the invention of differential calculus put on firm grounds classical mechanics, many important natural phenomena remained untractable by strict analytical methods. A shift towards a geometrical and topological viewpoint lead physicists and mathematicians to develop new and powerful techniques capable of addressing some of the complicated time evolutions occuring in nature. This modern approach constitutes the theory of dynamical systems. It uncovered profound and sometimes unexpected results, such as the intrinsic unpredictability of certain deterministic systems.
The present course is meant as an introduction to the theory of dynamical systems. Concepts such as chaos and strange attractors will be given a precise mathematical description. While concepts and results will be treated rigorously for the simplest class of dynamical systems (discrete-time dynamical systems), their relevance to problems arising in natural sciences will be stressed.
Hyperbolicity, stable and unstable manifolds. Symbolic dynamics, topological conjugacy. Chaos, strange attractors. Bifurcation theory. The period doubling route to chaos, Feigenbaum universality. Ergodic theory.
The course is intended for undergraduate students of mathematics, physics, and biology. Prior courses in advanced calculus and linear algebra are required (Diff.Int. 1-2 and Lineaarialgebra 1, or Mapu 1-2).
There exists a vast literature on dynamical systems. Below is a selected list relevant for the course. The following three books provide nice introductions to the mathematics and physics of dynamical systems and chaos. The first treats only discrete-time dynamical systems and is specifically aimed at mathematicians, whereas the third is aimed at physicists.
