Chaos and dynamical systems

These lectures concern deterministic dynamical systems, i.e. systems governed by coupled differential equations. Most dynamical problems in physics and elsewhere can be modeled that way. Interesting dynamics occur when the governing equations are nonlinear which is the usual case. In contrast to linear systems, nonlinear ones generally have multiple solutions of different qualitative nature, for instance different symmetries. The stability of these solutions is modified at bifurcation points when a control parameter of the system is changed. We will show that universal behaviors are observed in the vicinity of these bifurcations mostly dependent on symmetries. We will then show how a sequence of successive bifurcations can lead to chaos, i.e. a random dynamical behavior although the system is governed by deterministic rules.