Dynamical Systems

The course gives an introduction to dynamical systems with an emphasis on analytical and qualitative methods for linear and nonlinear ordinary differential equations. Topics covered: Linear and nonlinear systems, existence and uniqueness, continuous dependence, phase plane analysis, equilibria, limit cycles, stability, Lyapunov's direct method, index theory, the Poincaré-Bendixson theorem, and examples of applications. Additional topics that may change from year to year may include bifurcations, discrete dynamical systems and chaos, topics from calculus of variations and classical mechanics.

1. Knowledge: The student knows basic concepts and methods from the theory of dynamical systems, including analytical and geometrical techniques for the study of qualitative properties of solutions. In particular, the student is familiar with linear and nonlinear systems, existence and uniqueness, continuous dependence, phase plane analysis, equilibria, limit cycles, stability, Lyapunov's direct method, index theory, the Poincaré-Bendixson theorem, the additional topics and examples of applications.

2. Skills: The student is able to apply his or her knowledge to the study of concrete examples. The student masters central techniques of proof and is able to apply these to related problems.

Lectures and exercises.