The course consists of a detailed description of continuous and discrete dynamical systems. We shall combine the introduction to the general theory with the consideration of bifurcations and chaos, the most important subtopics. The analysis of appropriate mechanical, physical, economic and biological models is an essential part of almost every lecture of the course. To support the course numerical and computational toolbox will be used.
For further information see the academic catalog: IAM529
Students will acquire the ability to recognize and investigate the dynamics of the natural and social processes, applying differential equations and mappings.
Basics of discrete dynamical systems: mappings, time series, orbits, fixed points. The theory of flows, vector fields, equilibrium solutions, invariant manifolds, periodic solutions. Poincare-Bendixson theorem, Poincare maps. Stability, Lyapunov functions. Periodic attractors. quasiperiodic solutions, bifurcations, center manifold reduction; structural stability, chaos.
- Steven H. Strogatz, Nonlinear Dynamics and Chaos: with applications to Physics, Biology, Chemistry, and Engineering, Perseus Books Publishing, 1994
- Stephen Lynch, Dynamical Systems with Applications using MATLAB, Birkhäuser, 2004
- James D. Meiss, Differential Dynamical Systems, SIAM, 2007