Introduction to Dynamical Systems

• Determine phase portraits for linear ordinary differential equations.
• Analyze stability in systems.
• Determine and exploit hyperbolicity in the analysis of equilibrium points.
• Operate with stable, unstable and center manifolds.
• Topological equivalence and index theory.
• Apply Poincaré-Bendixon theorem and other results to decide the existence of limit cycles.
• Classify local bifurcations in general and find the possible local bifurcations in a specific system.
• Apply Hartman-Grobman theorem about linearisation to give a qualitative description of the dynamics in a vicinity of a hyperbolic equilibrium.
• Combine the aforementioned points to give a global phase portrait of certain dynamical systems.
• Poincaré maps and chaos.

Phase portraits of linear systems. Stability analysis including the Hartman-Grobman theorem. Stable, unstable and center manifolds. Local bifurcation theory. Theory of planar systems including Poincaré-Bendixon theorem. The mathematical techniques will be applied to problems from physics, chemistry, and biology.

01025/01034/01035/01037, Basic knowledge of linear algebra and linear differential equations

Lectures and classes.