*Instructor*: Dr. Péter SIMON

*Text*: L. Perko, Differential Equations and Dynamical Systems

*Prerequisite*: Calculus; linear algebra (linear spaces, basis, matrix operations, eigenvalues, eigenvectors); multivariable calculus (differentiation of functions in several variables), solving elementary differential equations.

*Course description:* This course provides an introduction into ordinary differential equations, dynamical systems and bifurcation theory. First, we study the basic concepts of discrete and continuous time dynamical systems and their relation to differential equations. Then linear systems of differential equations and their phase portraits are considered. The second part of the course is devoted to the modern qualitative theory of dynamical systems and to bifurcation theory.

*Topics:*

Discrete and continuous time dynamical systems, flows and maps.

Linear differential equations, the exponential of matrices.

Construction of phase portraits of dynamical systems, equilibria and periodic orbits.

One dimensional maps, symbolic dynamics, period doubling bifurcation, chaos.

Topological classification of dynamical systems.

Elementary bifurcations: saddle-node, Hopf.

*Special topics on request:*

Smale horseshoe.

Lorenz attractor.

Stability theory, Liapunov's direct method.

Poincare-Bendixson theorem.

Higher order linear differential equations, boundary value problems.

Structural stability.