*Instructor*: Dr. Péter SIMON

*Text*: C. Chicone, Ordinary Differential Equations with
Applications or 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, implicit function theorem).

*Course description:* This course provides an introduction into
ordinary differential equations, dynamical systems and bifurcation theory.
We show methods to solve certain equations, then the existence and uniqueness
of solutions is dealt with. The second part of the course is devoted to
the modern qualitative theory of dynamical systems and to bifurcation theory.

*Topics:*

Methods for solving differential equations.

Existence and uniqueness theory, Gronwall’s lemma, global solution.

Linear differential equations, the exponential of matrices.

Autonomous differential equations, dynamical systems.

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

Elementary bifurcations: saddle-node, Hopf.

*Special topics on request:*

One dimensional maps, symbolic dynamics.

Period doubling bifurcation, chaos.

Smale horseshoe.

Lorenz attractor.

Stability theory, Liapunov’s direct method.

Poincare-Bendixson theorem.

Higher order linear differential equations, boundary value problems.

Structural stability.