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, 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.