Dynamical Systems

Overview

Continuous dynamical systems
- Fundamental theory: existence, uniqueness and parameter dependence of solutions;
- Linear systems: constant coefficient systems and the matrix exponential; nonautonomous linear systems; periodic linear systems.
- Topological dynamics: invariant sets; limit sets; Lyapunov stability.
- Grobman-Hartman theorem.
- Stable, unstable and centre manifolds.
- Periodic orbits: Poincare-Bendixson theorem.
- Bifurcations
- Applications: the Van der Pol oscillator; the SIR compartmental model; the Lorenz system.
Discrete dynamical systems
- One-dimensional dynamics: the discrete logistic model; chaos; the Cantor middle-third set.

Learning Objectives

It is intended that students shall, on successful completion of the module: have a good understanding and ability to use the basics of dynamical systems; be able to perform computations pertaining to problems in these areas; have reached a good level of skill in manipulating basic and complex questions within this framework, and be able to reproduce, evaluate and extend logical arguments; be able to select suitable tools to solve a problem, and to communicate the mathematical reasoning accurately and confidently.

Skills

Analytic argument skills, computation, manipulation, problem solving, understanding of logical arguments.

Assessment

None