Introductory description

Dynamical Systems is one of the most active areas of modern mathematics. This course will be a broad introduction to the subject and will attempt to give some of the flavour of this important area.

Module web page

Module aims

Outline syllabus

This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.

We will cover some of the following topics:

1. Circle homeomorphisms and minimal homeomorphisms.
2. Expanding maps and Julia sets,
3. Horseshoe maps, toral automorphisms and other examples of hyperbolic maps.
4. Structural stability, shadowing, closing lemmas, Markov partitions and symbolic dynamics.
5. Conjugacy and topological entropy.
6. Strange attractors.

Learning outcomes

By the end of the module, students should be able to:

• Students who successfully complete the module will have developed an understanding of the modern qualitative theory of dynamical systems. This will include:a) familiarity with the basic concepts of topological dynamics, particularly minimality, topological transitivity, topological mixing and topological conjugacy and the ability to demonstrate that these properties are satisfied in appropriate examples;b) a familiarity with a range of examples of dynamical systems, including circle homeomorphisms, expanding maps, total automorphisms, horseshoes, solenoids, subshifts of finite type, and the relation of these examples to the concepts of hyperbolicity and structural stability;c) understanding of the concept of symbolic dynamical models and the proofs that such models exist in certain examples (e.g. the doubling maps, expanding circle maps, horseshoes)d) knowledge of the definition of topological entropy and the role it plays as an invariant of topological conjugacy, calculation of topological entropy in examples.

Subject specific skills

Students who have successfully taken the module will be equipped to use a variety of mathematical techniques to investigate and understand a range of dynamical systems exhibiting complex behaviour. They will have the background to carry of further study in the area, at graduate level, and to apply the techniques they have learned to various areas of applications where dynamical systems appear, such as control systems, engineering and meteorology. More generally, they will have had the opportunity to develop their analytic skills through the study of complex and abstract systems.

Transferable skills

Students who have successfully taken the module will be equipped to use a variety of mathematical techniques to investigate and understand a range of dynamical systems exhibiting complex behaviour. They will have the background to carry of further study in the area, at graduate level, and to apply the techniques they have learned to various areas of applications where dynamical systems appear, such as control systems, engineering and meteorology. More generally, they will have had the opportunity to develop their analytic skills through the study of complex and abstract systems.