Smooth dynamical systems are systems whose evolution is described by differential equations that depend continuously on time. Differential equations provide the basis for describing phenomenon observed in many areas of science, for example, biology, meteorology, astronomy and engineering. In the majority of applications, the underlying equations are inherently nonlinear. Nonlinear differential equations do not, in general, have closed-form solutions. The behaviour of the solutions can be rather complex, even unpredictable! However, the solutions can be characterised using an alternative approach to determine their long term qualitative behaviour. For example, do solutions approach a steady state, are they periodic in time or could they be chaotic?

This module will introduce the tools used to determine the qualitative behaviour of solutions to nonlinear ordinary differential equations, focusing on the existence and stability of steady states and periodic orbits in two-dimensional phase space. There will also be an introduction to bifurcation theory which describes how the structure of the solutions can change as parameters in the system are varied. In nonlinear systems, small changes in a parameter value (e.g. temperature) can lead to sudden changes in the phase space topology and therefore the behaviour of the system. We will explore the dynamics of systems near simple bifurcations.

If time allows, further topics will be discussed. These may include the effect of symmetries in systems of nonlinear ordinary differential equations or dynamics and bifurcations and chaos in discrete time dynamical systems.

This course provides an introduction to the concepts and techniques of perturbation analysis and its applications. Such techniques are important in almost every branch of applied mathematics especially those where exact analytic solutions are not available and numerical solutions are difficult to obtain. The course will be taught as a `methods' based course with practical examples to demonstrate the use of these methods.

Learning Outcomes

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

analyse simple dynamical systems to find and classify regular behaviour, sketch phase portraits and determine stable and unstable manifolds.

identify and classify codimension one bifurcations of fixed points and sketch bifurcation diagrams.

use the Centre Manifold Theorem to reduce the order of a system appropriately and bring the reduced system into normal form.

Demonstrate an understanding of the methods used to obtain asymptotic expansions of algebraic equations.

Demonstrate an understanding of the methods used to obtain asymptotic expansions of integrals.

Demonstrate an understanding of the methods used to obtain asymptotic expansions of differential equations, with solutions characterised by either boundary layers or rapid oscillations.