General description
This module is a sequel to MA31210 and develops the ideas to a more advanced level. The Poincare-Bendixson theory for two-dimensional systems is explained in detail, with emphasis on examples. Stability in the sense of Liapunov is studied and used in the investigation of three-dimensional systems.

Aims
To develop the fundamental ideas of the qualitative theory of differential equations introduced in MA31210. Particular emphasis will be placed on Poincare-Bendixson theory, Liapunov stability and the investigation of examples which arise in applications.

Learning outcomes
On completion of this module, a student should be able to:

• describe the properties of limit sets;
• define a flow and determine the limit sets of orbits in certain circumstances;
• use the Poincare-Bendixson theorem to study the existence of limit cycles of two-dimensional systems and obtaining information about their location;
• construct possible global phase-portraits of two-dimensional systems including those with limit cycles;
• describe the concept of stability and use the technique of Liapunov functions to determine the stability properties of a critical point;
• construct the phase-portrait of some three-dimensional systems.

Syllabus
1. Continuous dependence of solutions on initial conditions.
2. Flows on R^{n}: limit sets and their properties.
3. Poincare-Bendixson theory: the use of index arguments, the divergence criterion and the Poincare-Bendixson theorem to construct possible phse-portraits of two-dimensional systems.
4. Stability: Liapunov stability, asymptotic stability, global asymptotic stability; Liapunov functions, Zhubov's theorem; examples in R^{3}.
5. Three-dimensional systems.