Module Identifier | MAM4620 | ||

Module Title | NONLINEAR DIFFERENTIAL EQUATIONS 2 | ||

Academic Year | 2001/2002 | ||

Co-ordinator | Professor N Lloyd | ||

Semester | Semester 1 | ||

Pre-Requisite | MA31210 | ||

Course delivery | Lecture | 20 x 1hour lectures | |

Seminars / Tutorials | 7 x 1hour seminars | ||

Assessment | Exam | 2 Hours (written examination) | 100% |

Resit assessment | 2 Hours (written examination) | 100% |

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.

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.

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.

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.

D K Arrowsmith & C M Place. (1982)

D W Jordan & P Smith. (1987)

J Guckenheimer & P Holmes. (1983)

M Braun. (1995)