Module Identifier MAM4620  
Academic Year 2007/2008  
Co-ordinator Dr Robert J Douglas  
Semester Intended for use in future years  
Next year offered N/A  
Next semester offered N/A  
Pre-Requisite MA31210  
Course delivery Lecture   20 x 1hour lectures  
  Seminars / Tutorials   7 x 1hour seminars  
Assessment TypeAssessment Length/DetailsProportion
Semester Exam2 Hours (written examination)  100%
Supplementary Assessment2 Hours (written examination)  100%

Learning outcomes

On completion of this module, a student should be able to:
1. describe the properties of limit sets;
2. define a flow and determine the limit sets of orbits in certain circumstances;
3. use the Poincare-Bendixson theorem to study the existence of limit cycles of two-dimensional systems and obtaining information about their location;
4. construct possible global phase-portraits of two-dimensional systems including those with limit cycles;
5. describe the concept of stability and use the technique of Liapunov functions to determine the stability properties of a critical point;
6. construct the phase-portrait of some three-dimensional systems.

Brief 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.


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.


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.

Reading Lists

** Should Be Purchased
D K Arrowsmith & C M Place (1982) Ordinary differential equations Chapman & Hall 0412226103
** Recommended Text
D W Jordan & P Smith (1987) Nonlinear differential ordinary equations 2nd. Oxford University Press 0198596561
** Supplementary Text
J Guckenheimer & P Holmes (1983) Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields Springer 3540908196
M Braun (1995) Differential Equations and their Applications 4th. Springer 3540978941


This module is at CQFW Level 7