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%  

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:

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.

Reading Lists

Books
** 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