Module Identifier MA31210
Module Title NONLINEAR DIFFERENTIAL EQUATIONS 1
Co-ordinator Professor N Lloyd
Semester Semester 2
Pre-Requisite MA30210
Course delivery Lecture   19 x 1 hour lectures
Seminars / Tutorials   3 x 1 hour example classes
Assessment
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:
• interpret conditions for the existence and uniqueness of solutions of autonomous ordinary differential equations;
• explain what is meant by the maximal interval of existence of a solution;
• classify the critical points of one-dimensional systems;
• classify the critical points of linear two-dimensional systems;
• locate and classify the critical points of two-dimensional nonlinear systems;
• sketch possible phase portraits of two-dimensional nonlinear systems;
• describe simple ecological models and draw appropriate conclusions;
• explain the significance of the divergence of the vector field;
• determine the index of given paths and critical points;
• show how the possibility of limit cycles may be excluded under certain circumstances.

#### Brief description

A wide variety of phenomena can be modelled by means of ordinary differential equations. Very few such equations can be solved explicitly, and in the qualitative theory of differential equations methods have been developed to determine the behaviour of solutions directly from the equation itself. The subject was pioneered in the early part of the twentieth century by Poincare and then by Liapunov. The qualitative information obtained is used in conjunction with numerical methods and validates them. This module provides a thorough grounding in the theory of dynamical systems and nonlinear differential equations.

#### Aims

To provide an introduction to the qualitative theory of nonlinear differential equations, with particular emphasis on the construction of phase portraits of two-dimensional systems and applications.

#### Content

1. Existence and uniqueness of solutions; autonomous and non-autonomous systems.
2. One-dimensional systems; maximum interval of definition of solutions.
3. Two-dimensional linear systems: classification of critical points.
4. Critical points of two-dimensional nonlinear systems: location, classification; construction of possible phase portraits.
5. Modelling by means of two-dimensional nonlinear systems, eg predator-prey models, harvesting, chemostat.
6. Index: definition, calculation of the index of paths, index of critical points.
7. Limit cycles: consequences of index and the Poincare-Bendixson theorem.
8. The Bendixson criterion for the absence of limit cycles.

Books
** Should Be Purchased
P Glendinning (1994) Stability, instability and chaos : an introduction to the theory of nonlinear differential equations CUP 0-521-42566-2
** Recommended Text
D K Arrowsmith and C M Place (1982) Ordinary differential equations Chapman & Hall 0412226103
J Berry (1992) Introduction to Nonlinear Systems Arnold 0-340-67700-7
D W Jordan & P Smith (1987) Nonlinear ordinary differential equations 2nd. Oxford University Press 0198596561
** Supplementary Text
J Guckenheimer & P Holmes (1983) Nonlinear oscillations, dynamical systems & bifurcations of vector fields Springer 3540908196

#### Notes

This module is at CQFW Level 6