General 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 and its sequel, MA41610, provide a thorough grounding in the modern 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.

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.

Syllabus
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.
8. The Bendixson criterion for the absence of limit cycles.