|Module Title||NONLINEAR DIFFERENTIAL EQUATIONS 1|
|Co-ordinator||Professor N G Lloyd|
|Course delivery||Lecture||19 x 1hour lectures|
|Seminars / Tutorials||3 x 1hour example classes|
|Assessment||Exam||2 Hours (written examination)||100%|
|Resit assessment||2 Hours (written examination)||100%|
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.
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.
On completion of this module, a student should be able to:
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.
** Should Be Purchased
D K Arrowsmith and C M Place. Nonlinear Ordinary Differential Equations. 2nd edition. Chapman & Hall
** Recommended Text
D W Jordan & P Smith. Nonlinear Ordinary Equations. Oxford University Press
** Supplementary Text
J Guckenheimer & P Holmes. Nonlinear Oscillations, Dynamical Systems & Bifurcations of Vector Fields. Springer
M Braun. Differential Equations and their Applications. Springer