Please note that this module can be taken by 4th year students as a 10 credit module with the module code 0590301.

Aims

Dynamical systems describe the time evolution of systems which arise from physics, biology, chemsitry and other areas. As mathematical objects they are ordinary differential equations, usually nonlinear and therefore not usually able to be explicitly solved. The aim of the course is to see how to make a qualitative analysis of a dynamical system using many different analytic tools. By the end of the course students should be able to analyse planar systems to understand their global dynamics and how these might change as parameters of the system are varied.

Learning objectives

At the end of the module you should be able to...

• To understand the difference between linear and nonlinear systems, and the concept of local linearisation.
• To appreciate the difference between qualitative and quantitative analysis of dynamical systems.
• To be able to apply a range of qualitative techniques to the study of planar dynamical systems.
• To understand some possibilities of changes in dynamics in families of planar systems.

Syllabus

• Planar linear systems: explicit solution, classification diagram and phase portraits.
• Planar non-linear systems: fixed points, stability and linearisation, local and global phase portraits, conservative systems and ``energy'', damped systems and Liapunov functions, index theory, limit cycles and the Poincare-Bendixson theorem.
• Bifurcations of planar systems: saddle node, transcritical, pitchfork and Hopf bifurcations.

Recommended texts

• S H Strogatz, Nonlinear Dynamics and Chaos. Westview Press (Perseus), 1994 (York Library Code S7.38 STR).
• D K Arrowsmith and C M Place, An introduction to dynamical systems, Cambridge University Press, 1990. (S 7.38 ARR)

Teaching

• Autumn Term
• 27 hours, divided approximately into 21 lectures and 6 problems classes.

Assessment

Two hour closed examination in week 1 Spring Term.

Elective information

This module is not available as an elective.

Prerequisites

• Diagonalisation of 2x2 matrices Taught in: Matrices 
• General properties of second order linear systems Taught in: Differential Equations