For delivery to students completing Stage 1 before September 2016:
Pre-requisite: MA552 (Analysis) and MA553 (Linear Algebra)
Recommended: MA590 (Mathematical Modelling)
For delivery to students completing Stage 1 after September 2016:
Pre-requisite: MAST4006 (Mathematical Methods 1); MAST4007 (Mathematical Methods 2); MAST4004 (Linear Algebra) or MAST4005 (Linear Mathematics)
Recommended: MAST5012 (Ordinary Differential Equations)
Overview• Scalar autonomous nonlinear first-order ODEs. Review of steady states and their stability; the slope fields and phase lines.
• Autonomous systems of two nonlinear first-order ODEs. The phase plane; Equilibra and nullclines; Linearisation about equilibra; Stability analysis; Constructing phase portraits; Applications. Nondimensionalisation.
• Stability, instability and limit cycles. Liapunov functions and Liapunov's theorem; periodic solutions and limit cycles; Bendixson's Negative Criterion; The Dulac criterion; the Poincare-Bendixson theorem; Examples.
• Dynamics of first order difference equations. Linear first order difference equations; Simple models and cobwebbing: a graphical procedure of solution; Equilibrium points and their stability; Periodic solutions and cycles. The discrete logistic model and bifurcations.
This module appears in:
Method of assessment
80% examination, 20% coursework
Jordan, J. W., and Simth, P., Nonlinear Ordinary Differential Equations: an introduction for scientists and engineers, Oxford University Press, Fourth Edition, 2007
Elaydi, S., An introduction to difference equations, Springer, 1999
Murray, J. D., Mathematical Biology I: An Introduction, Springer, Third Edition, 2002
Glendinning, P. A., Stability, Instability and Chaos: An Introduction to the Qualitative Theory of Differential Equations, Cambridge University Press, 1994
Kaplan, D., and Glass, L., Understanding Nonlinear Dynamics, Springer, 1995
The intended subject specific learning outcomes. On successfully completing the level 6 module students will be able to:
1 demonstrate systematic understanding of key aspects of introductory nonlinear systems;
2 demonstrate the capability to deploy established approaches accurately to analyse and solve problems using a reasonable level of skill in calculation and manipulation of the material in the following areas: equilibra for both nonlinear differential and difference equations and their stability, phase portraits, the existence of limit cycles;
3 apply key aspects of nonlinear systems in well-defined contexts, showing judgement in the selection and application of tools and techniques;
4 show judgement in the selection and application of Maple.
The intended generic learning outcomes. On successfully completing the level 6 module students will be able to:
1 manage their own learning and make use of appropriate resources;
2 understand logical arguments, identifying the assumptions made and the conclusions drawn;
3 communicate straightforward arguments and conclusions reasonably accurately and clearly;
4 manage their time and use their organisational skills to plan and implement efficient and effective modes of working;
5 solve problems relating to qualitative and quantitative information;
6 make competent use of information technology skills such online resources (Moodle), internet communication;
7 communicate technical material competently;
8 demonstrate an increased level of skill in numeracy and computation;
9 demonstrate the acquisition of the study skills needed for continuing professional development.