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.
Level 7 Students only:
• Further applications of phase portraits and the Poincare-Bendixson theorem; Higher order difference equations.
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 7 module students will be able to:
1 demonstrate systematic understanding of qualitative analysis for nonlinear differential and difference equations;
2 demonstrate the capability to solve complex problems using a very good 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 a range of concepts and principles of nonlinear systems in loosely defined contexts, showing good judgment in the selection and application of tools and techniques;
4 make effective and well-considered use of Maple.
The intended generic learning outcomes. On successfully completing the level 7 module students will be able to:
1 work competently and independently, be aware of their own strengths and understand when help is needed;
2 demonstrate a high level of capability in developing and evaluating logical arguments;
3 communicate arguments confidently with the effective and accurate conveyance of conclusions;
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 effective use of information technology skills such as online resources (Moodle), internet communication;
7 communicate technical material effectively;
8 demonstrate an increased level of skill in numeracy and computation;
9 demonstrate the acquisition of the study skills needed for continuing professional development.