Module manager: Alastair Rucklidge
Email: A.M.Rucklidge@leeds.ac.uk
Taught: Semester 2 (Jan to Jun) View Timetable
Year running 2026/27
None
| MATH2400 | Mathematical Modelling |
MATH3396 Dynamical Systems MATH3397 Nonlinear Dynamics MATH5395M Advanced Dynamical Systems MATH5398M Advanced Nonlinear Dynamics
This module is not approved as a discovery module
Many applications, ranging from biology to physics to climate science and engineering, are described by nonlinear dynamical systems, in which a change in output is not proportional to a change in input. Nonlinear dynamical systems can exhibit sudden changes in behaviour as parameters are varied and even unpredictable chaotic dynamics. This module will provide students with the mathematical tools to analyse nonlinear dynamical systems, including identifying bifurcations and chaotic dynamics.
In this module you will develop tools for analysing a wide range of systems of nonlinear differential equations. Unlike in linear systems, explicit solutions are usually not available, so the module will take a qualitative approach to understanding the behaviour of nonlinear dynamical systems. Learning activities will focus on developing an intuition for how nonlinear systems can behave in applications. The module will treat both continuous nonlinear dynamical systems (ordinary differential equations) and discrete dynamical systems (maps), as well as the connection between the two.
On successful completion of the module, students will be able to: 1. Identify and explain equilibria in nonlinear differential equations and fixed points in nonlinear maps. 2. Analyse the stability of equilibria and fixed points. 3. Plot phase portraits with multiple equilibria in phase space. 4. Identify and explain nonlinear phenomena such as periodic orbits. 5. Identify bifurcations and draw bifurcation diagrams. 6. 7. Perform centre manifold reductions at bifurcation points. 8. Describe methods to analyse chaotic dynamics.
1. Motivation, definitions, terminology and background material for ordinary differential equations (ODEs) and maps. 2. Continuous dynamical systems: nonlinear ODEs, equilibrium points, stability, phase portraits, stable and unstable manifolds, topological equivalence. 3. Bifurcation theory for nonlinear ODEs and the centre manifold. 4. Hopf bifurcation, periodic orbits and the Poincare map. 5. Discrete dynamical systems: nonlinear maps, fixed points, stability and bifurcation theory. 6. Chaos in maps: Lyapunov exponent, sensitive dependence on initial conditions, period-three implies chaos. 7. Chaos in nonlinear ODEs: global bifurcations and the connection between ODEs and maps. Additional topics that build on these may be covered as time allows. Such topics may be drawn from the following, or similar: The period-doubling cascade, renormalisation and Feigenbaum's constant. The Bogdanov-Takens bifurcation. Numerical methods for nonlinear dynamics and chaos. Fractals and strange attractors. Applications and examples: models of climate change, population dynamics and disease dynamics.
| Delivery type | Number | Length hours | Student hours |
|---|---|---|---|
| Lecture | 44 | 1 | 44 |
| Private study hours | 156 | ||
| Total Contact hours | 44 | ||
| Total hours (100hr per 10 credits) | 200 | ||
Examples sheets will be provided fortnightly throughout the module. Students will be encouraged to consider problems independently, in collaboration with their peers, or making use of online resources. One lecture session per week will be devoted to examples, from the lecture notes and from the examples sheets. Students will have opportunities for receiving feedback on their work. Model solutions will be provided.
Check the module area in Minerva for your reading list
Last updated: 30/04/2026
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