MATH3397 Nonlinear Dynamics
15 Credits Class Size: 60
Module manager: Dr Jitse Niesen
Email: J.Niesen@leeds.ac.uk
Taught: Semester 1 (Sep to Jan) View Timetable
Year running
2023/24
Pre-requisite qualifications
MATH2391 or equivalent.
Mutually Exclusive
This module is not approved as a discovery module
Module summary
This module extends the study of nonlinear dynamics begun in MATH2391, and includes an in-depth study of bifurcation theory for systems of ordinary differential equations. Bifurcations occur when the structure of solutions change suddenly as a parameter is varied. Bifurcation theory has important consequences for many areas of science and engineering, where it is undesirable for small perturbations, for example due to noise, to have a large effect on solution behaviour.
Objectives
In this module you will develop tools for analysing a wide range of systems of nonlinear differential equations where explicit solutions are not available.
Learning outcomes
On completion of this module, students should be able to:
1. Use linearisation to determine the stability of fixed points in systems of nonlinear ODEs;
2. Define the stable and unstable manifolds of a fixed point;
3. Define what is meant by a hyperbolic fixed point;
4. State and apply the Routh-Hurwitz criteria to two and three dimensional systems of ODEs;
5. Identify codimension-one and two bifurcations in ODEs of arbitrary order;
6. Sketch bifurcation diagrams in one and two parameters;
7. Transform a nonlinear ODE with a bifurcation into its normal form;
8. Compute the extended centre manifold of systems of ODEs.
Syllabus
1. Definitions and terminology
2. Sketching phase-portraits and one-dimensional bifurcation diagrams (Saddle-node, Transcritical, Pitchfork)
3. Topological equivalence, local and global bifurcations
4. Bifurcations in n-dimensions, Jordan normal form
5. Routh-Hurwitz criteria in two and three dimensions
6. Hyperbolicity, Hartman-Grobman theorem, stable and unstable manifolds
7. Generic bifurcations, structural stability
8. Centre manifolds and extended centre manifolds
9. Codimension two Bogdanov-Takens bifurcation
and one or more of the following topics:
9. Turing instability and pattern formation
10. Poincare-Lindstedt theory
11. Bifurcations with symmetries
12. Applications
13. Numerical methods for continuation
Teaching Methods
| Delivery type |
Number |
Length hours |
Student hours |
| Lecture |
33 |
1 |
33 |
| Private study hours |
117 |
| Total Contact hours |
33 |
| Total hours (100hr per 10 credits) |
150 |
Opportunities for Formative Feedback
Regular examples sheets
Exams
| Exam type |
Exam duration |
% of formal assessment |
| Open Book exam |
2.0 Hrs 30 Mins |
100 |
| Total percentage (Assessment Exams) |
100 |
Normally resits will be assessed by the same methodology as the first attempt, unless otherwise stated
Reading List
The reading list is available from the Library website
Last updated: 8/18/2023
Errors, omissions, failed links etc should be notified to the Catalogue Team