2023/24 Undergraduate Module Catalogue

MATH3397 Nonlinear Dynamics

15 creditsClass 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.

This module is mutually exclusive with

MATH5398M

Advanced Nonlinear Dynamics

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	1.00	33.00
Private study hours			117.00
Total Contact hours			33.00
Total hours (100hr per 10 credits)			150.00

Opportunities for Formative Feedback

Regular examples sheets

Methods of assessment

Exams

Exam type	Exam duration	% of formal assessment
Open Book exam	2 hr 30 mins	100.00
Total percentage (Assessment Exams)		100.00

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: 18/08/2023

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