## MATH3397 Nonlinear Dynamics

### 15 creditsClass Size: 60

Module manager: Dr Jon Ward
Email: j.a.ward@leeds.ac.uk

Taught: Semester 1 View Timetable

Year running 2019/20

### Pre-requisite qualifications

MATH2391 or equivalent.

### This module is mutually exclusive with

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 Standard exam (closed essays, MCQs etc) 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