Module and Programme Catalogue

20 creditsClass Size: 30

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

Taught: Semester 1 (Sep to Jan) View Timetable

Year running 2021/22

Pre-requisite qualifications

MATH2391 or equivalent.

This module is mutually exclusive with

 MATH3397 Nonlinear Dynamics

This module is approved as an Elective

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. In this module you will develop tools for analysing a wide range of systems of nonlinear differential equations where explicit solutions are not available.

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;
9. Describe an advanced topic in the theory of continuous dynamical systems, for example: global bifurcations, chaotic dynamics, bifurcations in systems with symmetry, non-smooth dynamical systems.

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
10. Elementary bifurcations in maps
11. Homoclinic bifurcations

and one or more of the following topics:
12. Turing instability and pattern formation
13. Poincare-Lindstedt theory
14. Bifurcations with symmetries
15. Applications
16. Numerical methods for continuation

Teaching methods

Due to COVID-19, teaching and assessment activities are being kept under review - see module enrolment pages for information

 Delivery type Number Length hours Student hours Lecture 44 1.00 44.00 Private study hours 156.00 Total Contact hours 44.00 Total hours (100hr per 10 credits) 200.00

Private study

156 hours of private study:
- example sheets,
- revision for the exam.

Opportunities for Formative Feedback

There are five example sheets containing a mixture of pedagogical questions (feedback is given but the questions are not included in the assessment). The progress in the projects is discussed within the workshops.

Methods of assessment

Due to COVID-19, teaching and assessment activities are being kept under review - see module enrolment pages for information

Coursework
 Assessment type Notes % of formal assessment Project Report and Presentation 40.00 Total percentage (Assessment Coursework) 40.00

Normally resits will be assessed by the same methodology as the first attempt, unless otherwise stated

Exams
 Exam type Exam duration % of formal assessment Standard exam (closed essays, MCQs etc) 2 hr 00 mins 60.00 Total percentage (Assessment Exams) 60.00

Normally resits will be assessed by the same methodology as the first attempt, unless otherwise stated