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2023/24 Taught Postgraduate Module Catalogue
MATH5398M Advanced Nonlinear Dynamics
20 creditsClass Size: 30
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
| 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
| 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:- lecture reading and preparation,
- 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
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 |
| Open Book exam | 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
Reading list
The reading list is available from the Library websiteLast updated: 18/08/2023
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- Undergraduate module catalogue
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