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2013/14 Taught Postgraduate Module Catalogue

MATH5398M Advanced Nonlinear Dynamics

20 creditsClass Size: 30

Module manager: Dr J Ward
Email: J.A.Ward@leeds.ac.uk

Taught: Semester 1 (Sep to Jan) View Timetable

Year running 2013/14

Pre-requisite qualifications

MATH2391 or equivalent.

This module is mutually exclusive with

MATH3397

Nonlinear Dynamics

Module replaces

MATH5397M

This module is approved as an Elective

Module summary

This course extends the study of nonlinear dynamics begun in MATH2391. An in-depth study of local and global bifurcation theory and the transition to chaos is made in the context of ordinary differential equations.The course also uses bifurcation theory to understand the formation of patterns: stripes, squares, hexagons and more complex patterns.

Objectives

On completion of this module, students should be able to:

1. Identify codimension-one and two bifurcations in ODEs of arbitrary order, use the Centre Manifold Theorem to reduce the order of the ODEs appropriately, bring the reduced system into normal form, and sketch one and two-parameter bifurcation diagrams;
2. Analyse the formation and stability of one and two-dimensional patterns;
3. Identify global bifurcations in second and third-order ODEs, and construct and analyse Poincare maps that describe the dynamics (including chaotic dynamics) near the global bifurcation.

Learning outcomes
The aim of this module is to introduce the theory of bifurcations in dissipative nonlinear systems, treating both local and global bifurcations, and exploring the role of global bifurcations in the transition to chaos.

The course will also include an introduction to pattern formation and equivariant bifurcation theory. As well as being relevant to fluid dynamics experiments, there are important applications in many fields, including biology, chemistry, astrophysics and geophysics.

Syllabus

1. Introduction to the theory of bifurcations in dissipative systems. The Centre Manifold Theorem. Birkhoff normal form transformations. Local bifurcations of codimension-one and two. Normal form symmetry.

2. Introduction to pattern formation and equivariant bifurcation theory, and the role of symmetry. Linear stability of pattern-forming systems. Normal forms for one and two-dimensional patterns.

3. Global (homoclinic and heteroclinic) bifurcations. Construction and analysis of Poincare maps in second and third-order systems. The saddle index. The Lorenz equations. Transitions to chaos: the Lorenz and Shil'nikov mechanisms.

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) and assessed computer-based questions exploring different aspects of the dynamics of ODEs.

Methods of assessment

Exams

Exam type	Exam duration	% of formal assessment
Standard exam (closed essays, MCQs etc)	3 hr 00 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: 17/02/2014

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