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2013/14 Undergraduate Module Catalogue

MATH3397 Nonlinear Dynamics

15 creditsClass Size: 60

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

MATH5398M

Advanced Nonlinear Dynamics

This module is approved as an Elective

Module summary

This course extends the study of nonlinear dynamics begun in MATH 2391, and includes an in-depth study of bifurcation theory and the transition to chaos.

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. Classify Hopf bifurcations using Poincare-Lindstedt theory.
3. Analyse the formation and stability of one and two-dimensional patterns.

Learning outcomes
The aim of this module is to introduce the theory of bifurcations in dissipative nonlinear systems, treating mainly local bifurcations, and exploring the dynamics in the neighbourhood of bifurcations. As well as being relevant to fluid dynamics experiments, there are important applications in many fields, including biology, chemistry, astrophysics and geophysics.

Syllabus

1. Definitions and terminology. One-dimensional bifurcations. Saddle-node: Transcritical: Pitchfork.
2. General theory 1-D bifurcations. Structural stability and imperfections. Reduction to normal form.
3. Reflectional symmetry. Linear n-dimensional theory. Hartman-Grobman theorem. Hopf bifurcation.
4. Codimension 1 bifurcations. Regime diagrams. Routh-Hurwitz criteria.
5. Centre manifold theory. Extended centre manifolds. Normal form theory for Hopf bifurcation.
6. Supercritical/Subcritical Hopf bifurcation. Poincare-Lindstedt theory.
7. Limit cycle. Bifurcation in maps. Poincare map. Flouquet exponents.
8. Co-dimension 2 bifurcation. Takens-Bogdanov bifurcation. Idea of Global bifurcations.
9. Pattern formation. Swift-Hohenberg model. Dispersion relation. Rolls, squares and hexagons.
10. Analysis of coupled Landau equations for stability of squares/rolls.

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

Reading list

The reading list is available from the Library website

Last updated: 14/02/2014

Disclaimer

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