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2008/09 Taught Postgraduate Module Catalogue

MATH5396M Advanced Dynamical Systems

15 creditsClass Size: 50

Module manager: Dr T. Wagenknecht
Email: thomas@maths.leeds.ac.uk

Taught: Semester 2 (Jan to Jun) View Timetable

Year running 2008/09

Pre-requisite qualifications

MATH1035 and (MATH1015 or MATH1060 or MATH1331) and (MATH1915 or MATH1916 or MATH1917). MATH2391 is helpful but not essential.

This module is mutually exclusive with

MATH3396

Dynamical Systems

This module is not approved as an Elective

Module summary

This course continues the study of nonlinear dynamics begun in MATH 2391, but for maps rather than differential equations. Maps are the natural setting for understanding the nature of chaotic dynamics, which arise in a variety of contexts in biology, chemistry, physics, economics and engineering.

Objectives

On completing this module, students should be able to:
a) find fixed points, periodic orbits and other invariant sets in maps and compute their stability;
b) understand the structure of chaos in maps;
c) use a computer to investigate the behaviour of families of one-dimensional maps;
d) transform between the dynamics of a one-dimensional maps (the Lorenz map, the tent map and the logistic map) and symbolic dynamics;
e) identify codimension-one bifurcations in maps and sketch bifurcation diagrams;
f) use renormalisation techniques to understand the cascades of bifurcations involved in the transition to chaos.
g) analyze two-dimensional maps.

Syllabus

One-dimensional maps: fixed points, periodic points, asymptotic and Lyapunov stability, Lyapunov exponent, omega-limit sets, conjugate maps, topological entropy, topological chaos and horse-shoes, Period-three implies chaos, sensitive dependence on initial conditions, Schwartzian derivative, renormalisation, the period-doubling cascade and Feigenbaum's constant. Maple programs will be used throughout to demonstrate important principles. Two-dimensional maps: fixed points, stability, examples of bifurcations and chaotic dynamics.

Teaching methods

Delivery type	Number	Length hours	Student hours
Lecture	27	1.00	27.00
Tutorial	6	1.00	6.00
Private study hours			117.00
Total Contact hours			33.00
Total hours (100hr per 10 credits)			150.00

Private study

117 hours of private study: 2 hours per lecture reading and preparation, 8 hours for each of five example sheet, 23 hours 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

Coursework

Assessment type	Notes	% of formal assessment
Written Work	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.	15.00
Total percentage (Assessment Coursework)		15.00

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)	3 hr	85.00
Total percentage (Assessment Exams)		85.00

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: 28/07/2008

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