# Mathematics

## ECMM718 - Dynamical Systems and Chaos (2015)

MODULE TITLE CREDIT VALUE Dynamical Systems and Chaos 15 ECMM718 Dr Christian Bick (Coordinator)
DURATION: TERM 1 2 3
DURATION: WEEKS 0 11 weeks 0
 Number of Students Taking Module (anticipated) 18
DESCRIPTION - summary of the module content

Dynamical systems are mathematical models of real systems (for example, climate, brain, electronic circuits and lasers) that evolve in time according to definite (deterministic) rules. Given the set of rules, the purpose of this module is to explain the resulting behaviour as much as one can. The main three questions that a dynamical systems theory addresses are: what the long-term behaviours of such systems are, what their dependence on initial conditions are and what their dependence on the system parameters (bifurcations) are.

UG Students - pre-requisite module: ECM2707

AIMS - intentions of the module

The aim of this module is to expose you to qualitative and quantitative methods for dynamical systems, including nonlinear ordinary differential equations, maps, bifurcations and chaos. The phenomena you will study occur in many physical systems of interest.

INTENDED LEARNING OUTCOMES (ILOs) (see assessment section below for how ILOs will be assessed)

On successful completion of this module you should be able to:

Module Specific Skills and Knowledge:
1 understand the asymptotic behaviour of nonlinear dynamics, including an introduction to important areas of current research in dynamical systems theory, including bifurcations and deterministic chaos.
Discipline Specific Skills and Knowledge:
2 comprehend mathematical methods that can be used to analyse physical and biological problems.
Personal and Key Transferable/ Employment Skills and  Knowledge:
3 demonstrate enhanced modelling, problem-solving and computing skills, and will have acquired tools that are widely used in scientific research and modelling;
4 demonstrate appropriate use of learning resources;
5 demonstrate self management and time-management skills.

SYLLABUS PLAN - summary of the structure and academic content of the module

- asymptotic behaviour: asymptotic behaviour of autonomous and non-autonomous ODEs; omega and alpha limit sets; non-wandering set; phase space and stability of equilibria; limit cycles and Poincare map; index of equilibrium points;

- oscillations: examples from nonlinear oscillators; statement of Poincare - Bendixson theorem;

- multiple scales analysis and related methods: multiple time scales and method of averaging; application to oscillators; harmonic and subharmonic response for forced oscillations;

- bifurcations: stable manifold theorem; centre manifod theorem; bifurcations of equilibria for ODEs; normal forms and examples; statement of Hopf bifurcation theorem

- chaotic systems: chaotic ODEs and mappings; properties of the logistic map; period doubling; Cantor set, shift map and symbolic dynamics; horseshoes; Sharkovskii's theorem; period-three orbits imply chaos.

LEARNING AND TEACHING
LEARNING ACTIVITIES AND TEACHING METHODS (given in hours of study time)
 Scheduled Learning & Teaching Activities Guided Independent Study 33 117
DETAILS OF LEARNING ACTIVITIES AND TEACHING METHODS
 Category Hours of study time Description Scheduled learning and teaching activities 33 Lectures/example classes Guided independent study 117 Systematic lecture revision, basic and wider reading, coursework preparation (16 hours) and exam preparation. Exact time for each dependent upon individual student needs.

ASSESSMENT
FORMATIVE ASSESSMENT - for feedback and development purposes; does not count towards module grade
Form of Assessment Size of Assessment (e.g. duration/length) ILOs Assessed Feedback Method
Problem sheets Six one-hour tutorials 1-3 Verbal, on the spot

SUMMATIVE ASSESSMENT (% of credit)
 Coursework Written Exams 20 80
DETAILS OF SUMMATIVE ASSESSMENT
Form of Assessment % of Credit Size of Assessment (e.g. duration/length) ILOs Assessed Feedback Method
Written exam – closed book 80 2 hours 1-3 In line with CEMPS policy
Coursework – example sheet 1 10 4 hours 1-3 Written and verbal
Coursework – example sheet 2 10 4 hours 1-3 Written and verbal

DETAILS OF RE-ASSESSMENT (where required by referral or deferral)
Original Form of Assessment Form of Re-assessment ILOs Re-assessed Time Scale for Re-reassessment
All above Written exam (100%) All August Ref/Def period

RE-ASSESSMENT NOTES

If a module is normally assessed entirely by coursework, all referred/deferred assessments will normally be by assignment.

If a module is normally assessed by examination or examination plus coursework, referred and deferred assessment will normally be by examination. For referrals, only the examination will count, a mark of 40% being awarded if the examination is passed. For deferrals, candidates will be awarded the higher of the deferred examination mark or the deferred examination mark combined with the original coursework mark.

RESOURCES
INDICATIVE LEARNING RESOURCES - The following list is offered as an indication of the type & level of
information that you are expected to consult. Further guidance will be provided by the Module Convener

ELE: ELE – http://vle.exeter.ac.uk

Web based and Electronic Resources:

Reading list for this module:

Type Author Title Edition Publisher Year ISBN Search
Set Glendinning P.A. Stability, Instability and Chaos Cambridge University Press 1994 000-0-521-41553-5 [Library]
Set Devaney R.L. An Introduction to Chaotic Dynamical Systems Addison Wesley 2003 000-0-201-13046-7 [Library]
Set Drazin P.G. Nonlinear Systems Cambridge University Press 1992 000-0-521-40668-4 [Library]
Set Hasselblatt B. and Katok A. A first course in dynamics : with a panorama of recent developments Cambridge University Press 2003 000-0-521-58750-6 [Library]
Set Jordan D.W. & Smith P. Nonlinear Ordinary Differential Equations 3rd Oxford University Press 1999 000-0-198-56562-3 [Library]
Set Steven H Strogatz Nonlinear Dynamics and Chaos Perseus Books 2000 [Library]
CREDIT VALUE ECTS VALUE 15 7.5
PRE-REQUISITE MODULES ECM2707
NQF LEVEL (FHEQ) AVAILABLE AS DISTANCE LEARNING 7 No Friday 09 January 2015 Monday 12 January 2015
KEY WORDS SEARCH Dynamical systems; system parameters; bifurcations; nonlinear ordinary differential equations.