# Dynamical Systems and Chaos

## Module description

Dynamical systems are mathematical models of real systems (such as climate, brain, electronic circuits, lasers, etc.) that evolve in time according to definite (deterministic) rules.  Given the set of rules, the purpose of this module is to explain the resulting behavior as much as one can.  The main three questions that are addressed within dynamical systems theory are:

1) The long-term behaviours of such systems
2) Their dependence on initial conditions
3) Their dependence on the system parameters (bifurcations)

## Full module specification

Module title: Dynamical Systems and Chaos ECMM718 M 2014/5 Professor Sebastian Wieczorek - Convenor 15 7.5 Duration (weeks) - term 1: 0Duration (weeks) - term 2: 11Duration (weeks) - term 3: 0

## Module aims

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

## ILO: Module-specific skills

• 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.

## ILO: Discipline-specific skills

• 2. comprehend mathematical methods that can be used to analyse physical and biological problems.

## ILO: Personal and key skills

• 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.

## Learning activities and teaching methods (given in hours of study time)

Scheduled Learning and Teaching ActivitiesGuided independent studyPlacement / study abroad
331170

## Details of learning activities and teaching methods

CategoryHours of study timeDescription
Scheduled learning and teaching activities33Lectures/example classes
Guided independent study117Systematic lecture revision, basic and wider reading, coursework preparation (16 hours) and exam preparation. Exact time for each dependent upon individual student needs.

## Formative assessment

Form of assessmentSize of the assessment (eg length / duration)ILOs assessedFeedback method
Problem sheetsSix one-hour tutorials1-3Verbal, on the spot

## Summative assessment (% of credit)

CourseworkWritten examsPractical exams
20800

## Details of summative assessment

Form of assessment% of creditSize of the assessment (eg length / duration)ILOs assessedFeedback method
Written exam  closed book802 hours1-3In line with CEMPS policy
Coursework  example sheet 1104 hours1-3Written and verbal
Coursework  example sheet 2104 hours1-3Written and verbal

## Details of re-assessment (where required by referral or deferral)

Original form of assessmentForm of re-assessmentILOs re-assessedTimescale for re-assessment
All aboveWritten exam (100%)AllAugust 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.

## Syllabus plan

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. (5 lectures). Oscillations: Examples from nonlinear oscillators. Statement of Poincare - Bendixson theorem. (3 lectures). Multiple scales analysis and related methods: Multiple time scales and method of averaging. Application to oscillators. Harmonic and subharmonic response for forced oscillations. (6 lectures). Bifurcation: Bifurcation from equilibria for ODEs. Normal forms and examples. Statement of Hopf bifurcation theorem. Invariant manifolds. (7 lectures). Chaotic systems: Chaotic ODEs and mappings. Properties of the logistic map. Period doubling. Cantor set, shift map and symbolic dynamics. Sarkovskii theor em. E xamples of other aspects including ergodic properties. (9 lectures).

## Indicative learning resources - Basic reading

ELE: ELE – http://vle.exeter.ac.uk

Web based and Electronic Resources:

Other Resources:

1. ,Stability, Instability and Chaos,Glendinning P.A.,,Cambridge University Press,1994,515.355 GLE,000-0-521-41553-5,
2. ,An Introduction to Chaotic Dynamical Systems,Devaney R.L.,2nd,Addison Wesley,1989,515.352 DEV,000-0-201-13046-7,
3. ,Nonlinear Systems,Drazin P.G.,,Cambridge University Press,1992,515.355 DRA,000-0-521-40668-4,
4. ,A first course in dynamics : with a panorama of recent developments,Hasselblatt B. and Katok A.,,Cambridge University Press,2003,515.352 HAS,000-0-521-58750-6,
5. ,Nonlinear Ordinary Differential Equations,Jordan D.W. & Smith P.,3rd,Oxford University Press,1999,515.352 JOR,000-0-198-56562-3,

Yes

19/11/12

19/11/12