Study information

# Dynamical Systems and Chaos - 2023 entry

MODULE TITLE CREDIT VALUE Dynamical Systems and Chaos 15 MTHM018 Prof Jan Sieber (Coordinator)
DURATION: TERM 1 2 3
DURATION: WEEKS 11 0 0
 Number of Students Taking Module (anticipated) 33
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 expressed as nonlinear differential equations or iterated maps. Given the set of rules, the purpose of this module is to explain the resulting behaviour. The main questions that dynamical systems theory addresses are: What are the possible long-term behaviours of such systems? How do these depend on initial conditions? How do these depend on system parameters (bifurcations)? In particular, we highlight how seemingly random behaviour (chaos) is possible even in such deterministic systems.

Pre-requisite Module: MTH2003 Differential Equations or equivalent
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;

- Bifurcations: stable manifold theorem; centre manifold 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; Sharkovsky's theorem; period-three orbits imply chaos.

- Multiple Scales Analysis and Related Methods: multiple time scales and method of averaging; application to oscillators; response of forced oscillations;

LEARNING AND TEACHING
LEARNING ACTIVITIES AND TEACHING METHODS (given in hours of study time)
 Scheduled Learning & Teaching Activities Guided Independent Study Placement / Study Abroad 33 117 0
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
Coursework – Example Sheets (4) 4 hours per problem sheet 1-5 Written and Verbal

SUMMATIVE ASSESSMENT (% of credit)
 Coursework Written Exams Practical Exams 20 80 0
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 - Summery Exam Period All In line
Coursework - example sheet 1 10 4 hours All Written and verbal
Coursework - example sheet 2 10 4 hours All 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
Written Exam * Written Exam (2 hours) All August Ref/Def Period
Coursework 1 * Coursework 1 All August Ref/Def Period
Coursework 2 * Coursework 2 All August Ref/Def Period

*Please refer to reassessment notes for details on deferrals vs. Referral reassessment

RE-ASSESSMENT NOTES
Deferrals: Reassessment will be by coursework and/or written exam in the deferred element only. For deferred candidates, the module mark will be uncapped.

Referrals: Reassessment will be by a single written exam worth 100% of the module only. As it is a referral, the mark will be capped at 50%.

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

Web based and Electronic Resources:

Type Author Title Edition Publisher Year ISBN
Set Glendinning, P.A. Stability, Instability and Chaos Cambridge University Press 1994 000-0-521-41553-5
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
Set Devaney, R.L. An Introduction to Chaotic Dynamical Systems Addison Wesley 2003 000-0-201-13046-7
Set Jordan, D.W. & Smith, P. Nonlinear Ordinary Differential Equations 3rd Oxford University Press 1999 000-0-198-56562-3
Set Drazin, P.G. Nonlinear Systems Cambridge University Press 1992 000-0-521-40668-4
Set Strogatz, S.H. Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry and Engineering Perseus Books 2000
CREDIT VALUE ECTS VALUE 15 7.5
PRE-REQUISITE MODULES MTH2003
NQF LEVEL (FHEQ) AVAILABLE AS DISTANCE LEARNING 7 No Tuesday 10th July 2018 Thursday 26th January 2023
KEY WORDS SEARCH Dynamical Systems; Chaos; Bifurcations; Nonlinear Ordinary Differential Equations; Iterated Maps