Dynamical Systems and Chaos
2) Their dependence on initial conditions
3) Their dependence on the system parameters (bifurcations)
Full module specification
|Module title:||Dynamical Systems and Chaos|
|Duration of module:||
Duration (weeks) - term 1: |
0Duration (weeks) - term 2:
11Duration (weeks) - term 3:
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 Activities||Guided independent study||Placement / study abroad|
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.|
|Form of assessment||Size of the assessment (eg length / duration)||ILOs assessed||Feedback method|
|Problem sheets||Six one-hour tutorials||1-3||Verbal, on the spot|
Summative assessment (% of credit)
|Coursework||Written exams||Practical exams|
Details of summative assessment
|Form of assessment||% of credit||Size of the assessment (eg length / duration)||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||Timescale for re-assessment|
|All above||Written exam (100%)||All||August Ref/Def period|
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.
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:
- ,Stability, Instability and Chaos,Glendinning P.A.,,Cambridge University Press,1994,515.355 GLE,000-0-521-41553-5,
- ,An Introduction to Chaotic Dynamical Systems,Devaney R.L.,2nd,Addison Wesley,1989,515.352 DEV,000-0-201-13046-7,
- ,Nonlinear Systems,Drazin P.G.,,Cambridge University Press,1992,515.355 DRA,000-0-521-40668-4,
- ,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,
- ,Nonlinear Ordinary Differential Equations,Jordan D.W. & Smith P.,3rd,Oxford University Press,1999,515.352 JOR,000-0-198-56562-3,
Module has an active ELE page?
Last revision date