Dynamical Systems and Chaos - 2020 entry

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.

Note that part of the material of this module will be delivered via a number of short videos that will need to be viewed during the semester, independently of the timetabled lectures.

UG Students - pre-requisite module: MTH2003 (formerly ECM2702)

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.

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.
 

- 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.
 

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 50% 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.

ELE: ELE – http://vle.exeter.ac.uk

Web based and Electronic Resources:

http://www.mat.univie.ac.at/~gerald/ftp/book-ode/
 

Reading list for this module: