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MC183 Dynamical Systems

MC183 Dynamical Systems

Credits: 10 Convenor: Dr Ruslan Davidchack Semester: 2
Prerequisites: essential: None desirable:
Assessment: Continual assessment: (30% coursework problems, 30% lab reports, 40% MATLAB project) $=$ 100% Examination: 0%
Lectures: 6 Problem Classes: none
Tutorials: 6 Private Study: 51
Labs: 12 Seminars: 0
Project: none Other: none
Surgeries: none Total: 75

Course Description

The extraordinary visual beauty of fractal images and their applications in chaos theory have made these endlessly repeating geometric figures widely familiar. Yet they are more than just appealing visual patterns and have proved to have wide range of uses. Chaos dynamics and fractal geometry are important and exciting topics in contemporary mathematics. This course introduces these topics using a combination of hands-on computer experimentation and simple mathematics. Students are led through a series of experiments that produce fascinating images of Julia sets, Mandelbrot sets, and fractals. The basic ideas of dynamics - iteration, stability, and chaos - are illustrated via computer projects.

Aims

To present an overview of the exciting area of nonlinear dynamical systems at a level accessible to first year undergraduates.

Objectives

Understand basic concepts of the theory of dynamical systems.

Know typical mechanisms by which simple systems generate complicated dynamics and fractal structures.

Develop skills for modelling simple dynamical systems and studying their properties.

Transferable Skills

Use of the scientific computing environment MATLAB, scientific report writing, modelling skills.

Syllabus

Dynamical systems. Orbits. Fixed and periodic points. Graphical analysis. The logistic function. Stable and unstable orbits. Chaotic orbits. Sensitive dependence on initial conditions. Lyapunov exponent. Period doubling bifurcation. Bifurcation diagrams.

Self-similarity and fractals. Cantor sets. Constructing fractals. Fractal dimension. Julia sets. Parameter spaces and Mandelbrot sets. Fractal basins of the Newton-Raphson method.

Reading list

Recommended:

R. L. Devaney, Chaos, Fractals, and Dynamics: Computer Experiments in Mathematics, Addison-Wesley.

H. Lauwerier, Fractals: Enlessly Repeated Geometrical Figures, Penguin Books.

P. S. Addison, Fractals and Chaos: an Illustrated Course, Institute of Physics Publishing.

M. Barnsley, Fractals Everywhere, Academic Press.

B. Fraser, The Non-Linear Lab, http://www.apmaths.uwo.ca/~ bfraser/version1/nonlinearlab.html,

Details of Assessment

There is no examination for this module. The module assessment will be as follows: 30% for a coursework problems, 30% for lab reports, 40% for a MATLAB project.

Next: MC192 Information Processing Up: Year 1 Previous: MC160 Probability

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