	Department of Mathematics & Computer Science

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MA1251 Chaos and Fractals

MA1251(=MC183) Chaos and Fractals

Credits: 10

Convenor: Dr Ruslan Davidchack

Semester: 2

Prerequisites:	essential: None	desirable:
Assessment:	Continual assessment: (30% coursework problems, 70% lab reports) 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, 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

Details of Assessment

There is no examination for this module. The module assessment will be as follows: 30% for a coursework problems and 70% for lab reports.