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MA1251 Chaos and Fractals

MA1251 Chaos and Fractals

Credits: 10 Convenor: Dr. Ruslan Davidchack Semester: 2 (weeks 21 to 26)
Prerequisites: desirable: MA1002
Assessment: Lab reports and coursework: 100% Examination: 0%
Lectures: 6 Problem Classes: 6
Tutorials: none Private Study: 51
Labs: 12 Seminars: none
Project: none Other: none
Surgeries: none Total: 75

Subject Knowledge

Aims

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

Learning Outcomes

Students should understand basic concepts of the theory of dynamical systems and know typical mechanisms by which simple systems generate complicated dynamics and fractal structures.

Methods

Class sessions, computer labs, and problem classes

Assessment

Marked problem sheets and lab reports

Subject Skills

Aims

Students should develop skills for modelling simple dynamical systems and studying their properties.

Learning Outcomes

Students will learn to use the scientific computing environment MATLAB and to model simple dynamical system within this environment. In addition to the problem solving skills, the students will also learn to write a report of the outcome of their computer experiments. Of the general IT skills, the students will have the opportunity to learn file creation, management and storage; use of electronic resourses, such as internet, email and library catalogues; word processing.

Methods

Class sessions, computer lab sessions, and problem classes

Assessment

Marked problem sheets and lab reports

Explanation of Pre-requisites

From the module MA1002, students should be familiar with the concepts of vectors and differential equations.

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.

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.

Resources

Lecture rooms, computer labs, internet-based lab material presentation, problem sheets, additional handouts.

Module Evaluation

Module questionnaires, module review, year review.

Next: MA1271 Geometry of the Plane Up: ModuleGuide03-04 Previous: MA1221 Pure Mathematics at Work

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Last updated: 2004-02-21
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