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MA4001 Dynamical Systems

MA4001 Dynamical Systems

Credits: 20 Convenor: Dr Ruslan Davidchack Semester: 1
Prerequisites: essential: MA1001(=MC126), MA1002(=MC127), MA1152(=MC147), MA2001(=MC224), MA2102(=MC241 desirable: MA1061(=MC160), MA2151(=MC240)
Assessment: Coursework: 10% Three hour exam: 90%
Lectures: 36 Problem Classes: 10
Tutorials: none Private Study: 104
Labs: none Seminars: none
Project: none Other: none
Surgeries: none Total: 150

Subject Knowledge

Aims

To learn fundamentals and basic tools of the science of chaotic dynamics.

Learning Outcomes

By the end of the course students should have a general understanding of the language and concepts in the theory of chaos, which will allow them to explore further and understand new developments in the rapidly changing field.

Subject Skills

Aims

Within the past two decades workers in many disciplines have realized that a large variety of systems exhibit complicated evolution with time, which can be studied within the context of nonlinear dynamics. Therefore, the knowledge of the properties of nonlinear dynamical systems becomes essential for further progress in almost every branch of science and technology.

Explanation of Pre-requisites

The study of nonlinear dynamical systems requires basic knowledge of several branches of mathematics: calculus (MA1001), ordinary differential equations (MA1002), linear algebra (MA1152, MA2102), vector calculus (MA2001). Understanding of some essential concepts of chaos will be easier for those who studied abstract analysis (MA2151) and are familiar with the basics of probability and statistics (MA1061).

Course Description

This is an introduction into the subject that captures imagination with words like 'chaos', 'fractals', 'strange attractors', 'devil's staircase', etc. The main focus will be on providing an overall view of the subject, without dwelling too much on specific areas or mathematical details.

The subject of dynamical systems is truly interdisciplinary, and its concepts and methods are currently used in all fields of science, such as physics, engineering, chemistry, biology, physiology, economics, and sociology. Therefore, the module should be interesting not only to mathematicians, but also to physicists, engineers, chemists and everybody interested in the subject.

Syllabus

Dynamical systems, continuous and discrete. Poincaré surface of section. Limit sets, attractors. Basin of attraction. Sensitive dependence on initial conditions.

One-dimensional maps. Fixed points, periodic orbits and their stability. Piecewise linear maps. Symbolic dynamics. Logistic map. Bifurcations. Universal scaling of period doubling bifurcations in quadratic maps. Other types of bifurcations in one-dimensional maps. Measure, ergodicity and Lyapunov exponents for one-dimensional maps.

Strange attractors and fractal sets. Cantor set. Lebesgue measure. Box-counting dimension. Generalised baker's map. Natural measure and dimension spectrum. Determination of fractal dimension in experiments. Embedding.

Horseshoe map and symbolic dynamics. Linear stability analysis. Invariant subspaces and manifolds. Homoclinic and heteroclinic intersections. Lyapunov spectrum. Metric and topological entropies.

Quasiperiodicity. Circle map. Arnold tongues. Hopf bifurcation. Hamiltonian systems. Symplectic structure. Canonical transformations. Integrable systems. Perturbation of integrable systems. KAM (Kolmogorov-Arnold-Moser) theorem. Resonant tori. Chaotic transitions. Intermittency. Crises.

Reading list

Essential:

E. Ott, Chaos in Dynamical Systems, Cambridge University Press.

Recommended:

K. Alligood, T. D. Sauer, J. A. Yorke, Chaos: An Introduction to Dynamical Systems, Springer Verlag.

S. H. Strogatz, Nonlinear Dynamics and Chaos, Perseus Books.

R. L. Devaney, An Introduction to Chaotic Dynamical Systems, Addison-Wesley.

S. Wiggins, Introduction to Applied Nonlinear Dynamical, Springer Verlag.

Module Evaluation

Module questionnaires, module review, year review.

Next: MA4011 Finite Element Methods for Partial Up: ModuleGuide03-04 Previous: MA3521 Computational Mathematics Project

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Last updated: 2004-02-21
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