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MC451 Dynamical Systems

MC451 Dynamical Systems

Credits: 20 Convenor: Dr. R. L. Davidchack Semester: 1

Prerequisites: essential: MC126, MC127, MC147, MC224, MC241 desirable: MC160, 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

Explanation of Pre-requisites

The study of nonlinear dynamical systems requires basic knowledge of several branches of mathematics: calculus (MC126), ordinary differential equations (MC127), linear algebra (MC147, MC241), vector calculus (MC224). Understanding of some essential consepts of chaos will be easier for those who studied abstract analysis (MC240) and are familiar with the basics of probability and statistics (MC160).

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.

Aims

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

Objectives

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 the new developments in the rapidly changing field.

Transferable Skills

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.

Syllabus

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

One-dimensional maps. Tent map. Fixed points, periodic orbits and their stability. Bifurcations. Logistic map. 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. Natural measure and dimension spectrum.

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

Hamiltonian systems. Symplectic structure. Canonical transformations. Integrable systems. Perturbation of integrable systems. The KAM theorem. Resonant tori. Strongly chaotic systems. Classification of increasingly random systems.

Chaotic transitions. Intermittency. Crises.

Reading list

Recommended:

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

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

D. Ruelle, Chaotic Evolution and Strange Attractors, Cambridge University Press.

R. Z. Sagdeev, D. A. Usikov, and G. M. Zaslavsky, Nonlinear Physics, Harwood Academic Publishers.

Details of Assessment

The final assessment of this module will consist of 10% coursework and 90% from a three hour examination during the January exam period. The 10% coursework contribution will be determined by students' solutions to coursework problems. The examination paper will contain 6 questions with full marks on the paper obtainable from 4 complete answers.

Next: MC480 Mathematics Project Up: Year 4 Previous: MC450 Finite Element Methods for Partial

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Last updated: 10/4/2000
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