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Courses in Mathematics and Statistics Level 1 Modules Level 2 Modules Level 3 Modules Level 4 Modules Level 5 Modules Honours timetable 2011/2012 Sem. 1 2011/2012 Sem. 2 2012/2013 Sem. 1 & Sem. 2 2013/2014 Sem. 1 & Sem. 2

MT5808 ADVANCED DYNAMICAL SYSTEMS This module consists of the Module MT4508 with the addition of directed reading on more advanced and technical aspects of the subject and a requirement for the students to carry out a detailed analytical or numerical investigation of a more sophisticated problem relevant to this course. Such a problem could be one of the Challenges at the end of each chapter of the textbook by Alligood, Sauer and Yorke (see below). Aims To introduce students to the basic ideas of the modern theory of dynamical systems and to the concepts of chaos and strange attractors. Objectives By the end of the course students are expected to - understand the concept of an attracting set for a mapping or system of differential equations, and the distinction between fixed points, periodic orbits and chaotic attractors. - be able to analyse the linear stability of fixed points. - understand the concept of the Poincaré section. - be familiar with the behaviour of period doubling bifurcations and have a knowledge of the Feigenbaum scaling for them. - understand how chaotic attractors can be characterised by the Lyapunov exponent and by various types of dimension. - understand the way in which transversal homoclinic or heteroclinic points give rise to chaos. - understand the idealisation of the behaviour in the Smale horseshoe mapping and, in broad outline, how this is related to symbolic dynamics. - understand the concept of a bifurcation and be familiar with the standard bifurcations. - have some appreciation of the range of physical and biological problems to which this theory is applicable. Syllabus - Discrete and continuous dynamical systems. - One and two dimensional maps as discrete dynamical systems. - Fixed points, periodic points and stability. - Chaos, Lyapunov exponents and chaotic attractors. - Differential equations as continuous dynamical systems. - Periodic orbits and limit sets. - Bifurcations. Textbooks Chaos - An Introduction to Dynamical Systems: K.T. Alligood, T.D. Sauer and J.A. Yorke, Springer; 1997. Chaos in Dynamical Systems: E. Ott, Cambridge University Press; 1993. 2nd edition 2002. Assessment 2 hour examination (75 %) plus project (25 %) Prerequisites MT2001 or MT2101 (Mathematics) Antirequisites MT4508 Availability Academic year 2003/4 in semester 2: Tuesdays, Thursdays and even Mondays at 10 Lecturer Dr T Neukirch Click here for access to past examination papers for this module. Click here to see the University Course Catalogue entry. Revised: JOC (September 2003)