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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 2010/2011 Sem. 1 2010/2011 Sem. 2 2011/2012 Sem. 1 & Sem. 2 2012/2013 Sem. 1 & Sem. 2

MT4508 DYNAMICAL SYSTEMS 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. Nonlinear Dynamics and Chaos - With Applications to Physics, Biology, Chemistry and Engineering: S.H. Strogatz, Westview Press; 2000. Assessment 2 Hour Examination = 100% Prerequisites MT3504 Availability Academic year 2011/12 in semester 2 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: PMH (September 2011)