Module Code - Title: MS4018 - DYNAMICAL SYSTEMS Year Last Offered: 2017/8 Hours Per Week Grading Type: N Prerequisite Modules: MS4403 Rationale and Purpose of the Module: To demonstrate to the student how dynamical techniques can be applied to the analysis of nonlinear and chaotic models, data and systems. Syllabus: One dimensional flows: flows on the line, fixed points and stability; bifurcations, flows on the circle. Two dimensional flows: Linear systems, classification of fixed points; phase plane, linearisation, stability and Lyapunov functions. Limit cycles, oscillators. Bifurcations in the plane, Hopf bifurcations, global bifurcations of cycles, quasi-periodicity. Poincare maps. Chaos : Lorenz equations; strange attractors; control of chaos. One dimensional maps : fixed points, periodic points and stability; bifurcations, the logistic map -- numerics and analysis, period-doubling and intermittency; Lyapunov exponents, renormalisation and Feigenbaum numbers. Introduction to time series applications. Fractals : dimensions; strange attractors revisited. Lecture Lab Tutorial Other Private Credits 2 1 1 0 6 6 Learning Outcomes: Cognitive (Knowledge, Understanding, Application, Analysis, Evaluation, Synthesis) On successful completion of this module students will (will be able to): 1. Compute the fixed points of a 1-d flow and determine their stability properties. Assessment by written exam. 2. Compute the fixed points/periodic points of a 1-d map and determine their stability properties. Assessment by written exam. 3. Determine whether a 2-d flow has a periodic orbit using index theory, DulacÆs criterion, PoincarÚ-Bendixson theorem or PoincarÚ Map as appropriate. Assessment by written exam. 4. Compute the stable, unstable and centre invariant manifolds of 2-d flows and maps. Assessment by written exam. 5. Construct Lyapunov functions and use them to estimate basins of attraction for flows and maps. Assessment by written exam. 6. Perform local bifurcation analyses on 1-d flows and maps. Assessment by written exam. 7. Investigate whether a 2-d flow can exhibit a Hopf bifurcation or a 2-d map a Neimark-Sacker bifurcation. Assessment by written exam. 8. Determine whether 1-d map exhibits chaotic behaviour. Assessment by written exam. 9. Using Maple plot orbits, phase plane plots, invariant manifolds and bifurcation diagrams of dynamical systems. Assessment by course work. Affective (Attitudes and Values) N/A Psychomotor (Physical Skills) N/A How the Module will be Taught and what will be the Learning Experiences of the Students: Research Findings Incorporated in to the Syllabus (If Relevant): Prime Texts: Strogatz, S. H. (2001) Nonlinear Dynamics and Chaos, Cambridge, Mass.: Perseus Books Group . Robinson, R. C. (2004) ) An Introduction to Dynamical Systems: Continuous and Discrete, Upper Saddle River, New Jersey: Prentice-Hall. Other Texts: Alligood, K. T., Sauer, T. D. and Yorke, J. A. (2000) Chaos : An introduction to Dynamical Systems, New York: Springer Verlag Programmes Semester - Year to be First Offered: Spring - 08/09 Module Leader: Generic PRS