MODULE TITLE    Dynamical Systems    MODULE CODE       MS4018                                            

PRE-REQUISITE MODULES    MS4403,  MS4407

GRADING TYPE                  Normal                         CREDITS                   6

AIMS/OBJECTIVES

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].

Two dimensional Maps.

[Introduction to time series applications].

[Fractals : dimensions; strange attractors revisited].

PRIME TEXT(S)

Steven S Strogatz, 1994,  Nonlinear Dynamics and Chaos, Addison-Wesley .

K Alligood, T Sauer, & J A Yorke, 1997,  Chaos : An introduction to Dynamical Systems, Springer Verlag

R Clark Robinson, 2004, An Introduction to Dynamical Systems, Prentice Hall