Dynamical Systems

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The availability of powerful computers in the sixties and seventies of the previous century enabled researchers to attempt numerical solutions of problems from the physical sciences which hitherto had been ignored either because they contained awkward nonlinear effects or because they were too complex. This gave rise to an exciting new discipline in science which goes by the name of 'Dynamical Systems'. This new science concerns itself with the description and characterisation of different modes of evolution of time-dependent systems, some of which are so complex as to be called 'chaotic' motion.

A research program in Dynamical Systems has been followed in this Department for more than a decade. It was initially concerned largely with the numerical investigation of the dynamics of specific model systems in both the classical and quantum regimes, i.e. with the calculation of phase diagrams, Lyapunov exponents and bifurcation diagrams. Latterly, the program has shifted in the direction of the control of chaotic systems.

Quite paradoxically, the inherent instability of a chaotic dynamical system facilitates efficient control techniques. The reason for this lies in the fact that the motion of such a system is sensitively dependent on initial conditions. Very small perturbations of a control parameter can therefore effect disproportionately large changes in the evolution of the system. To an engineer this approach is particularly attractive because of the energy economy implied.