The Control of Chaos

The complicated behaviours seen in chaotic dynamical systems can be tamed in extremely subtle ways. The application of miniscule bounded parameter perturbations can transform chaos into order, provided those perturbations are appropriately chosen. Many methods have been derived to control chaotic behaviour in a wide range of systems. Moreover, they often require no knowledge of the underlying dynamical equations describing the system. A simple estimate of the dynamics at particular points on the chaotic attractor suffices in order to effect control. In this thesis a well known method for the control of chaos is reviewed and its convergence properties analysed. From this analysis, several more powerful methods are proposed. Robust methods are considered which are less sensitive to errors in the estimation of system dynamics and in the presence of noise. Before systems can be brought under control a chaotic transient is observed. The length of this transient depends sensitively upon initial conditions and on the size of the permitted maximum perturbation. By extending the region within which parameter perturbations can be activated, transient times can be reduced. The adverse effects of noise can also be combatted. Targeting, that is, rapidly changing the state of a system from a given initial condition to some desired state, can also reduce transient times. Both of these avenues for the reduction of transient times are explored here. Synchronising the dynamics of two independent chaotic oscillators can often be achieved by coupling the systems in a particular way. If the coupling is not strong enough, synchronised behaviour will not result. In such a case, it is shown that control methods can be used to induce a synchronous behaviour in systems and to maintain synchronous behaviour as desired.