Controlling Chaos in Dynamical Systems Described by Maps

The purpose of this thesis is to study methods for the control of chaos in dynamical systems described by maps. Two simple model dynamical systems, a one-dimensional map, the logistic map and a two-dimensional map, the prey-predator map are treated first. A feedback control method is introduced, in which an independent parameter of the system is perturbed. The chaotic behavior is successfully suppressed and the maps are stabilized about an unstable fixed point. Next, a more complex two-dimensional map is studied: the dynamics of a marker particle advected by the flow field generated by a blinking vortex. This flow field is produced by two blinking vortices in an inviscid fluid. The two vortices are turned on and off consecutively, at a constant period T. It is known that the particle motion in such a flow becomes chaotic when the period of blinking is increased beyond a critical value. In order to control or enhance chaos in this system, a feedback control acting on the period of blinking or the location of the vortices is introduced. In this case, partial success was achieved. The motion of the marker particle can be controlled (stabilized) in one space direction (x-direction) but the motion in the other direction (y-direction) cannot be successfully controlled using the control laws proposed. On the other hand, it is possible to enhance the chaotic behavior (anti-control) using the proposed method. Additional control laws and methods should be tried in future work