A New Approach for Controlling Chaos in Lorenz System
Ali Sanayei, Theodore E. Simos, George Psihoyios, Ch. Tsitouras · AIP conference proceedings · 2009
Is there a need for chaos? In order to answer to this important question, first, we should answer to “what chaos is?” Does “chaos” mean anarchy and confusion, or it means “randomness”? In order to answer to the second question, one may briefly consider that “chaos” means “far from the equilibrium.” It is true that in a random behavior, we have “far from the equilibrium” phenomenon, but in the chaotic behavior, however, the trajectory goes far from the equilibrium, but it moves in a bounded basin. Therefore, chaos differs from randomness. In order to answer to the first question, we distinguish two states from each other. Chaos could be dangerous in many states, e.g. for an aircraft in the sky. Therefore, we should control it and return the system from the chaotic mood. But, in some states it is useful. Suppose that we have a periode‐2 behavior system. If we intend to change its period, what should we do? One of the best techniques in order to change a system behavior is reaching the system into the chaotic mood for a short time, and then, by controlling chaos which is based on the feedback law, we could return the system into the desired period. Further, the control of chaos is also a way to manipulate the natural systems that are already chaotic. In this paper, we can imagine each mentioned states for chaos. Our goal is the control of a very famous system in the chaotic mood, in order to stabilize it and change its behavior into the desired behavior. We will achieved to this goal using OGY method which is based on the discrete dynamical system concept, and find the stabilized state by a new approach which is based on the generalized Routh‐Hurwitz criterion.