Energy saving controlling chaos

An energy saving control of unstable periodic orbits embedded in a hybrid chaotic system is proposed. The conventional controlling chaos methods utilize small perturbations of states or parameters as control input, however, quick time responses cannot be expected since the corresponding basins of attractions for higher periodic solutions become tiny. While If one allows a large perturbation to improve the time response, rather the total controlling energy which is proposed to the distance between the target orbit and the current orbit may increases. In this paper, when we consider the chaotic hybrid system, we noticed that we could utilize the perturbation of the referenced value for controlling, i.e., only a threshold value (Poincaré mapping surface) is updated in control. No control input as a perturbation of the state or parameter value is applied to the system. In fact, the threshold value is used instantly when the feedback system determines the next updated threshold value. The variation of the threshold value can be obtained numerically by computing variational equations, and the control matrix is designed with the linear control theory. Since no affection to the state and parameters, it is emphasized that the total behavior of the controlled system is different from the conventional methods, especially it is unlike the impulsive control methods. We demonstrate this control method in a simple hybrid system and show that a large basin of attraction for the control is realized.