Controlling Chaos of Hybrid Systems by Variable Threshold Values

Abstract

We try to stabilize unstable periodic orbits embedded in a given chaotic hybrid dynamical system by a perturbation of a threshold value. In conventional chaos control methods, a control input is designed by state-feedback, which is proportional to the difference between the target orbit and the current state, and it is applied to a specific system parameter or the state as a small perturbation. During a transition state, the control system consumes a certain control energy given by the integration of such perturbations. In our method, we change the threshold value dynamically to control the chaotic orbit. Unlike the OGY method and the delayed feedback control, no actual control input is added into the system. The state-feedback is utilized only to determine the dynamic threshold value, thus the orbit starting from the current threshold value reaches the next controlled threshold value without any control energy. We obtain the variation of the threshold value from the composite Poincaré map, and the controller is designed by the linear feedback theory with this variation. We demonstrate this method in simple hybrid chaotic systems and show its control performances by evaluating basins of attraction.