Threshold Control for Stabilization of Unstable Periodic Orbits in Chaotic Hybrid Systems

Stabilization of unstable periodic orbits within a given chaotic hybrid dynamical system is realized by a variable threshold value. In the conventional chaos control methods, a control input is proportional to the difference between the target orbit and the current state and it is added to a specific system parameter or the state as a small perturbation. Thus the whole system consumes certain control energy even if small as the amount of such input values during the transition state. We propose a new control method that changing the threshold value dynamically to stabilize the chaotic orbit. No actual control input is added into the system unlike the OGY method and the delayed feedback control. When the orbit hits the threshold, the state-feedback only determines the next threshold value to convey the controlled orbit to the target unstable periodic orbit eventually. Thus the orbit starting from the current threshold value reaches the next controlled threshold value without any direct control energy. We obtain the variation of the threshold value from the composite Poincare 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 with evaluating basins of attraction.