IA new control algorithm based on the partial control method has been developed. The general situation we are considering is an orbit starting in a certain phase space region Q having a chaotic transient behavior affected by a bounded noise, so that the orbit will definitely escape from Q in an unpredictable number of iterations. Thus, the goal of the algorithm is to control in a predictable manner when to escape. While partial control has been used as a way to avoid escapes, here we want to adapt it to force the escape in a controlled manner. We have introduced new tools such as escape functions and escape sets that once computed makes the control of the orbit straightforward. The partial control method aims to avoid the escape of orbits from a phase space region Q where the transient chaotic dynamics takes place. The technique is based on finding a special subset of Q called the safe set. The chaotic orbit can be sustained in the safe set with a minimum amount of control. We have developed a control strategy to gradually lead any chaotic orbit in Q to the safe set by using the safety function. With the technique proposed here, the safe set can be converted into a global attractor of Q. In addition, we deal with the Hénon and the Lozi maps for a choice of parameters where they show transient chaos, and we compute their safety functions showing the strong dependence of the safety function with the strength of the bounded noise affecting the maps, drastically impacting the controlled orbits. This is a joint work with G. Alfaro and R. Capeáns from URJC, Spain.
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[2] Gaspar Alfaro, Rubén Capeáns , and Miguel A. F. Sanjuán. Forcing the escape: Partial control of escaping orbits from a transient chaotic region. Nonlinear Dynamics 104, 1603—1612 (2021).
[3] Rubén Capeáns and Miguel A. F. Sanjuán. Beyond partial control: Controlling chaotic transients with the safety function. Nonlinear Dynamics 107, 2903—2910 (2022)
[4] Rubén Capeáns and Miguel A. F. Sanjuán. Controlling chaotic transients in the Hénon and the Lozi map with the safety function. Journal of Difference Equations and Applications (2023), DOI: 10.1080/10236198.2022.2063050