An approach for stabilizing periodic orbits in nonlinear systems

A standard target in controlling chaos is the stabilization of one of the unstable periodic orbits (UPOs) embedded in the chaotic attractor. In this paper a new approach for the design of controllers ensuring small signal input-output L₂-stability of periodic orbits in a class of nonlinear systems is proposed. A classical criterion due to Freedman and Zames is tailored to our setting in order to provide a criterion for designing a controller that improves stability of the periodic orbits with respect to the uncontrolled case. Since such criterion takes into account also the rate of time-variation of the system, it is expected that the results obtained in the same context via previously employed frequency domain criteria could be outperformed. An application example is presented to illustrate the features of the proposed approach.