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On the stability of cycles by delayed feedback control

We present a delayed feedback control (DFC) mechanism for stabilizing cycles of one-dimensional discrete time systems. In particular, we consider a DFC for stabilizing -cycles of a differentiable function of the form where with . Following an approach of Morgül, we construct a map whose fixed points correspond to -cycles of . We then analyse the local stability of the above DFC mechanism by evaluating the stability of the corresponding equilibrium points of . We associate to each periodic orbit of an explicit polynomial whose Schur stability corresponds to the stability of the DFC on that orbit. This polynomial is the characteristic polynomial of a Jacobian matrix that lies in a large class of matrices that encompasses the usual ‘companion matrices’ found in linear algebra; the primary purpose of this paper is to show that this polynomial may be expressed in a surprisingly simple form. An example indicating the efficacy of this method is provided.