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We construct a family of polynomials with real coefficients that contains as a particular case the Fejér and Suffridge polynomials. These polynomials allow us to suggest a robust algorithm to search for cycles of arbitrary length in non-linear autonomous discrete dynamical systems. Numeric examples are included.

We present a delayed feedback control (DFC) mechanism for stabilizing cycles of one dimensional discrete time systems. In particular, we consider a delayed feedback control for stabilizing T -cycles of a differentiable function f : R → R of the formwith a 1 + · · · + a N = 1. Following an approach of Morgül, we construct a map F : R T +1 → R T +1 whose fixed points correspond to T -cycles of f . We then analyze the local stability of the above DFC mechanism by evaluating the stability of the corresponding equilibrum points of F . We associate to each periodic orbit of f an explicit polynomial whose Schur stability corresponds to the stability of the DFC on that orbit. An example indicating the efficacy of this method is provided.2010 Mathematics Subject Classification. Primary 93B52, 93B60.

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