Inducing stable periodic behaviour in a class of dynamical systems by parametric perturbations

We develop an analytic approach to the problem of the behaviour stabilization of dynamical systems by parametric perturbations. First, questions of realization of the stable dynamics in non-chaotic systems with continuous time are studied. It is analytically shown that by means of parametric perturbations it is possible to obtain the stable periodic behaviour in systems which do not possess stable oscillations in the autonomous case. Then, on the basis of these results we advance the following conjecture: assume that a system has a chaotic attractor. Then, if we successfully choose a parametric perturbation of such a system in those regions where its behaviour is chaotic, then one can expect that this perturbation leads to the appearance of the stable periodic orbits which are either unstable or non-existent in the initial unperturbed system. We present rigorous results which assert that for some discrete time dynamical systems such a conjecture is valid: for certain families of mappings it is possible to find perturbations that lead to stabilization of their chaotic dynamics. In the framework of such an approach we offer a goal-oriented non-feedback way for stabilization of the desired stable periodic behaviour.