Parametric Perturbations and Dynamic System Control

The article analyzes dynamical systems with externally applied periodic perturbations in a general setting. We provide a rigorous justification of an approach that reduces such systems to autonomous systems and thus simplifies the analysis. The behavior of families of quadratic one-dimensional maps and circle maps in the presence of parametric perturbations is studied in detail. We prove the existence of periodic perturbations acting strictly on a chaotic subset that stabilize the dynamics and induce the emergence of stable cycles in initially chaotic maps. The analytical results are supplemented with numerical data. It is shown that chaos may be suppressed by a sufficiently complex periodic perturbation.