Existing Conditions of Fractal Boundaries of Invariant Sets of a Class of One-Dimensional Nonlinear Discrete-Time Systems

Abstract

In this paper, fractal basin boundaries are investigated in connection with a class of one-dimensional nonlinear discrete-time systems.Noting that the basin is an invariant set of the nonlinear function, describing the system dynamics, conditions, under which the Hausdorff dimension of boundaries of invariant sets is positive, are obtained and the existence of fractal basin boundaries is shown. Furthermore, it is demonstrated that if periodic points with period three exist, then fractal basin boundaries appear.Secondly, the result obtained is applied to explore, the basin boundary of a class of one-dimensional nonlinear sampled-data control systems. Illustrative examples together with numerical experiments show the existence of fractal basin boundaries. These theoretical and numerical results reveal that in order to determine the sampling period, it is necessary to take into account the structure of the basin of the equilibrium.