Hausdorff Dimension of Fractal Basin Boundaries of a Class of One-dimensional Nonlinear Sampled-data Control Systems

Abstract

In this paper, by calculating the Hausdorff dimension of basin boundaries, existing conditions of fractal basin boundaries are shown, concerned with a class of one-dimensional discrete nonlinear dynamical systems. The theoretical contribution of this paper is to clarify that if periodic points with period three exist, then fractal basin boundaries appear. By using this result, the fractal basin boundary of a class of nonlinear sampled-data control systems is explored. Illustrative examples together with numerical experiments show that if the sampling period becomes long, then the basin of an asymptotically stable equilibrium has fractal boundaries.