Invariant Measures of Tunable Chaotic Sources: Robustness Analysis and Efficient Estimation

Abstract

In this paper, a theoretical approach for studying the robustness of the chaotic statistics of piecewise affine maps with respect to parameter perturbations is discussed. The approach is oriented toward the study of the effects that the nonidealities derived from the circuit implementation of these chaotic systems have on their dynamics. The ergodic behavior of these systems is discussed in detail, adopting the approach developed by Boyarsky and Gora, with particular reference to the family of sawtooth maps, and the robustness of their invariant measures is studied. Although this paper is particularly focused on this specific family of maps, the proposed approach can be generalized to other piecewise affine maps considered in the literature for information and communications technology applications. Moreover, in this paper, an efficient method for estimating the unique invariant density for stochastically stable piecewise affine maps is proposed. The method is an alternative to Monte Carlo methods and to other methods based on the discretization of the Frobenius-Perron operator.