How to Implement Nonlinear Dynamical Systems on Digital Computers: The Bernoulli Shift Map Case

Abstract

The Bernoulli shift map is one of the nonlinear dynamical systems that show chaotic responses. The Bernoulli shift map is a piecewise linear dynamic system, but because of its simplicity of description, it is used in various situations. On the other hand, the essence of the dynamics of the Bernoulli shift map is that it doubles the state value. In other words, we must be careful when we implement the Bernoulli shift map on the current digital computers that use binary systems. In this article, we first describe an issue in the implementation of the response of the Bernoulli shift map with a digital computer. To solve this issue, we introduced two methods and investigated their performance: the first one is to give a small displacement to the slope of the map and the second one is to give a small displacement to the state value of the map. We evaluated these two methods by estimating Lyapunov exponents and invariant measures. In addition, we generated chaotic time series generated from the Bernoulli shift map by the methods and investigated the performance of the obtained chaotic time series as pseudorandom numbers, presenting the results of applying the NIST test to the chaotic times series.