Analysis of relationship between chaotic dynamics and stochastic processes

This paper presents an equivalence relation between chaotic and stochastic systems based on a combinatorial study of chaotic dynamics and stochastic processes. The study focuses on the dynamic density of chaotic systems and Brownian motion in stochastic systems. The equivalency in invariant measure and ergodicity between the two systems is defined. Based on the definition, the equivalence relation is studied and the equivalent stochastic system model of the chaotic system is derived analytically, which leads to an equivalent stochastic system model. The concept of the equivalency can be used to achieve system reconstruction using time series of chaotic systems and to develop a control strategy for control of chaotic dynamics.