Network analysis of the state space of chaotic map in digital domain

Complex dynamics of chaotic maps under an infinite-precision mathematical framework have been well known. The case in a finite-precision computer remains to be further explored. Previous work treated a digital chaotic map as a black box and gave different explanations according to the test results of the output. Using the Logistic map as a typical example, we disclose some dynamical properties of chaotic maps in fixed-point arithmetic by studying its corresponding state network, where every possible value is considered as a node and every possible mapping relation between a pair of nodes works as a directed edge. The scale-free properties of the state network are quantitatively proven. The obtained results can be extended to the scenario of floating-point arithmetic and to other chaotic maps. Understanding the real network structure of the state space of a chaotic map in the digital domain will help evaluate and improve the randomness of pseudo-random number sequences generated by chaotic maps.