Dynamic analysis of chaotic maps as complex networks in the digital domain

Chaotic dynamics is widely used to design random number generators. This paper aims to study the dynamics of chaotic maps in a digital finite-precision domain. Differing from the traditional approaches treating a digital chaotic map as a black box with different explanations according to the test results of the output, the dynamical properties of such chaotic maps are explored in an fixed-point arithmetic domain, using the Logistic map and the Tent map as representative examples, from a new perspective with a corresponding state-mapping network (SMN). In SMN, every possible value is considered as a node and the mapping relationship between any pair of nodes works just like a directed edge. The scale-free properties of SMN are proved. The analytic results can be further extended to the scenario of floating-point arithmetic and for other chaotic maps. Understanding the network structure of SMN of a chaotic map in the digital computers can facilitate counteracting the undesirable dynamics degenerations of digital chaotic maps in finite-precision domains, helping also classify and improve the randomness of pseudo-random number sequences generated by iterating chaotic maps.