Network analysis of dynamics of chaotic maps in digital domain

Dynamics of chaotic maps in an infinite-precision domain have been established complete mathematical framework. The case in finite-precision computer remains to be further explored. The related previous works treated a digital chaotic map as a black box and gave different explanations according to the test results of the output. Using the Logistic and Tent maps as typical examples, we disclose some dynamical properties of chaotic maps in fixed-point arithmetic by studying its corresponding state-mapping network (SMN), where every possible value is considered as a node and the mapping relation existing between any pair of nodes works as a directed edge. The scale-free properties of SMN 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 SMN of a chaotic map in the digital computer can facilitate counteracting dynamics degeneration of digital chaotic maps, which also help evaluate and improve the randomness of pseudo-random number sequences generated by iterating chaotic maps.