Zhongwei Hou∗ and Dengyuan Xu∗∗
2D chaotic cat map, finite state space, period distribution, Galoisring, Hensel lifting
The chaotic dynamical systems over continuous-state space have found a multitude of real-world applications, and most of them are implemented by computers over a finite-state space. However, due to limited computation, memory and communication capabilities, the discretization process during implementation may give rise to inaccuracy. In addition, the chaotic cat map has found applications in various areas such as cryptography and steganography. Dynamical behavior is usually performed as a period solution owing to finite- state space. Hence, it is worthy to consider the dynamics of chaotic dynamical systems over finite-state space. In this paper, the period distribution of the 2D chaotic cat map at field size N = 3e was considered. Full knowledge of the period distribution is obtained using the Hensel lifting method. Our results not only lead to advances in chaos theory but also broaden the applications of cat maps as they provide design strategy where the knowledge of the periods is required in advance. Two examples are illustrated to validate our findings.
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