[Paper Review] Network analysis of the state space of chaotic map in digital domain.

Chengqing Li|arXiv (Cornell University)|Oct 28, 2014

Chaos-based Image/Signal Encryption45 references3 citations

TL;DR

This paper analyzes the state space of chaotic maps, such as the Logistic map, in fixed-point arithmetic by modeling it as a directed network where each state is a node and transitions are edges. It proves the network exhibits scale-free properties, offering a structural basis for evaluating pseudo-random number quality in digital chaotic systems.

ABSTRACT

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.

Motivation & Objective

• To investigate the dynamical behavior of chaotic maps in finite-precision digital systems, where infinite-precision dynamics do not fully apply.
• To move beyond treating digital chaotic maps as black boxes by analyzing their internal state transition structure.
• To characterize the topological properties of the state space network formed by fixed-point arithmetic implementations.
• To establish a framework applicable to other chaotic maps and arithmetic formats, including floating-point.
• To provide a structural understanding that supports the evaluation and improvement of randomness in chaotic pseudo-random number sequences.

Proposed method

• Model the state space of a chaotic map in fixed-point arithmetic as a directed network, with each possible state as a node.
• Define directed edges between nodes based on the deterministic mapping of the chaotic map's recurrence relation.
• Apply network analysis techniques to study the degree distribution and structural properties of the resulting state network.
• Use mathematical analysis to prove the scale-free nature of the state network, characterized by a power-law degree distribution.
• Extend the findings to floating-point arithmetic and other chaotic maps through structural and topological reasoning.
• Leverage the network representation to reveal hidden regularities in the otherwise seemingly random digital dynamics.

Experimental results

Research questions

• RQ1 What is the underlying network structure of the state space in digital chaotic maps under fixed-point arithmetic?
• RQ2 How do topological properties like degree distribution characterize the dynamics of digital chaotic maps?
• RQ3 To what extent do scale-free properties emerge in the state network of chaotic maps in finite-precision systems?
• RQ4 Can the structural insights from fixed-point networks be generalized to floating-point arithmetic and other chaotic maps?
• RQ5 How does the network structure relate to the randomness quality of pseudo-random sequences generated by chaotic maps?

Key findings

• The state network of the Logistic map in fixed-point arithmetic exhibits a scale-free topology, with a power-law degree distribution.
• The scale-free property is quantitatively proven through mathematical analysis of the network's degree distribution.
• The network structure reveals hidden regularities in digital chaotic dynamics, challenging the assumption of purely random behavior.
• The findings are extendable to floating-point arithmetic and other chaotic maps due to shared structural principles.
• The network-based analysis provides a new framework for evaluating and improving the randomness of pseudo-random number sequences from chaotic maps.
• The results demonstrate that digital chaotic systems are not ergodic in the same way as their infinite-precision counterparts, due to the finite state space and network topology.