Dynamic Analysis of Digital Chaotic Maps via State-Mapping Networks (1410.7694v5)

Abstract: Chaotic dynamics is widely used to design pseudo-random number generators and for other applications such as secure communications and encryption. This paper aims to study the dynamics of discrete-time chaotic maps in the digital (i.e., 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 first explored with a fixed-point arithmetic, using the Logistic map and the Tent map as two representative examples, from a new perspective with the corresponding state-mapping networks (SMNs). In an SMN, every possible value in the digital domain is considered as a node and the mapping relationship between any pair of nodes is a directed edge. The scale-free properties of the Logistic map's SMN are proved. The analytic results are further extended to the scenario of floating-point arithmetic and for other chaotic maps. Understanding the network structure of a chaotic map's SMN in digital computers can facilitate counteracting the undesirable degeneration of chaotic dynamics in finite-precision domains, helping also classify and improve the randomness of pseudo-random number sequences generated by iterating chaotic maps.

• The paper introduces state-mapping networks (SMNs) as a novel framework to analyze the dynamic behavior of digital chaotic maps under finite precision.
• This study maps digital values as nodes and their transitions as edges in an SMN, revealing structural properties like scale-free characteristics in the digital Logistic map, contrasting previous assumptions.
• Insights from this analysis can improve the understanding of dynamics degradation in digital systems and enhance the design and evaluation of pseudo-random number generators (PRNGs) for cryptographic applications.

Dynamic Analysis of Digital Chaotic Maps via State-Mapping Networks

This paper provides a comprehensive paper on the dynamics of discrete-time chaotic maps within digital domains, employing state-mapping networks (SMNs) as a novel investigative framework. Chaotic systems in digital applications often suffer from dynamics degradation when finite precision is invoked, leading to deviations from their theoretical continuous counterparts. This investigation leverages fixed-point and floating-point arithmetic to elucidate the nuanced behavior of chaotic maps like the Logistic and Tent maps by distilling their digital structural characteristics.

Overview of Findings

Contrasting traditional approaches that viewed digital chaotic maps as opaque entities characterized via post-hoc output analyses, the paper prioritizes the networked relationships amongst possible state values. The SMN model considers all feasible digital values as nodes, with mapping relationships portrayed as directed edges. The Logistic map's SMN is proven to exhibit scale-free properties, a significant insight implying an inherent robustness akin to natural and man-made complex systems. The analytic framework extends to chaotic maps under floating-point arithmetic, illustrating strength in generalizing the results beyond fixed-precision constraints.

Numerical Results and Contradictory Claims

A marked finding is the quantitative characterization of the Logistic map as having more extensive and dominant connectivity amongst weakly connected components in its SMN than previously assumed. This contrasts established beliefs around its dynamics in limited precision settings. Specific properties — in-degree distribution and cumulative in-degree distribution under varying arithmetic precisions — are meticulously derived, reinforcing the theoretical underpinnings with empirical data from simulations across different precision environments.

Implications and Speculations

The paper's implications reverberate across both theoretical and practical spectrums. Theoretically, it advances the understanding of SMNs' role as fingerprints of digital chaotic systems, offering a structured route to combating dynamics degradation. Practically, insights derived can significantly bolster the efficacy of pseudo-random number generators (PRNGs) pivotal to cryptographic operations, by enabling more refined evaluations of randomness via SMNs.

For future developments, exploring higher-dimensional chaotic systems through the SMN lens presents a promising avenue. Additionally, refining SMN analyses to optimize computational efficiency can broaden applications in real-time digital signal processing and secure communications.

Conclusion

By elucidating the dynamics of digital chaotic maps through state-mapping networks, this paper contributes foundational insights into chaos theory's application within computational domains. It highlights the critical synthesis of complex network approaches to decode the structural nuances of finite-precision chaotic systems, echoing broader implications for digital system design and security. Continued advancement in this area is anticipated to enhance both mathematical rigor and practical utility in complex systems analysis.