Singularity, Observability and Statistical Independence in the Context of Chaotic Systems

Abstract

Pseudorandom number generators (PRNGs) have always been a central research topic in data science, and chaotic dynamical systems are one of the means to obtain scientifically proven data. Chaotic dynamical systems have the property that they have a seemingly unpredictable and random behavior obtained by making use of deterministic laws. The current paper will show how several notions used in the study of chaotic systems—statistical independence, singularity, and observability—can be used together as a suite of test methods for chaotic systems with high potential of being used in the PRNG or cryptography fields. In order to address these topics, we relied on the adaptation of the observability coefficient used in previous papers of the authors, we calculated the singularity areas for the chaotic systems considered, and we evaluated the selected chaotic maps from a statistical independence point of view. By making use of the three notions above, we managed to find strong correlations between the methods proposed, thus supporting the idea that the resulting test procedure is consistent. Future research directions consist of applying the proposed test procedure to other chaotic systems in order to gather more data and formalize the approach in a test suite that can be used by the data scientist when selecting the best chaotic system for a specific use (PRNG, cryptography, etc.).