Leveraging improved approaches for the investigation of patterns and randomness in digital chaos

• Autor(a) principal: Bispo Junior, Altamir Gomes
• Data de Publicação: 2024
• Tipo de documento: Tese
• Idioma: eng
• Título da fonte: Biblioteca Digital de Teses e Dissertações da USP
• Texto Completo: https://www.teses.usp.br/teses/disponiveis/76/76132/tde-26082024-085906/
• topic: Caos; Chaos; Dynamical systems; Pattern recognition; Reconhecimento de padrões; Sistemas dinâmicos

Dynamical deterministic systems with chaotic properties have been actively studied and new applications are established as the qualities of these systems are tested and proved. The k-logistic map is a variation of the logistic map that presents interesting properties for applications in generating pseudo-random numbers (PRNGs) and encryption. Given that the orbits generated by the k-logistic map present characteristics of high entropy and uniform distribution under statistical tests, question arises on which transformations and projections on the orbit of the k-logistic map are able to reveal patterns that are imperceptible in the original space. And another question also arises about which combinations of computer-based and mathematics-based techniques and methods of the arsenal of mathematics should be included into the toolset. This proposal has applications in the areas of cryptoanalysis, dynamical systems analysis and pattern recognition. The following methods were employed: generation of orbits with the k-logistic map, statistical test suites, local density measurements, topological graphs on objects contained in metric spaces, Dynamic Time Warping, spectral analysis, random matrix theory and machine learning. When projecting an orbit or distributions of orbits in metric spaces, and constructing topological graph distributions from these projections, even simple and conventional statistical tests revealed previously imperceptible patterns that were considered to be mere random noises. The results that were found from this development have relevance in improving test batteries for number sequences and analysis of dynamical systems.