A Novel Four-Stage Synchronized Chaotic Map: Design and Statistical Characterization

Abstract

Digital implementations of chaotic systems often suffer from inherent degradation, limiting their long-term performance and statistical quality. To address this challenge, we propose a novel four-stage synchronized piecewise linear chaotic map. This new map is meticulously designed with four independent segments, each possessing its own control parameters, specifically engineered to mitigate the natural degradation observed in digitally realized dynamical systems. We characterize its behavior using established tools from nonlinear dynamics, including bifurcation diagrams and graphical analysis, which provide a comprehensive qualitative understanding of its complex dynamics. To rigorously validate the statistical features of the generated sequences, we employed the National Institute of Standards and Technology (NIST) statistical testing suite. A substantial 100 MB dataset, comprising sequences produced by the proposed map, was generated via a Matlab script and subjected to this rigorous battery of tests. Our results demonstrate that the proposed map exhibits superior statistical properties compared to the classic Bernoulli map, successfully passing all NIST tests where the traditional map did not. This research confirms the proposed map's potential as a robust and high-quality source for chaotic sequence generation.