A New Approach for Designing Chaotic Systems Without Linear Terms Based on a Modified Thomas Cyclic System

Abstract

This study presents a novel approach to designing chaotic systems devoid of linear terms, leveraging a modified Thomas circulant system. By introducing a nonlinear dissipation in the original Thomas system and adopting a strictly nonlinear function, a new family of chaotic systems without linear terms is constructed. Accordingly, five new examples of such systems are presented by selecting five different nonlinear functions. We conduct a comprehensive theoretical analysis of a prototypal system, the model with piecewise cubic nonlinearities, focusing on fixed points and bifurcation behaviors to uncover the dynamical features of the system. Notably, we explore the impact of bidirectional connections between variables, which enhances the complexity and richness of the chaotic dynamics. Additionally, we investigate the effects of circulant symmetry breaking, revealing new insights into stability and bifurcation patterns. The findings indicate potential pathways for generating chaos in systems typically constrained by linear interactions. To validate the theoretical framework, we implement the modified systems on an Arduino module, demonstrating their practical applicability. This implementation not only confirms the theoretical predictions but also paves the way for future explorations in chaotic system design, particularly in fields requiring nonlinear dynamics. Our work offers significant contributions to both theoretical and applied aspects of chaotic systems, with implications for engineering and computational applications.