article · International Journal of Bifurcation and Chaos
A new method has been developed to construct chaotic systems that operate entirely without linear terms, built upon a modified Thomas circulant framework. By incorporating nonlinear dissipation alongside strictly nonlinear functions into the original structure, a new class of chaotic systems was created, yielding five distinct models. Detailed theoretical analysis of a representative system with piecewise cubic nonlinearities explored fixed points and bifurcation characteristics to reveal key dynamical features. The study also examined how bidirectional connections between variables heighten dynamic complexity and how breaking circulant symmetry influences stability patterns. To verify the theoretical design, the modified systems were physically deployed on an Arduino module. This microcontroller demonstration confirmed the theoretical behaviour, illustrating practical routes for creating chaos in systems that are otherwise restricted by linear interactions.
Chaotic systems are essential components in complex computational and engineering technologies. By proving that chaos can be generated without linear terms, this research broadens the mathematical methods available for designing dynamical systems. Furthermore, demonstrating these models on standard microcontroller hardware shows that such non-standard chaotic behaviours can be practically realised in accessible electronic environments.
The research sits at an applied, laboratory-tested stage, having moved from mathematical formulation to a practical proof of concept on an Arduino module. Potential users include engineers working on computational systems that require nonlinear dynamics. However, the abstract does not specify concrete market end-uses or commercial products, indicating that the technology remains at an early developmental phase prior to commercial uptake.
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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.
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DOI: 10.1142/s0218127426502147
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