Analyzing the Chaotic Dynamics of a Fractional-Order Dadras–Momeni System Using Relaxed Contractions

This paper is inspired by cutting-edge advancements in chaos theory, fractional calculus, and fixed point theory, which together provide a powerful framework for examining the dynamics of complex systems. At the heart of our research is the fractional-order Dadras–Momeni chaotic system, a pivotal model in chaos theory celebrated for its intricate, multi-scroll dynamics. Leveraging the Atangana–Baleanu fractional derivative, we extend fractional computation to chaotic systems, offering deeper insights into their behavior. To fortify the mathematical foundation of our analysis, we employ the relaxed θ rational contractions in the realm of metric spaces, enabling a more precise exploration of the system’s dynamics. A key goal of this work is to simplify the definition of the function class Θ while maintaining the existence and uniqueness of fixed points under θ-relaxed contractions, integrating this framework with the established literature on complete metric spaces. We explore the system’s behavior across six distinct cases by varying δ with a fixed fractional order of ℏ=0.98. In the first case, a single scroll forms, while successive cases lead to increased scrolls—reaching up to four by the sixth case. Phase portraits and time series analyses reveal a progression in complexity and chaos, with denser, intertwined scrolls as δ increases. This behavior highlights the system’s heightened sensitivity to parameter variations, demonstrating how fractional parameters influence the chaotic dynamics. Our results offer meaningful contributions to both the theoretical foundations and practical applications of chaos theory and fractional calculus, advancing the understanding of chaotic systems in new and impacted ways.