Adaptation of non-quadratic chaotic discrete dynamical systems to the synchronization scheme

This paper addresses the challenge of synchronizing non-quadratic chaotic discrete dynamical systems, focusing on the Zeraoulia-Sprott map, which includes a fractional term that complicates the synchronization process. The presence of this fractional component in the first system has been a significant obstacle for existing synchronization schemes. To overcome this issue, we propose a novel control strategy that eliminates the fractional term in the controller vector, allowing for a more effective synchronization approach. The legitimacy and robustness of the proposed controller design are rigorously demonstrated using Lyapunov stability theory, ensuring the convergence of synchronization errors to zero. In parallel with the study of chaos in discrete dynamical systems, synchronization has emerged as a fundamental research direction in mathematics and physics. Early works by Yamada and Fujisaka introduced a local approach to chaotic synchronization, while Afraimovich et al. expanded on the theoretical foundations. Subsequently, Pecora and Carroll formalized identical synchronization using coupled chaotic systems, introducing the concepts of master and slave dynamics. More recently, the generalized synchronization method, as developed by Rulkov et al., has further advanced this research area. Our study contributes to this rich body of work by adapting non-quadratic chaotic systems to synchronization schemes through a tailored control approach. The proposed method opens new pathways for synchronizing complex discrete dynamical systems, offering potential applications in secure communications, signal processing, and complex system analysis.