Nonlinear dynamics of the complex periodic coupled system via a proportional generalized fractional derivative

The objective of this work is to study the intricate dynamics of nonlinear periodic coupled systems, introducing a novel approach based on the proportional fractional generalized derivative. We establish and rigorously derive sufficient conditions for the existence, uniqueness, and stability of solutions for these systems. This ensures the mathematical validity of the systems, making them reliable for simulations, predictions, and control design. This represents a significant advancement in the field of fractional-order systems. Our analysis utilizes the Banach contraction mapping principle and the Leray-Schauder alternative to ensure the well-posedness of the system. We present a detailed mathematical analysis to discuss the stability outcomes, making the results accessible and readily applicable to a wide range of problems. Furthermore, to showcase the versatility and practical implications of our approach, we present a concrete example. This demonstration highlights the novelty and impact of our research, underscoring the power of the Caputo generalized proportional fractional derivative-based periodic coupled system.