Qualitative Analysis of a Coupled Fractional-Order System Under Generalized Weighted Caputo Operators

This paper investigates a new class of coupled fractional-order systems driven by the generalized weighted fractional operator in the Caputo sense, recently introduced with a modified Mittag-Leffler kernel and an auxiliary strictly increasing function φ. This operator provides a unified framework that recovers many well-known fractional derivatives and integrals as particular cases, covering both singular and nonsingular kernel models. We treat both linear and nonlinear coupled systems: first, we derive explicit series representations that yield analytical insight and a convenient basis for numerical implementation. Next, we establish existence and uniqueness results using Banach’s contraction principle and Schaefer’s fixed point theorem under appropriate Lipschitz and growth conditions. In addition, we study Ulam–Hyers and Ulam–Hyers–Rassias stability, demonstrating the robustness of solutions to perturbations. Several numerical examples and simulations are presented to illustrate the theoretical findings and to highlight the influence of the governing parameters on the system dynamics.