On Caputo Delta q−Fractional Dynamical Systems: Lyapunov Stability

The investigation of dynamic systems that incorporate Caputo delta q−fractional derivatives has garnered significant interest due to their practicality in diverse scientific and engineering fields. This paper studies the stability of a dynamic system with the Caputo delta q−fractional derivative using Lyapunov's direct method. The motivation behind our work stems from the necessity to comprehend the dynamics and resilience of systems defined by Caputo delta q−fractional derivatives, which exemplify a category of operators that are both non-local and non-singular. This unique fractional derivative, which accounts for memory effects and long-range interactions, adds a level of complexity that calls for a thorough study of stability properties. Expanding upon previous scholarly works, we fill a significant research void by presenting a series of criteria that determine the stability, asymptotic stability, and uniform stability of dynamic systems with Caputo delta q−fractional derivatives. Through the utilization of Lyapunov's direct method, we establish a meticulous framework for examining the stability of these systems, providing a valuable understanding of their dynamic behavior.