Nonlinear Variable-order Fractional System with Application to Chaotic Dynamics and Adaptive Fixed-time Fractional-order Sliding Mode Control

In this paper, a theoretical and computational analysis is established for a new general class of variable order (VO) system of fractional differential equations (FDEs). The theoretical findings are based on the literature on the fixed point theorems (FPTs) and stability of Hyers-Ulam (HU). Banach’s fixed point theorem (FPT) is used to check the uniqueness of the solutions of the presumed system. Leray-Schauder’s (LS) approach is utilized to study solution existence. A computational iterative numerical technique is developed. In addition, a control approach is devised for the fractional-order dynamical power system, employing adaptive fixed-time fractional-order sliding mode control. The main objective is to effectively control the chaotic power system. Therefore, a fractional-order sliding mode control method is devised with the objective of achieving superior tracking performance and ensuring smooth control inputs. Subsequently, an adaptivemethod is employed to mitigate the effects of the unknown disturbance. The Lyapunov theorem is employed to investigate the overall dynamical system. The comparative results are demonstrated to give a deeper understanding of the study and show that the suggested approach has better tracking control and convergence capabilities as process innovation.