Investigation of Initial Time Difference Mittag–Leffler Stability for Fractional Perturbed Systems

This study investigates the Mittag–Leffler-type stability properties of fractional perturbed systems with respect to their unperturbed counterparts by incorporating initial time differences into the analysis. In contrast to many existing studies in which initial time effects are neglected, the proposed framework explicitly considers time shifts together with the memory-dependent nature of fractional-order systems. Using Caputo fractional derivatives and Lyapunov-type functionals, new sufficient conditions are established for the stability behavior of perturbed systems relative to the corresponding unperturbed systems under shifted initial times. The obtained results extend existing stability criteria by simultaneously addressing fractional memory effects, perturbation terms, and variations in the initial time. To illustrate the applicability and effectiveness of the theoretical findings, representative examples, numerical simulations, graphical comparisons, and global error analyses are presented. The numerical part is based on the Caputo framework and is further supported by benchmark comparisons involving Riemann–Liouville and shifted Grünwald–Letnikov approaches. The proposed results provide a useful framework for the stability analysis of memory-dependent dynamical systems arising in engineering and applied sciences.