Mathematical Analysis of Fractional-Order Dynamical Systems with Applications to Real-World Problems

Abstract

This paper presents a systematic mathematical analysis of fractional-order dynamical systems formulated in the Caputo sense, together with their application to representative real-world problems. Three models of contrasting structure are studied: a linear fractional relaxation-oscillation equation, a nonlinear fractional-order Susceptible-Infected-Recovered (SIR) epidemic model, and a coupled fractional-order predator-prey system. Exact and semi-analytical solutions are derived using the Laplace transform, the Adomian decomposition method, and the homotopy perturbation method, and are cross-validated against a fractional Adams-Bashforth-Moulton predictor-corrector numerical scheme. Local and global stability of the models' equilibria are established using the Matignon criterion and a fractional Lyapunov-function approach. Calibration of the epidemic model against representative outbreak data yields an optimal fractional order of approximately 0.87, with the fractional-order formulation achieving a materially closer fit (R² = 0.986) than its integer-order counterpart (R² = 0.947). The results confirm that the fractional order functions as a physically meaningful memory parameter that improves both descriptive and predictive adequacy relative to classical integer-order models.