Fractional Stochastic Modeling of Nonlinear Dynamical Systems: Application to an Electromechanical Process with Memory Effects

In this study, a comprehensive stochastic and fractional-order modeling framework is developed to investigate the dynamic behavior of a shunt DC motor under random disturbances and memory effects. The motor dynamics are formulated as a system of stochastic differential equations incorporating Gaussian noise to represent uncertainties in the electrical and mechanical subsystems. The existence, stochastic ultimate boundedness, stationary distribution, and ergodic properties of the proposed model are established. To further enhance modeling capabilities, a modified Atangana–Baleanu–Caputo (mABC) fractional operator is introduced, enabling the incorporation of nonlocal memory effects inherent in electromechanical systems. The series solution is derived using the Laplace transform and the Adomian decomposition method to handle nonlinearities. Qualitative analysis of the solution is performed through fixed-point theory, while stability assessments utilize the T-Picard method. The results of the numerical simulation indicate that the stochastic model exhibits limited variability around the operating regimes, whereas the fractional-order representation is more effective at smoothing transient responses and limiting oscillatory behavior. The study proposes a realistic and adaptable method to analyze the dynamics of shunt DC motors with uncertainty and also presents useful information for the design and control of electromechanical systems.