Review of Fractional-Order Dynamical Systems: Mathematical Developments and Applications in Science and Engineering

Abstract

Fractional-order dynamical systems have gained considerable attention in recent decades due to their superior ability to model complex phenomena involving memory, hereditary effects, non-local interactions, and anomalous diffusion that cannot be accurately described using traditional integer-order differential equations. This review paper presents a comprehensive analysis of recent mathematical developments in fractional-order dynamical systems and their diverse applications across science and engineering. The study examines the theoretical foundations of fractional calculus, including commonly used fractional derivatives such as Caputo, Riemann–Liouville, and Atangana–Baleanu operators, highlighting their mathematical characteristics and modelling capabilities. It further reviews major analytical approaches, including existence and uniqueness of solutions, stability analysis, bifurcation behaviour, controllability, and numerical approximation techniques employed for solving fractional differential equations. The paper also explores the growing application of fractional-order models in engineering systems, biomedical sciences, control theory, electrical circuits, signal processing, viscoelastic materials, finance, environmental modelling, and epidemiology, where memory-dependent and nonlinear dynamics play a significant role. Recent advances in computational algorithms and numerical simulation methods have expanded the practical implementation of these models, improving prediction accuracy and system representation. Despite these developments, challenges remain in parameter estimation, computational efficiency, model validation, and the development of robust numerical methods for high-dimensional problems. The review identifies current research trends and emerging opportunities, particularly the integration of fractional calculus with artificial intelligence, machine learning, and data-driven modelling to enhance predictive performance in complex systems. By synthesising contemporary literature, this paper provides a consolidated understanding of the mathematical developments, computational techniques, and multidisciplinary applications of fractional-order dynamical systems while outlining future research directions that can further strengthen their theoretical foundations and practical relevance in modern scientific and engineering research.