Nonlinear Dynamics and Stability Analysis of a Fractional Coupled Spring-Pendulum System: Historical Perspective, Comparative Methodology, and Future Directions

Abstract

Fractional calculus provides a powerful mathematical framework for modeling memory-dependent phenomena in dynamical systems; however, its application to coupled multi-body configurations with internal resonance remains largely unexplored. This paper develops a consistent fractional Hamiltonian framework for a coupled spring-pendulum system using Caputo fractional derivatives, incorporating fractional damping characterized by orders between zero and one and fractional inertia described by higher-order fractional parameters ranging from one to two. The governing equations are analyzed through three complementary methodologies: the Optimal Homotopy Perturbation Method (OHPM), the Multi-step Differential Transform Method (Ms-DTM), and Grünwald-Letnikov numerical simulations as a benchmark. Validation against exact analytical solutions for fractional Duffing, pendulum, and Mathews-Lakshmanan oscillators demonstrates that the Continuous Parameter Linearization Method (CPLM) achieves exceptional accuracy, significantly outperforming conventional amplitude-frequency formulations. The analysis reveals three characteristic effects of fractional-order dynamics: power-law energy dissipation governed by the Mittag-Leffler function, extended stabilization times associated with memory-dependent damping, and the emergence of quasi-periodic oscillations below a critical fractional-order threshold. Internal resonance analysis at one-to-one and two-to-one frequency ratios shows that fractional orders can substantially amplify inter-mode energy transfer, providing an additional mechanism for resonance control. This consistent fractional Hamiltonian framework offers a pathway toward treating fixed material properties as effectively tunable parameters within the modeled regime, suggesting potential applications in adaptive vibration control, energy harvesting, soft robotics, and seismic isolation systems.