Parametric resonance and nonlinear dynamics in a coupled double-pendulum system

Nonlinear dynamics plays a significant role in interdisciplinary fields spanning biology, engineering, mathematics, and physics. Under small-amplitude approximations, certain nonlinear systems can be effectively described by the linear Mathieu equation. Within the Lagrangian mechanics framework, here we systematically investigate nonlinear dynamics and parametric resonance phenomena in a collision-coupled double pendulum system. The complete dynamical model successfully captures the system’s nonlinear behavior under large angular displacements. Numerical simulations and experimental validations demonstrate parametric resonance within specific frequency ratio ranges, with resonance boundaries showing good agreement with theoretical predictions. The collision coupling mechanism enables unique energy transfer properties, while system nonlinearity ultimately leads to amplitude saturation. Notably, frequency modulation techniques achieve perturbation-free excitation of parametric resonance. We propose a novel example in nonlinear dynamics, with a methodological framework applicable to broader collision-coupled systems.