Limit cycles and period-doubling bifurcations in a van der pol–type triple pendulum with spring coupling

This paper presents the analytical and experimental analysis of a spring-coupled triple-pendulum system in which a free spring is attached to the end of the second pendulum. This configuration, which has not been addressed in prior literature, introduces a hybrid mechanical system that combines rigid multi-arm dynamics with nonlinear coupling. We derive the full equations of motion and establish well-posedness and boundedness of the solutions. We prove the existence of nontrivial limit cycles and establish their orbital stability via Floquet theory. We then derive the Poincaré return map and identify a critical parameter value at which the system undergoes a period-doubling (flip) bifurcation. To validate the theoretical results, we construct a physical prototype of the proposed system using three-pendulum set up and a freely suspended elastic spring. Time series data, phase portraits, and Poincaré sections confirm the presence of stable periodic motion, bifurcation phenomena, and chaotic behavior.