Dynamics and non-integrability of the variable-length double pendulum: Exploring chaos and periodicity via the Lyapunov refined maps

This paper extends our previous work (Szumiński and Maciejewski, 2024), where we explored the dynamics and integrability of the double-spring pendulum. Here, we investigate the variable-length double pendulum , a three-degree-of-freedom Hamiltonian system combining features of the classic double pendulum and the swinging Atwood machine. With its intricate dynamics, this system is crucial for studying nonlinear phenomena such as high-order resonances, chaos, and bifurcations. We address the challenges posed by high-dimensional phase spaces using a novel tool, the Lyapunov refined maps , which integrates Poincaré sections , phase-parametric diagrams, and Lyapunov exponents . This framework comprehensively analyzes periodic, quasi-periodic, and chaotic behaviors. By measuring the strength of chaos, it also offers insights into the system’s dynamical structure. Additionally, we apply Morales-Ramis theory to examine integrability, leveraging the differential Galois group of variational equations to establish non-integrability conditions. The Kovacic algorithm is used to analyze the solvability of higher-dimensional differential equations , complemented by Lyapunov exponent diagrams to exclude integrable dynamics under certain parameters. Our findings advance the fundamental understanding of variable-length pendulum dynamics, offering new insights and methodologies for further research with potential applications in adaptive robotics, energy harvesting , and biomechanics . Additionally, this work represents a significant step toward proving the long-sought non-integrability of the classical double pendulum. • A novel model of a variable-length pendulum system is examined. • Lyapunov’s refined map is a new integrated method for global analysis of high-dimensional phase spaces. • Lyapunov diagrams are linked with first integrals and integrable dynamical behavior. • Morales-Ramis theory and the Kovacic algorithm prove non-integrability of the system.