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

Abstract

This paper extends our previous work~(Szumi\'nski 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 \textit{Lyapunov refined maps}, which integrates Poincar\'e 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.