Dynamics and non-integrability of the Swinging Atwood Machine with a massive string: chaos, periodic orbits and resonance structures

Abstract

Building upon our previous studies on nonlinear variable-length pendulum systems, we investigate the Swinging Atwood Machine with a massive string. In contrast to the classical model, string inertia introduces a configuration-dependent moment of inertia, leading to a modified Hamiltonian structure and substantially richer dynamics. To uncover the global organization of the phase space, we combine Poincaré sections, bifurcation diagrams, and Lyapunov exponent maps with our recently developed numerical framework, ,,Lyapunov Refined Maps". This approach provides a unified visualization of periodic, quasi-periodic, chaotic, and terminating motions, revealing intricate resonance networks and high-order periodic structures. We investigate the influence of the string mass, system parameters, and energy by constructing Lyapunov maps in parameter and initial-condition spaces and on fixed-energy surfaces. Liouville integrability is studied within the Morales--Ramis theory. By analyzing the normal variational equations along explicit non-stationary radial solutions and applying the Kovacic algorithm, we prove that the differential Galois group is generically SL(2,C), providing a rigorous obstruction to meromorphic Liouville integrability for every nonzero string mass. Thus, the exceptional integrable case of the classical Swinging Atwood Machine is destroyed by the inclusion of string inertia.