Dynamics and non-integrability of the double spring pendulum

TL;DR

This study explores the complex dynamics and non-integrability of the double spring pendulum using advanced numerical and analytical methods, revealing chaotic behaviors and identifying conditions for different motion types.

Contribution

It introduces a novel integration of numerical tools and an analytical proof of non-integrability for the double spring pendulum, advancing understanding of Hamiltonian system dynamics.

Findings

Identification of parameter regions with hyper-chaotic, chaotic, quasi-periodic, and periodic motions.

Demonstration of Lyapunov exponents as indicators of integrability.

Analytical proof of non-integrability using differential Galois theory.

Abstract

This paper investigates the dynamics and integrability of the double spring pendulum, which has great importance in studying nonlinear dynamics, chaos, and bifurcations. Being a Hamiltonian system with three degrees of freedom, its analysis presents a significant challenge. To gain insight into the system's dynamics, we employ various numerical methods, including Lyapunov exponents spectra, phase-parametric diagrams, and Poincar\'e cross-sections. The novelty of our work lies in the integration of these three numerical methods into one powerful tool. We provide a comprehensive understanding of the system's dynamics by identifying parameter values or initial conditions that lead to hyper-chaotic, chaotic, quasi-periodic, and periodic motion, which is a novel contribution in the context of Hamiltonian systems. In the absence of gravitational potential, the system exhibits $S^1$ symmetry, and the presence of an additional first integral was identified using Lyapunov exponents diagrams. We demonstrate the effective utilisation of Lyapunov exponents as a potential indicator of first integrals and integrable dynamics. The numerical analysis is complemented by an analytical proof regarding the non-integrability of the system. This proof relies on the analysis of properties of the differential Galois group of variational equations along specific solutions of the system. To facilitate this analysis, we utilised a newly developed extension of the Kovacic algorithm specifically designed for fourth-order differential equations. Overall, our study sheds light on the intricate dynamics and integrability of the double spring pendulum, offering new insights and methodologies for further research in this field.