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~(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 \textit{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.

Figures (15)

• Figure 1: The geometry of a variable-length double pendulum with a counterweight mass.
• Figure 2: (Color online) The Lyapunov exponents spectra $\Lambda=\{\lambda_1,\ldots, \lambda_6\}$ of system \ref{['eq:rhs']}, computed for the constant values of parameters $\mu_1=3, \mu_2=0.2$, and initial condition \ref{['eq:ini']}. For a sufficient amount of time steps, the convergences of the Lyapunov exponents are ensured.
• Figure 3: (Color online) Three-dimensional Lyapunov's exponents' diagrams of the system \ref{['eq:rhs']} depicted in $(\mu_1,\mu_2,\lambda)$-space, and the projections of $\lambda_1$ and $\lambda_2$ onto $(\mu_1,\mu_2)$-plane. The colorful diagrams were obtained by numerical integrations of Lyapunov's exponent's spectra on a grid of $400\times 400$ values of the parameters $(\mu_1,\mu_2)$, under the respective initial conditions eq:ini. The color scales are logarithmic, corresponding to the magnitudes of $\lambda_1$ and $\lambda_2$, respectively.
• Figure 4: The largest Lyapunov exponent in the plane of the initial swing angles $(\varphi_{10},\varphi_{20})$, constructed for $\mu_1=3$ with varying $\mu_2$. The numerical integrations were performed successively for a uniform grid of $400\times 400$ values of $(\varphi_{10},\varphi_{20})\in(-\pi,\pi)$, with $\ell_0=0.2$ and zero initial velocities. The color scale is logarithmic, corresponding to the magnitude of $\lambda_1$. The plots visualize two zones: regular and chaotic. Blue regions with $\lambda_1\approx 0$ indicate regular dynamics, while the rest of the domain is responsible for the system’s chaotic behavior.
• Figure 5: (Color online) The Lyapunov diagrams for system \ref{['eq:rhs']} in the polar plane $(\ell_0, \varphi_{10})$ with $\varphi_2=0$ are constructed for the initial conditions \ref{['eq:ini_2']}, with varying values of $\mu_2$. In the radial direction, we measure $\ell_0 \in (0, 0.5]$, and in the angular direction, we measure $\varphi_{10} \in (-\pi, \pi)$. The color scale is logarithmic, corresponding to the magnitude of $\lambda_1$. The plots reveal two zones: regular and chaotic. Blue regions indicate regular dynamics, while regions with $\lambda_1 > 0$ correspond to the system’s hyperchaotic behavior.

Theorems & Definitions (17)

• Theorem 1: Morales--Ramis (1999)
• Theorem 2
• proof
• Lemma 1
• proof
• Lemma 2
• proof
• Conjecture 1
• Theorem 3
• proof