Source data for: A new model of variable-length coupled pendulums – from hyperchaos to superintegrability

This paper studies the dynamics and integrability of a variable-length coupled pendulum system. The complexity of the model is presented by joining various numerical methods, such as the Poincar\'e cross-sections, phase-parametric diagrams, and Lyapunov exponents spectra. We show that the presented model is hyperchaotic, which ensures its nonintegrability. We gave analytical proof of this fact analyzing properties of the differential Galois group of variational equations along certain particular solutions of the system. We employ the Kovacic algorithm and its extension to dimension four to analyze the differential Galois group. Amazingly enough, in the absence of the gravitational potential and for certain values of the parameters, the system can exhibit chaotic, integrable, as well as superintegrable dynamics. To the best of our knowledge, this is the first attempt to use the method of Lyapunov exponents in the systematic search for the first integrals of the system. We show how to effectively apply the Lyapunov exponents as an indicator of integrable dynamics. The explicit forms of integrable and superintegrable systems are given.

本文研究了变长度耦合摆系统的动力学特性与可积性。本文通过结合庞加莱截面（Poincaré cross-sections）、相参数图、李雅普诺夫指数谱（Lyapunov exponents spectra）等多种数值方法，展现了该模型的复杂性。研究表明，所提出的模型具备超混沌特性，这证明其不可积性。我们通过分析该系统特定特解沿线上变分方程（variational equations）的微分伽罗瓦群（differential Galois group）性质，对这一结论给出了解析证明。我们采用科瓦西奇算法（Kovacic algorithm）及其四维推广形式来分析微分伽罗瓦群。值得注意的是，在无引力势（gravitational potential）且参数取特定值时，该系统可呈现混沌、可积乃至超可积的动力学行为。据我们所知，这是首次尝试利用李雅普诺夫指数方法系统性地搜索该系统的首次积分（first integrals）。我们展示了如何将李雅普诺夫指数作为可积动力学的有效指示指标。本文给出了可积与超可积系统的显式形式。