The symmetric harmonic three-mass system with finite rest lengths, despite its apparent simplicity, displays a wide array of interesting dynamics for different energy values. At low energy the system shows regular behavior that produces a deformation-induced rotation with a constant averaged angular velocity. As the energy is increased this behavior makes way to a chaotic regime with rotational behavior statistically resembling Lévy walks and random walks. At high enough energies, where the rest lengths become negligible, the chaotic signature vanishes and the system returns to regularity, with a single dominant frequency. The transition to and from chaos, as well as the anomalous power-law statistics measured for the angular displacement of the harmonic three-mass system are largely governed by the structure of regular solutions of this mixed Hamiltonian system. Thus, a deeper understating of the system's irregular behavior requires mapping out its regular solutions. In this work we provide a comprehensive analysis of the system's regular regimes of motion, using perturbative methods to derive analytical expressions of the system as almost-integrable in its low- and high-energy extremes. The compatibility of this description with the full system is shown numerically. In the low-energy regime, the Birkhoff normal form method is utilized to circumvent the low-order 1:1 resonance of the system, and the conditions for Kolmogorov-Arnold-Moser theory are shown to hold. The integrable approximations provide the back-bone structure around which the behavior of the full nonlinear system is organized and provide a pathway to understanding the origin of the power-law statistics measured in the system.

中文翻译：

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三次谐波系统的规则形式

尽管具有明显的静止长度，但具有有限静止长度的对称谐波三质量系统却显示出各种有趣的动力学，涉及不同的能量值。在低能量下，系统显示出规律的行为，该行为会产生变形诱导的旋转，并具有恒定的平均角速度。随着能量的增加，这种行为逐渐演变为混沌状态，其旋转行为在统计上类似于莱维步行和随机步行。在足够高的能量下，其余长度可以忽略不计，混沌信号消失，系统以单一主导频率恢复规律。进入和摆脱混乱，以及针对谐波三质量系统角位移测得的异常幂律统计在很大程度上受该混合哈密顿系统的正则解的结构支配。因此，对系统的不规则行为的更深层次的低估需要制定出其规则的解决方案。在这项工作中，我们使用扰动方法得出系统的运动规律的综合分析，得出系统的分析表达式在其低能量和高能量极端情况下几乎可以积分。此说明与整个系统的兼容性以数字形式显示。在低能状态下，利用Birkhoff范式方法规避了系统的低阶1：1共振，并证明了Kolmogorov-Arnold-Moser理论的条件成立。