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The symmetric harmonic three-mass system with finite rest lengths, despiteits apparent simplicity, displays a wide array of interesting dynamics fordifferent energy values. At low energy the system shows regular behavior thatproduces a deformation-induced rotation with a constant averaged angularvelocity. As the energy is increased this behavior makes way to a chaoticregime with rotational behavior statistically resembling L\'evy walks andrandom walks. At high enough energies, where the rest lengths becomenegligible, the chaotic signature vanishes and the system returns toregularity, with a single dominant frequency. The transition to and from chaos,as well as the anomalous power law statistics measured for the angulardisplacement of the harmonic three mass system are largely governed by thestructure of regular solutions of this mixed Hamiltonian system. Thus a deeperunderstating of the system's irregular behavior requires mapping out itsregular solutions.  In this work we provide a comprehensive analysis of the system's regularregimes of motion, using perturbative methods to derive analytical expressionsof the system as almost-integrable in its low- and high-energy extremes. Thecompatibility of this description with the full system is shown numerically. Inthe low-energy regime, the Birkhoff normal form method is utilized tocircumvent the low-order 1:1 resonance of the system, and the conditions forKolmogorov-Arnold-Moser theory are shown to hold. The integrable approximationsprovide the back-bone structure around which the behavior of the fullnon-linear system is organized, and provide a pathway to understanding theorigin of the power-law statistics measured in the system.

Regular Regimes of the Three Body Harmonic System