Regular regimes of the harmonic three-mass system

Area of Science:

• Classical Mechanics
• Nonlinear Dynamics
• Statistical Physics

Background:

• The symmetric harmonic three-mass system exhibits complex dynamics across various energy levels.
• Its behavior ranges from regular, deformation-induced rotation at low energies to chaotic regimes resembling Lévy walks at higher energies.

Purpose of the Study:

• To comprehensively analyze the regular motion regimes of the three-mass system.
• To understand the origin of anomalous power-law statistics observed in its angular displacement.
• To map out regular solutions governing the transitions to and from chaos.

Main Methods:

• Perturbative methods to derive analytical expressions for almost-integrable low- and high-energy extremes.
• Numerical verification of the analytical descriptions.
• Birkhoff normal form method to address 1:1 resonance in the low-energy regime.
• Application of Kolmogorov-Arnold-Moser theory conditions.

Main Results:

• Analytical expressions for low- and high-energy almost-integrable regimes were derived.
• The system returns to regularity at high energies with a single dominant frequency.
• Regular solutions provide the structural backbone for the nonlinear system's behavior.

Conclusions:

• The study provides a pathway to understanding the origin of power-law statistics in the system.
• Integrable approximations are crucial for organizing the full nonlinear system's dynamics.
• A deeper understanding of irregular behavior is achieved by mapping regular solutions.