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Classical three rotor problem: Periodic solutions, stability and chaos.

Chaos (Woodbury, N.Y.)·2020
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Ergodicity, mixing, and recurrence in the three rotor problem.

Govind S Krishnaswami1, Himalaya Senapati1

  • 1Physics Department, Chennai Mathematical Institute, SIPCOT IT Park, Siruseri, Chennai 603103, India.

Chaos (Woodbury, N.Y.)
|May 3, 2020
PubMed
Summary

This study numerically confirms ergodicity and mixing in a chaotic band of the three rotor problem. Trajectories uniformly distribute on energy hypersurfaces, demonstrating ergodic and mixing dynamics within this specific energy range.

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Area of Science:

  • Classical Mechanics
  • Dynamical Systems
  • Statistical Mechanics

Background:

  • The three rotor problem involves three equal point masses on a circle with cosine potentials.
  • Previous research identified an order-chaos-order transition and a global chaos band (5.33g≤E≤5.6g).

Purpose of the Study:

  • To provide numerical evidence for ergodicity and mixing within the global chaos band of the three rotor problem.
  • To investigate the behavior of trajectories and distributions outside this chaotic band.

Main Methods:

  • Numerical simulations of the three rotor problem.
  • Analysis of trajectory distributions (relative angles, angular momenta) over time.
  • Examination of recurrence time distributions within the chaotic band.

Main Results:

  • Ergodicity and mixing are numerically confirmed within the 5.33g≤E≤5.6g chaos band.
  • Trajectory distributions approach Liouville measure distributions on constant energy hypersurfaces.
  • Outside the band, ergodicity and mixing fail, with phase transitions observed in angular momentum distributions.

Conclusions:

  • The identified band of global chaos in the three rotor problem exhibits ergodic and mixing dynamics.
  • Recurrence time distributions in the chaos band follow an exponential law, consistent with ergodicity.
  • The study highlights distinct dynamical behaviors inside and outside the global chaos band.