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Ergodicity, mixing, and recurrence in the three rotor problem.

Chaos (Woodbury, N.Y.)·2020

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Magnetically Induced Rotating Rayleigh-Taylor Instability

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

Govind S Krishnaswami¹, Himalaya Senapati¹

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

Chaos (Woodbury, N.Y.)

|January 3, 2020

Summary

This study explores the dynamics of three coupled rotors, revealing an order-chaos-order behavior. The system transitions from regular to chaotic motion as energy increases, with periodic solutions found at various energy levels.

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Area of Science:

Classical mechanics
Dynamical systems theory
Nonlinear dynamics

Background:

Investigates the classical dynamics of three coupled rotors with attractive cosine potentials.
This system models coupled Josephson junctions in its classical limit.
Energy (E) is the sole free parameter governing the relative motion.

Purpose of the Study:

To identify periodic solutions within the three-rotor system.
To characterize the transition from regular to chaotic dynamics.
To analyze the impact of energy on system stability and symmetry.

Main Methods:

Analysis of periodic solutions, including pendulum, isosceles, and choreographies.
Examination of Poincaré surfaces to identify chaotic regions and symmetry breaking.
Application of the Jacobi-Maupertuis metric to assess curvature and chaos.

Main Results:

Families of periodic solutions (pendulum, isosceles, choreographies) exist across different energy levels.
The system exhibits an order-chaos-order behavior, transitioning to chaos around E≈4.
Pendulum solutions exhibit alternating stability, and discrete symmetries break during the transition to chaos.

Conclusions:

The three-rotor system demonstrates complex dynamics with a clear transition to chaos.
Energy level is critical in determining the system's regularity, chaoticity, and stability.
The findings offer insights into nonlinear dynamics and the behavior of coupled systems.