Stability and chaos in the classical three rotor problem

Research paper by Govind S. Krishnaswami, Himalaya Senapati

Indexed on: 02 Oct '18Published on: 02 Oct '18Published in: arXiv - Nonlinear Sciences - Chaotic Dynamics


We study the equal-mass classical three rotor problem, a variant of the three body problem of celestial mechanics. The quantum N-rotor problem arises via a partial continuum limit of the Wick rotated XY model. In units of the coupling, the energy serves as a control parameter. We find periodic `pendulum' and `breather' orbits at all energies and choreographies at relatively low energies. They furnish analogs of the Euler-Lagrange and figure-8 solutions of the planar three body problem. Integrability at very low energies gives way to a rather marked transition to chaos at $E_c \approx 4$, followed by a gradual return to regularity as $E \to \infty$. We find four signatures of this transition: (a) the fraction of the area of Poincar\'e surfaces occupied by chaotic sections rises sharply at $E_c$, (b) discrete symmetries are spontaneously broken at $E_c$, (c) $E=4$ is an accumulation point of stable to unstable transitions in pendulum solutions and (d) the Jacobi-Maupertuis curvature goes from being positive to having both signs above $E=4$. Moreover, Poincar\'e plots also reveal a regime of global chaos slightly above $E_c$.