2018/10/31 by Govind S. Krishnaswami, Himalaya Senapati · 2 citations
Mathematics · Physics and Astronomy · #math-ph #math.DS #math.MP #nlin.CD
paper · pdf · doi:10.29195/iascs.02.01.0020
published as Indian Academy of Sciences Conference Series 2(1), pp. 139-143 (2019) · 8 pages, 8 figures, Presented at the Conference on Nonlinear Systems and Dynamics, New Delhi, Oct 2018
arxiv created 2019/09/24 · arxiv updated 2019/09/25
We study the equal-mass classical three rotor problem, a variant of the three body problem of celestial mechanics. The quantum N-rotor problem has been used to model chains of coupled Josephson junctions and also 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 Ec ≈ 4, followed by a gradual return to regularity as E → ∞. We find four signatures of this transition: (a) the fraction of the area of Poincaré surfaces occupied by chaotic sections rises sharply at Ec, (b) discrete symmetries are spontaneously broken at Ec, (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, Poincaré plots also reveal a regime of global chaos slightly above Ec.