Chaos in the three-rotor problem

We study the classical dynamics of three coupled rotors: equal masses moving on a circle subject to attractive cosine inter-particle potentials. It is a simpler variant of the gravitational three-body problem and also arises as the classical limit of a model of coupled Josephson junctions. The energy E serves as a control parameter. We find analogues of the Euler-Lagrange family of periodic solutions: pendulum and isosceles solutions at all energies and choreographies up to moderate energies. The model displays order-chaos-order behavior and undergoes a fairly sharp transition to chaos at a critical energy. We present several manifestations of this transition: (a) a dramatic rise in the fraction of Poincare surfaces occupied by chaotic sections, (b) spontaneous breaking of discrete symmetries, (c) a geometric cascade of stability transitions in pendulum solutions and (d) a change in sign of the Jacobi-Maupertuis curvature. Examination of Poincare sections also indicates global chaos in a band of energies slightly above this transition where we provide numerical evidence for ergodicity and mixing and study the statistics of recurrence times. 

This talk is based on joint work with Himalaya Senapati.