2:00 pm, Lecture Hall 5
Instabilities, chaos and ergodicity in the three rotor problem.
Chennai Mathematical Institute.
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. 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 around 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 accumulation of stable-to-unstable 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 critical energy, where we provide numerical evidence for ergodicity and mixing.