Classical three rotor problem: periodic solutions, stability and chaos

Abstract

This paper concerns 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. Unlike in the gravitational problem, there are no singularities (neither collisional nor non-collisional), leading to global existence and uniqueness of solutions. In appropriate units, the non-negative energy $E$ of the relative motion is the only free 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: it is integrable at zero and infinitely high energies but displays a fairly sharp transition from regular to chaotic behavior as $E$ is increased beyond $E_c \approx 4$ and a more gradual return to regularity. The transition to chaos is manifested in a dramatic rise of the fraction of the area of the Hill region of Poincar\'e surfaces occupied by chaotic sections and also in the spontaneous breaking of discrete symmetries of Poincar\'e sections present at lower energies. Interestingly, the above pendulum solutions alternate between being stable and unstable, with the transition energies cascading geometrically from either side at $E = 4$. The transition to chaos is also reflected in the curvature of the Jacobi-Maupertuis metric that ceases to be everywhere positive when $E$ exceeds four. Examination of Poincar\'e sections also indicates global chaos in a band of energies $(5.33 \lesssim E \lesssim 5.6)$ slightly above this transition.