Bifurcation cascade, self-similarity, and duality in the three-rotor problem

Abstract

The three-rotor system concerns equally massive point particles moving on a circle subject to attractive cosine potentials of strength g. The quantum theory models chains of coupled Josephson junctions. Classically, it displays order–chaos–order behavior with increasing energy E along with a seemingly globally chaotic phase for 5.33g≲E≲5.6g. It is also known to admit pendulum and isosceles breather families of periodic orbits at all energies. While pendula display a doubly infinite sequence of stability transitions accumulating at their libration to rotation threshold at E=4g, breathers undergo only one stability transition. Here, we show that these stability transitions are associated with forward and reverse fork-like isochronous and period-doubling bifurcations. The new family of periodic orbits born at each of these bifurcations is found using an efficient search algorithm starting from a transverse perturbation to the parent orbit. The graphs of stability indices of various classes of orbits born at pendulum bifurcations meet at E=4g forming “fans.” The transitions in the librational and rotational phases are related by an asymptotic duality between bifurcation energies and shapes of newly born periodic orbits. The latter are captured by solutions to a Lamé equation. We also find and numerically validate values of scaling constants for self-similarity in (a) stability indices of librational and rotational pendula and (b) shapes of newly born orbits as E→4g. Finally, we argue that none of the infinitely many families of periodic orbits we have found is stable for 5.33g≲E≲5.6g, providing further evidence for global chaos in this energy band.