Double Hopf Bifurcation with Huygens Symmetry

Abstract

Double Hopf bifurcations have been studied prior to this work in the generic nonresonant case and in certain strongly resonant cases, including 1:1 resonance. In this paper, the case of symmetrically coupled identical oscillators, motivated by the classic problem of synchronization of Huygens' clocks, is studied using the codimension-three Elphick--Huygens equivariant normal form presented here. The focus is on the effect that the Huygens symmetry assumption has on the dynamic behavior of the system. Periodic solutions include the classical in-phase and anti-phase normal modes that are forced by the symmetry, as well as pairs of mixed mode phase-locked periodic solutions. The escapement paradox is explained. A theorem based on topological degree theory establishes the existence of quasi-periodic solutions in an invariant 3-torus that resembles a 2-torus slightly thickened to a solid toroidal shell, with the two principal radii of the 2-torus slowly modulated in time---that is, a toroidal breather. Secondary bifurcations from the in-phase and anti-phase normal modes are explored, of codimension one and two, and it is shown that an Arnold tongue plays a fundamental role in the determination of whether secondary bifurcation gives birth to phase-locked periodic solutions or to quasi-periodic solutions. Detailed numerical analysis, using MATLAB, extends the local bifurcation analysis to a more global picture that includes coexistence of multiple stable solutions and a “swallowtail” bifurcation of periodic solutions.