Hopf bifurcation with the symmetry of the square

Four identical nonlinear oscillators, coupled with the symmetry of a square, can undergo a symmetric version of the standard Hopf bifurcation. Golubitsky and Stewart (1986) have studied the case of N oscillators coupled in a ring with nearest-neighbour coupling. Their results are incomplete for the square case (N=4) because they only considered periodic solutions which have 'maximal' symmetry. Here the author studies the dynamics of all possible square-symmetric Hopf bifurcations; these codimension-one bifurcations are parametrised by the three complex cubic coefficients in the normal form. He finds that invariant tori (quasiperiodic solutions with two frequencies) and periodic solutions with 'minimal' symmetry bifurcate from the origin for open regions of the parameter space of cubic coefficients. The coefficients can be chosen so that the invariant tori are the only asymptotically stable solutions near the origin. Thus a direct transition from a stable fixed point to flow on a stable invariant torus is expected in certain laboratory experiments with only one adjustable parameter (provided the square symmetry is accurate enough). Furthermore, it is conjectured that there are attracting chaotic solutions arbitrarily close to the bifurcation in a certain codimension-two case.