Hopf bifurcation on the hexagonal lattice with small frequency

Abstract

Hopf bifurcations with symmetry are typically treated in a two-step process: First the center manifold theorem is used to reduce the equations to a finite dimensional system, and then the resulting ordinary differential equations are reduced to Birkhoff normal form. The latter step involves transformations where the frequency appears in the denominator. In this paper, we are interested in situations where the frequency at the bifurcation point is small, and it would be of interest to consider bifurcated solutions with amplitudes on the same order as the frequency. In this case, the Birkhoff normal form breaks down, and the bifurcation equations become significantly more complex. We consider the Hopf bifurcation on the hexagonal lattice in this context. A possible application arises in the two-layer Bénard problem.