Hopf bifurcation on the hexagonal lattice with small frequency (Q1907690)

A system of six complex, cubic ODE's for \(z_i \in \mathbb{C}\), \(i = 1, 2, \dots, 6\), is considered which has the symmetry of a hexagonal lattice. The amplitudes \(z_i\) are related to six eigenmodes in the \((x,y)\)-plane, \[ z_{1,4} : e^{\pm i \alpha z}, \quad z_{2,3,5,6} : e^{i \alpha {\pm x \pm \sqrt 3 y \over 2}}. \] These equations arise by a center manifold reduction for the two-layer Bénard problem. Reduced systems for solutions of a prescribed symmetry for steady states and traveling waves (rolls, hexagons, triangles, rectangles, etc.) are discussed. The paper aims on Hopf bifurcation with a small frequency leading to periodic solutions with high period. In this case the Birkhoff normal form reduction is not possible.