Hopf bifurcation on a square superlattice (Q2722667)

This paper concerns Hopf bifurcation equivariant under the group \(\Gamma:= D_4\ltimes T^2\), which acts \(\Gamma\)-simply on the phase space \(\mathbb{C}^8\). The problem is considered to be the centre manifold reduction of an evolution equation, invariant under the Euclidean group of planar translations, rotations and reflections, where the solutions are supposed to lie on a square lattice.NEWLINENEWLINENEWLINEThe normal form of the bifurcation problem is computed, and conditions for the stability of 17 \(\mathbb{C}\)-axial branches of periodic solutions are given. Moreover, numerical integrations indicate the existence of chaotic dynamics in the third-order truncation of the normal form (in open regions of the coefficient space, but without breaking the normal form symmetry).