Discrete Rotating Waves in Neutral Functional Differential Equations: Symmetric Centre Manifolds and Bifurcations
In this paper, we consider a class of equivariant neutral functional differential equations (NFDEs) with stable $D$ operator. We show the existence of a centre manifold near periodic solutions with finite spatio-temporal symmetry group (a.k.a discrete rotating waves) invariant with respect to the spatio-temporal symmetry group. This is done by extending a construction of integral manifold near periodic solutions of NFDEs (autonomous and non-autonomous) of Hale and Weedermann (J. Diff Eqs, 197 (2004)) to the equivariant class. Using this, we show that the symmetry-breaking bifurcation theory for periodic solutions with finite spatio-temporal symmetry group of Lamb and Melbourne (Arch. Rat. Mech. Anal., 149 (1999)) can be extended from ordinary differential equations to NFDEs. We apply our result to discrete rotating waves obtained in the context of symmetric rings of Lang-Kobayashi equations which are rate equations for laser dynamics.