Theory and applications of equivariant normal forms and Hopf bifurcation for semilinear FDEs in Banach spaces (Q2078892)

The paper extends existing methods for the analysis of autonomous delay differential equations on the existence of invariant manifolds to semilinear functional differential equations. The author summarises the paper in a succinct and comprehensive way: `We show that in the neighborhood of trivial solutions, variables can be chosen so that the form of the reduced vector field relies not only on the information of the linearized system at the critical point but also on the inherent symmetry. We observe that the normal forms give critical information about dynamical properties, such as generic local branching spatiotemporal patterns of equilibria and periodic solutions. As an important application of equivariant normal forms, we not only establish equivariant Hopf bifurcation theorem for semilinear FDEs in general Banach spaces, but also in a natural way derive criteria for the existence, stability, and bifurcation direction of branches of bifurcating periodic solutions. We employ these general results to obtain the existence of infinite many small-amplitude wave solutions for a delayed Ginzburg-Landau equation on a two-dimensional disk with the homogeneous Dirichlet boundary condition.' Sections of the paper on decomposition of the phase space, equivariant normal form, and Hopf bifurcation with symmetry provide detail and useful discussion.