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Theory and applications of equivariant normal forms and Hopf bifurcation for semilinear FDEs in Banach spaces. (English) Zbl 1496.34112

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.

MSC:

34K30 Functional-differential equations in abstract spaces
34K17 Transformation and reduction of functional-differential equations and systems, normal forms
34K18 Bifurcation theory of functional-differential equations
34K13 Periodic solutions to functional-differential equations
34K20 Stability theory of functional-differential equations
34K19 Invariant manifolds of functional-differential equations
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