Hopf Bifurcations of Functional Differential Equations with Dihedral Symmetries

A short treatise on the equivariant degree theory and its applications

“…The importance of S 1 -basic maps was indicated in [8] (see also [38,68,12]), where the computational formulae for the S 1 -degree were established (see also [37,68]). Twisted subgroups of Γ × S 1 were studied in [4,5,9,11,12,7,8,53,49,50,66,67,102] (see also [13]). The twisted equivariant degree was considered in [5,9,8].…”

Applied equivariant degree. part II: Symmetric Hopf bifurcations of functional differential equations

“…Take F defined by (19) and construct Ω according to (21). Let ς : Ω → R be a G-invariant auxiliary function (see (22)) and let F ς be defined by (23).…”

“…The equivariant degree method that was used for studying symmetric Hopf bifurcation problems in [21,22,24,33,34,35] (see also [12,23,18]), led to the results based on partial computations only. This was mainly due to technical difficulties related to the absence of a general computational scheme and elaborated algebraic calculations.…”

Equivariant normal forms for neutral functional differential equations

“…An important contribution was made by Ruelle [17] early, and recent progress in the normal form theory seems mainly due to physicists, for example, Coullet and Spiegel [3], and Elphick et al [4]. There are some studies for the calculation of normal forms and for their applications to bifurcations for Functional Differential Equations (FDEs) with symmetries, for example, [1,2,[13][14][15]21], but these studies are restricted to particular classes of Retarded FDEs and usually involve the immediate step of reduction to center manifolds before calculating and using the normal form. Normal forms have also been developed by Weedermann [19,20] or Wang and Wei [18] for NFDEs without symmetry.…”