An Equivariant Degree with Applications to Symmetric Bifurcation Problems. Part 1: Construction of the Degree

Applications of equivariant degree for gradient maps to symmetric Newtonian systems

“…The equivariant degree for G-orthogonal maps is constructed using the primary equivariant degree with one free parameter. We show that the G-orthogonal degree extends the degree for G-gradiernt maps (in the case G = Γ × S 1 ) introduced by K. Gȩba in [19]. The obtained computational results are applied to a Γ-symmetric autonomous Newtonian system for which we study the existence of 2π-periodic solutions.…”

“…[14,16,18,19,21,27], see also [2,8,15,24,28,29,30,31,34]), which are important tools of the equivariant analysis, provide an effective alternative to such methods as Conley index, Morse theory, minimax techniques and singularity theory. The main difficulty related to the usage of the equivariant degree seems to be its complicated construction relying on the notions from the equivariant topology, homotopy theory and algebraic topology.…”

Hopf Bifurcations of Functional Differential Equations with Dihedral Symmetries

“…We call G-Deg( f, 0) the primary degree 1 of f in 0. We will write G-Deg( f, 0)= (H ) n H (H ), where the summation is taken over all the primary orbit types in V. The primary degree was introduced independently of the work of Ize et al in [5] (see also [21,22]). The primary degree of f : 0 Ä V can be expressed by an analytic formula: Approximate f by a regular normal map (i.e., a map satisfying certain normality and transversality conditions, see [21] for more details) g:…”

Equivariant Hopf bifurcation for neutral functional differential equations

“…In [12], Krawcewicz, Ma and Wu employ equivariant degree theory developed by Geba et al (cf. [4]) to study the existence, multiplicity and global continuations of symmetric periodic solutions for a one-parameter family of NFDEs with dihedral symmetry.…”