Applied equivariant degree, part I: An axiomatic approach to primary degree

Abstract

An axiomatic approach to the primary equivariant degree is discussed and a construction of the primary equivariant degree via fundamental domains is presented. For a class of equivariant maps, which naturally appear in one-parameter equivariant Hopf bifurcation, effective computational primary degree formulae are established.

On the space of equivariant local maps

“…The following result is an immediate consequence of the invariant tubular neighbourhood theorem (see for example section 2.4.3 in [2]).…”

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

“…We say that S xo is positively oriented if the orientation of the slice followed by the orientation of the orbit W (H)x o (induced by the fixed orientation of W (H)) coincides with the (fixed) orientation of R ⊕ V H (see, for instance, [1]). 3.3.…”

“…This is the second paper in a series devoted to the equivariant degree theory and its applications to non-linear problems admitting a certain (in general, non-abelian) compact Lie group of symmetries (cf. [1]). Our main goal is to study, by means of the equivariant degree theory, the occurrence of Hopf bifurcations in a symmetric system of delayed functional differential equations.…”

“…The equivariant degree theory (cf. [1,2,4,5,14,18,21,25,26]) provides the most effective method for a full analysis of symmetric Hopf bifurcation problems (cf. [5,6,12,18,24,34,35]).…”

“…Section 2 contains the equivariant topology and analysis background together with algebraic constructions frequently used throughout the paper. In section 3, we introduce the primary equivariant degree, following an axiomatic approach developed in [1], discuss the multiplicativity property and the Splitting Lemma. In section 4, we present a general parameterized system of symmetric delayed functional differential equations, introduce the notion of the so-called isotypical crossing number, and construct the Γ × S 1 -equivariant mapping F ς associated with the Hopf bifurcation problem.…”