Three faces of equivariant degree
Topological methods based on the usage of degree theory have proved
themselves to be an important tool for qualitative studying of solutions to
nonlinear differential systems (including such problems as existence,
uniqueness, multiplicity, bifurcation, etc.).
During the last twenty years the equivariant degree theory emerged in Non-
linear Analysis. In short, the equivariant degree is a topological tool
allowing “counting” orbits of solutions to symmetric equations in the same
way as the usual Brouwer degree does, but according to their symmetry
properties. This method is an alternative and/or complement to the
equivariant singularity theory developed by M. Golubitsky et al., as well as
to a variety of methods rooted in Morse theory/Lusternik–Schnirelman theory.
In fact, the equivariant degree has different faces reflecting a diversity of
symmetric equations related to applications. In the two talks, I will discuss
three variants of the equivariant degree: (i) non-parameter equivariant
degree, (ii) twisted equivariant degree with one parameter, and (iii)
gradient equivariant degree. Each of the three variants of equivariant degree
will be illustrated by appropriate examples of applications: (i) boundary
value problems for vector symmetric pendulum equation, (ii) Hopf bifurcation
in symmetric neural networks (simulation of legged locomotion), and (iii)
bifurcation of relative equilibria in Lennard-Jones three-body problem.
The talk is addressed to a general audience, without any special knowledge
of the subject.