# 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.

- Last modified: 8/12/2014