# Equilibrium Index of Invariant Sets and Global Static Bifurcation for Nonlinear Evolution Equations

## TL;DR

This paper develops an equilibrium index for invariant sets in nonlinear evolution equations, providing a new global bifurcation theorem that applies even when the crossing number is even, with applications to boundary value problems.

## Contribution

It introduces a novel equilibrium index for invariant sets and establishes a global static bifurcation theorem applicable to cases with even crossing numbers.

## Key findings

- Established an index formula for bifurcating invariant sets.
- Proved a global bifurcation theorem valid for even crossing numbers.
- Provided an example involving bifurcations in boundary value problems.

## Abstract

We introduce the notion of equilibrium index for statically isolated invariant sets of the system $u_t+A u=f_\lambda(u)$ on Banach space $X$ (where $A$ is a sectorial operator with compact resolvent) and present a reduction theorem and an index formula for bifurcating invariant sets near equilibrium points. Then we prove a new global static bifurcation theorem where the crossing number $\mathfrak{m}$ may be even. In particular, in case $\mathfrak{m}=2$, we show that the system undergoes either an attractor/repeller bifurcation, or a global static bifurcation. An illustrating example is also given by considering the bifurcations of the periodic boundary value problem of second-order differential equations.