# Bifurcation of heteroclinic orbits via an index theory

## TL;DR

This paper introduces a new $ extbf{Z}_2$-index and develops an index theory for heteroclinic orbits in nonautonomous vector fields, leading to novel bifurcation results under standard assumptions.

## Contribution

The paper defines a new $ extbf{Z}_2$-index and constructs an index theory for heteroclinic orbits, connecting it to the parity invariant and deriving new bifurcation results.

## Key findings

- Established the $ extbf{Z}_2$-index equals the parity under transversality.
- Proved an index theorem linking the $ extbf{Z}_2$-index to bifurcation phenomena.
- Derived a new bifurcation result for heteroclinic orbits in nonautonomous systems.

## Abstract

Heteroclinic orbits for one-parameter families of nonautonomous vectorfields appear in a very natural way in many physical applications. Inspired by some recent bifurcation results for homoclinic trajectories of nonautonomous vectorfield, we define a new $\mathbf Z_2$-index and we construct a index theory for heteroclinic orbits of nonautonomous vectorfield. We prove an index theorem, by showing that, under some standard transversality assumptions, the $\mathbf Z_2$-index is equal to the parity, a homotopy invariant for paths of Fredholm operators of index 0. As a direct consequence of the index theory developed in this paper, we get a new bifurcation result for heteroclinic orbits.