Bifurcation of heteroclinic orbits via an index theory

Abstract

Heteroclinic orbits for one-parameter families of nonautonomous vectorfields appear in a very natural way in many physical applications. Inspired by a recent bifurcation result for homoclinic trajectories of nonautonomous vectorfield proved by author in [13], 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.