Mathematische Zeitschrift

, Volume 292, Issue 1–2, pp 705–723 | Cite as

Bifurcation of heteroclinic orbits via an index theory

  • Xijun Hu
  • Alessandro PortaluriEmail author


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.


K-theory Index bundle \(\mathbf {Z}_2\)-index Parity of Fredholm operators Heteroclinic orbits 

AMS Subject Classification

34C37 37C29 47J15 53D12 70K44 



We thank the anonymous referee for fixing some inaccuracies in the preliminary version and for his/her comments which contributed to improving the quality of the publication.


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© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsShandong UniversityJinanChina
  2. 2.DISAFA, Università di TorinoGrugliascoItaly

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