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Index theory for heteroclinic orbits of Hamiltonian systems

  • Xijun HuEmail author
  • Alessandro Portaluri
Article

Abstract

Index theory revealed its outstanding role in the study of periodic orbits of Hamiltonian systems and the dynamical consequences of this theory are enormous. Although the index theory in the periodic case is well-established, very few results are known in the case of homoclinic orbits of Hamiltonian systems. Moreover, to the authors’ knowledge, no results have been yet proved in the case of heteroclinic and halfclinic (i.e. parametrized by a half-line) orbits. Motivated by the importance played by these motions in understanding several challenging problems in Classical Mechanics, we develop a new index theory and we prove at once a general spectral flow formula for heteroclinic, homoclinic and halfclinic trajectories. Finally we show how this index theory can be used to recover all the (classical) existing results on orbits parametrized by bounded intervals.

Mathematics Subject Classification

53D12 58J30 34C37 37C29 

Notes

Acknowledgements

We thank the anonymous referee for fixing some typos and for the comments that improved the presentation of the paper. We are grateful to proff. Nils Waterstraat, Jacobo Pejsachowicz and Chaofeng Zhu, for many stimulating discussions on this subject.

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Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Department of MathematicsShandong UniversityJinanThe People’s Republic of China
  2. 2.DISAFAUniversità degli Studi di TorinoTorinoItaly

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