# Index theory for heteroclinic orbits of Hamiltonian systems

@article{Hu2017IndexTF, title={Index theory for heteroclinic orbits of Hamiltonian systems}, author={Xijun Hu and Alessandro Portaluri}, journal={Calculus of Variations and Partial Differential Equations}, year={2017}, volume={56}, pages={1-24} }

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…

## 26 Citations

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## References

SHOWING 1-10 OF 41 REFERENCES

Index and Stability of Symmetric Periodic Orbits in Hamiltonian Systems with Application to Figure-Eight Orbit

- Mathematics
- 2009

In this paper, using the Maslov index theory in symplectic geometry, we build up some stability criteria for symmetric periodic orbits in a Hamiltonian system, which is motivated by the recent…

Spectral flow, crossing forms and homoclinics of Hamiltonian systems

- Mathematics
- 2014

We prove a spectral flow formula for one‐parameter families of Hamiltonian systems under homoclinic boundary conditions, which relates the spectral flow to the relative Maslov index of a pair of…

Morse index and stability of elliptic Lagrangian solutions in the planar three-body problem

- Mathematics
- 2010

Spectral Flow and Bifurcation of Critical Points of Strongly Indefinite Functionals

- Mathematics
- 2000

Abstract Our main results here are as follows: Let X λ be a family of 2 π -periodic Hamiltonian vectorfields that depend smoothly on a real parameter λ in [ a , b ] and has a known, trivial, branch…

Morse Index and Linear Stability of the Lagrangian Circular Orbit in a Three-Body-Type Problem Via Index Theory

- Mathematics
- 2014

It is well known that the linear stability of the Lagrangian elliptic solutions in the classical planar three-body problem depends on a mass parameter β and on the eccentricity e of the orbit. We…

MASLOV-TYPE INDEX THEORY FOR SYMPLECTIC PATHS AND SPECTRAL FLOW （II）

- Mathematics
- 1999

Based on the spectral flow and the stratification structures of the symplectic group Sp(2n, C),the Maslov-type index theory and its generalization, the ω-index theory parameterized by all ω on the…