Index theory for heteroclinic orbits of Hamiltonian systems

  title={Index theory for heteroclinic orbits of Hamiltonian systems},
  author={Xijun Hu and Alessandro Portaluri},
  journal={Calculus of Variations and Partial Differential Equations},
  • Xijun Hu, A. Portaluri
  • Published 11 March 2017
  • Mathematics
  • Calculus of Variations and Partial Differential Equations
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… 
Morse index theorem for heteroclinic orbits of Lagrangian systems
The classical Morse Index Theorem plays a central role in Lagrangian dynamics and differential geometry. Although many generalization of this result are well-known, in the case of orbits of
Bifurcation of heteroclinic orbits via an index theory
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
Mean Index for Non-periodic Orbits in Hamiltonian Systems
  • Xijun Hu, Li Wu
  • Mathematics
    Acta Mathematica Sinica, English Series
  • 2022
In this paper, we define mean index for non-periodic orbits in Hamiltonian systems and study its properties. In general, the mean index is an interval in ℝ which is uniformly continuous on the
On the Fredholm Lagrangian Grassmannian, spectral flow and ODEs in Hilbert spaces
Linear instability for periodic orbits of non-autonomous Lagrangian systems
Inspired by the classical Poincaré criterion about the instability of orientation preserving minimizing closed geodesics on surfaces, we investigate the relation intertwining the instability and the
Linear instability of periodic orbits of free period Lagrangian systems
In this paper we provide a sufficient condition for the linear instability of a periodic orbit for a free period Lagrangian system on a Riemannian manifold. The main result establish a general
Renormalized Oscillation Theory for Linear Hamiltonian Systems on [0, 1] Via the Maslov Index
Working with a general class of linear Hamiltonian systems on $[0, 1]$, we show that renormalized oscillation results can be obtained in a natural way through consideration of the Maslov index
Keplerian Orbits Through the Conley–Zehnder Index
It was discovered by Gordon (Am J Math 99(5):961–971, 1977) that Keplerian ellipses in the plane are minimizers of the Lagrangian action and spectrally stable as periodic points of the associated
An Index Theory for Zero Energy Solutions of the Planar Anisotropic Kepler Problem
In the variational study of singular Lagrange systems, the zero energy solutions play an important role. The anisotropic Kepler problem is such a singular system introduced by physicist M. Gutzwiller


Index and Stability of Symmetric Periodic Orbits in Hamiltonian Systems with Application to Figure-Eight Orbit
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
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
Spectral Flow and Bifurcation of Critical Points of Strongly Indefinite Functionals
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
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
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