# Linear instability for periodic orbits of non-autonomous Lagrangian systems

@article{Portaluri2019LinearIF, title={Linear instability for periodic orbits of non-autonomous Lagrangian systems}, author={Alessandro Portaluri and Li Wu and Ran Yang}, journal={Nonlinearity}, year={2019}, volume={34}, pages={237 - 272} }

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 variational properties of periodic solutions of a non-autonomous Lagrangian on a finite dimensional Riemannian manifold. We establish a general criterion for a priori detecting the linear instability of a periodic orbit on a Riemannian manifold for a (maybe not Legendre convex) non-autonomous…

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## References

SHOWING 1-10 OF 59 REFERENCES

A Morse index theorem for perturbed geodesics on semi-Riemannian manifolds

- Mathematics
- 2005

Perturbed geodesics are trajectories of particles moving on
a semi-Riemannian manifold in the presence of a potential. Our
purpose here is to extend to perturbed geodesics on
semi-Riemannian…

Spectral Flow, Maslov Index and Bifurcation of Semi-Riemannian Geodesics

- Mathematics
- 2002

We give a functional analytical proof of the equalitybetween the Maslov index of a semi-Riemannian geodesicand the spectral flow of the path of self-adjointFredholm operators obtained from the index…

Instability of semi-Riemannian closed geodesics

- MathematicsNonlinearity
- 2019

A celebrated result due to Poincare affirms that a closed non-degenerate minimizing geodesic $\gamma$ on an oriented Riemannian surface is hyperbolic. Starting from this classical theorem, our first…

An index theory for asymptotic motions under singular potentials

- Mathematics
- 2017

We develop an index theory for parabolic and collision solutions to the classical n-body problem and we prove sufficient conditions for the finiteness of the spectral index valid in a large class of…

High action orbits for Tonelli Lagrangians and superlinear Hamiltonians on compact configuration spaces

- Mathematics
- 2006

Abstract Multiplicity results for solutions of various boundary value problems are known for dynamical systems on compact configuration manifolds, given by Lagrangians or Hamiltonians which have…

MORSE INDEX AND BIFURCATION OF p-GEODESICS ON SEMI RIEMANNIAN MANIFOLDS

- Mathematics
- 2007

Given a one-parameter family {gλ : λ ∈ (a, b)} of semi Riemannian metrics on an n- dimensional manifold M , a family of time-dependent potentials {Vλ : λ ∈ (a, b)} and a family {σλ : λ ∈ (a, b)} of…

Computation of the Maslov index and the spectral flow via partial signatures

- Mathematics, Physics
- 2004

Given a smooth Lagrangian path, both in the finite and in the infinite di- mensional (Fredholm) case, we introduce the notion of partial signatures at each isolated intersection of the path with the…

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…

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…

Index theory for heteroclinic orbits of Hamiltonian systems

- Mathematics
- 2017

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…