# Spectral flow, crossing forms and homoclinics of Hamiltonian systems

@article{Waterstraat2014SpectralFC, title={Spectral flow, crossing forms and homoclinics of Hamiltonian systems}, author={Nils Waterstraat}, journal={arXiv: Dynamical Systems}, year={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 curves of Lagrangians induced by the stable and unstable subspaces, respectively. Finally, we deduce sufficient conditions for bifurcation of homoclinic trajectories of one-parameter families of nonautonomous Hamiltonian vector fields.

## 15 Citations

A K-theoretical invariant and bifurcation for homoclinics of Hamiltonian systems

- Mathematics
- 2016

We revisit a K-theoretical invariant that was invented by the first author some years ago for studying multiparameter bifurcation of branches of critical points of functionals. Our main aim is to…

Spectral flow and bifurcation for a class of strongly indefinite elliptic systems

- MathematicsProceedings of the Royal Society of Edinburgh: Section A Mathematics
- 2017

We consider bifurcation of solutions from a given trivial branch for a class of strongly indefinite elliptic systems via the spectral flow. Our main results establish bifurcation invariants that can…

The Maslov index and the spectral flow—revisited

- MathematicsFixed Point Theory and Applications
- 2019

We give an elementary proof of a celebrated theorem of Cappell, Lee and Miller which relates the Maslov index of a pair of paths of Lagrangian subspaces to the spectral flow of an associated path of…

On the Fredholm Lagrangian Grassmannian, spectral flow and ODEs in Hilbert spaces

- MathematicsJournal of Differential Equations
- 2021

We consider homoclinic solutions for Hamiltonian systems in symplectic Hilbert spaces and generalise spectral flow formulas that were proved by Pejsachowicz and the author in finite dimensions some…

A dihedral Bott-type iteration formula and stability of symmetric periodic orbits

- Mathematics
- 2017

Motivated by the recent discoveries on the stability properties of symmetric periodic solutions of Hamiltonian systems, we establish a Bott-type iteration formula for dihedraly equivariant…

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…

A Morse-Smale index theorem for indefinite elliptic systems and bifurcation

- Mathematics
- 2014

We generalise the semi-Riemannian Morse index theorem to elliptic systems of partial differential equations on star-shaped domains. Moreover, we apply our theorem to bifurcation from a branch of…

Linear instability of periodic orbits of free period Lagrangian systems

- Mathematics
- 2021

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…

Linear instability for periodic orbits of non-autonomous Lagrangian systems

- Physics, Mathematics
- 2019

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…

On a Comparison Principle and the Uniqueness of Spectral Flow

- Mathematics
- 2019

The spectral flow is a well-known quantity in spectral theory that measures the variation of spectra about $0$ along paths of selfadjoint Fredholm operators. The aim of this work is twofold. Firstly,…

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