233 research outputs found
Index theory for heteroclinic orbits of Hamiltonian systems
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. parametrised 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 parametrised by
bounded intervals.Comment: 24 pages, 4 figure
Instability of semi-Riemannian closed geodesics
A celebrated result due to Poincar\'e affirms that a closed non-degenerate
minimizing geodesic on an oriented Riemannian surface is hyperbolic.
Starting from this classical theorem, our first main result is a general
instability criterion for timelike and spacelike closed semi-Riemannian
geodesics on a (non)oriented manifold. A key role is played by the spectral
index, a new topological invariant that we define through the spectral flow
(being the Morse index truly infinite) of a path of selfadjoint Fredholm
operators. A major step in the proof of this result is a em new spectral flow
formula. Bott's iteration formula, introduced by author in 1956, relates in a
clear way the Morse index of an iterated closed Riemannian geodesic and the
so-called -Morse indices. Our second result is a semi-Riemannian
generalization of the famous Bott-type iteration formula in the case of closed
(resp. timelike closed) Riemannian (resp. Lorentzian) geodesics. Our last
result is a strong instability result obtained by controlling the Morse index
of the geodesic and of all of its iterations.Comment: 33 pages, 2 figures. Fixed some typos and updated references. arXiv
admin note: text overlap with arXiv:1705.0917
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