  A celebrated result due to Poincar\'e affirms that a closed non-degenerate
minimizing geodesic $\gamma$ 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 $\omega$-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.
