Instability of semi-Riemannian closed geodesics

Decomposition of spectral flow and Bott-type iteration formula

“…and we could give the decomposition of spectral flow from Theorem 1.2. This case include the Bott-type formula of Semi-Riemann manifold [17]. Let c be a space-like or time-like closed geodesic on n + 1 dimension Semi-Riemann manifold (M, g) with period T .…”

“…For ω ∈ U, let [17] that there exist s 0 sufficiently large such that for s ≥ s 0 , A ω s is nondegenerate. The ω spectral index of c is defined by…”

Linear instability for periodic orbits of non-autonomous Lagrangian systems

“…The literature on this criterion is quite broad. We refer the interested reader to [8,7,18] and references therein. Loosely speaking, a closed geodesic is termed linearly stable if the monodromy matrix associated to it splits into two-dimensional rotations.…”

“…Several years later, the authors in [20] proved a generalization of the aforementioned result, dropping the non-degeneracy assumption. Very recently, the authors in [18], by using a mix of variational and symplectic techniques, were able to establish, amongst other results, a general criterion to detect the (linear) instability of a closed geodesic on a finite dimensional semi-Riemannian manifold, by controlling the parity of an integer which naturally replaces, in this setting, the classical Morse index (which is, in general, infinite). In fact, in the semi-Riemannian (not Riemannian world) the critical points of the geodesic energy functional have, in general, an infinite Morse index and co-index.…”

“…A natural question (which was on the main motivations for writing this paper) is how the results proved by the authors in [18] can be carried over in the case of non-autonomous (not Legendre convex) Lagrangian functions on Riemannian manifolds.…”