Morse Index, Stability and Spectral Bounds for Periodic Orbits in Reversible Lagrangian Systems

We prove quantitative stability and instability estimates for reversible periodic solutions of strictly Legendre-convex Lagrangian systems. If the configuration dimension is $n$, the fixed-period Morse index is $k$, and $k_D,{\nu }_D$ are the Dirichlet index and nullity on a half-period, set $$\tilde{k}=k-2k_D-{\nu }_D\mathrm{.}$$ Then the algebraic number ${\nu }_{+}$ of positive real Floquet multipliers satisfies $${\nu }_{+}\ge 2n-2\tilde{k},$$ and we obtain corresponding bounds for the positive real spectrum away from $1$, allowing arbitrary Jordan chains at the unit multiplier. Hence at most $2\tilde{k}$ Floquet multipliers can lie outside the positive real axis. In particular, when $k=0$ we recover the positive-real-spectrum theorem for reversible minimizers, while for nonconstant autonomous brake orbits the bounds improve by two. If, moreover, $n\ge 2$ and the algebraic multiplicity of $1$ is two, an index-one brake orbit is transversely hyperbolic, with $n-1$ stable and $n-1$ unstable directions. Decoupled oscillators show that the refined bounds can be sharp even when the estimate based on $k$ alone is vacuous, and the Hénon–Heiles straight-line orbit provides a nonlinear illustration of the half-period mechanism.