Morse Index, Stability and Spectral Bounds for Periodic Orbits in Reversible Lagrangian Systems
Abstract
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 kD,νD are the Dirichlet index and nullity on a half-period, set k̃=k−2kD−νD. Then the algebraic number ν+ of positive real Floquet multipliers satisfies ν+≥ 2n−2k̃, and we obtain corresponding bounds for the positive real spectrum away from $1$, allowing arbitrary Jordan chains at the unit multiplier. Hence at most 2k̃ 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≥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.
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BibTeX
@misc{portaluri2026morse,
title = {{Morse Index, Stability and Spectral Bounds for Periodic Orbits in Reversible Lagrangian Systems}},
author = {Alessandro Portaluri and Li Wu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.24070},
eprint = {2609.24070},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.24070},
}
RIS
TY - UNPB
TI - Morse Index, Stability and Spectral Bounds for Periodic Orbits in Reversible Lagrangian Systems
AU - Alessandro Portaluri
AU - Li Wu
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2609.24070
UR - https://arxiv.org/abs/2609.24070
AB - 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,ν_D are the Dirichlet index and nullity on a half-period, set k̃=k−2k_D−ν_D. Then the algebraic number ν_+ of positive real Floquet multipliers satisfies ν_+≥ 2n−2k̃, and we obtain corresponding bounds for the positive real spectrum away from $1$, allowing arbitrary Jordan chains at the unit multiplier. Hence at most 2k̃ 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≥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.
ER -