AI 中文总结
本文针对可逆拉格朗日系统，利用莫尔斯指标与狄利克雷指标给出周期轨道弗洛凯乘子的定量谱界，证明指标一刹车轨道的横向双曲性，并展示精细界的尖锐性。
AI 中文摘要
我们证明了严格勒让德凸拉格朗日系统中可逆周期解的定量稳定性与不稳定性估计。若配置维数为$n$，固定周期的莫尔斯指标为$k$，且$k_D,\nu_D$为半周期上的狄利克雷指标与零化度，令\\[ \widetilde k=k-2k_D-\nu_D. \\] 则正实弗洛凯乘子的代数个数$\nu_+$满足\\[ \nu_+\ge 2n-2\widetilde k, \\] 并且我们得到远离$1$的正实谱的相应界，允许单位乘子处有任意约当链。因此，至多有$2\widetilde k$个弗洛凯乘子可以位于正实轴之外。特别地，当$k=0$时，我们恢复了可逆极小化子的正实谱定理，而对于非恒定自治刹车轨道，该界改进为加二。此外，若$n\ge2$且$1$的代数重数为二，则指标为一的刹车轨道是横向双曲的，具有$n-1$个稳定方向和$n-1$个不稳定方向。解耦振子表明，即使仅基于$k$的估计是无效的，精细界也可能是尖锐的，而Hénon--Heiles直线轨道提供了半周期机制的非线性例证。
英文摘要
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 \[ \widetilde k=k-2k_D-ν_D. \] Then the algebraic number $ν_+$ of positive real Floquet multipliers satisfies \[ ν_+\ge 2n-2\widetilde k, \] and we obtain corresponding bounds for the positive real spectrum away from $1$, allowing arbitrary Jordan chains at the unit multiplier. Hence at most $2\widetilde k$ 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, $n\ge2$ 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.
Comments21 pages, 2 figures
