二次频率比须环面上分离面的指数小分裂和横截性
Exponentially small splitting of separatrices and transversality associated to whiskered tori with quadratic frequency ratio
摘要：
研究了近可积哈密顿系统中具有两个频率的须状（双曲）环面的不变流形的分裂问题，其中双曲部分由摆给出。我们考虑具有快速频率向量$ω/\sqrt\varepsilon$的环面，其中$ω=(1,Ω)$的频率比$Ω$是二次无理数。应用Poincaré-Melnikov方法，我们对Melnikov势的主谐波进行了仔细的研究。这使我们能够提供一个渐近估计的最大分裂距离，并显示存在横向同宿轨道的须环面的横向分裂的渐近估计。这两个估计在$\varepsilon$中都是指数小的，指数中的函数相对于$\ln\varepsilon$是周期性的，并且可以从$Ω$的连分数中显式地构造。通过这种方式，我们强调了我们的结果对$Ω$的算术性质的强依赖性。特别是，二次比率$Ω$与1-周期或2-周期的连分数（称为金属和金属色的比率分别），我们提供了准确的上限和下界的分裂。最大分裂距离的估计对于$\varepsilon$的所有足够小的值都是有效的，并且对于$\varepsilon$的大多数值可以建立横向性，不包括在谐波的主导发生变化的一些过渡值周围的小间隔，并且可能发生分叉。
英文摘要：
The splitting of invariant manifolds of whiskered (hyperbolic) tori with two frequencies in a nearly-integrable Hamiltonian system, whose hyperbolic part is given by a pendulum, is studied. We consider a torus with a fast frequency vector $ω/\sqrt\varepsilon$, with $ω=(1,Ω)$ where the frequency ratio $Ω$ is a quadratic irrational number. Applying the Poincaré-Melnikov method, we carry out a careful study of the dominant harmonics of the Melnikov potential. This allows us to provide an asymptotic estimate for the maximal splitting distance, and show the existence of transverse homoclinic orbits to the whiskered tori with an asymptotic estimate for the transversality of the splitting. Both estimates are exponentially small in $\varepsilon$, with the functions in the exponents being periodic with respect to $\ln\varepsilon$, and can be explicitly constructed from the continued fraction of $Ω$. In this way, we emphasize the strong dependence of our results on the arithmetic properties of $Ω$. In particular, for quadratic ratios $Ω$ with a 1-periodic or 2-periodic continued fraction (called metallic and metallic-colored ratios respectively), we provide accurate upper and lower bounds for the splitting. The estimate for the maximal splitting distance is valid for all sufficiently small values of $\varepsilon$, and the transversality can be established for a majority of values of $\varepsilon$, excluding small intervals around some transition values where changes in the dominance of the harmonics take place, and bifurcations could occur.
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