三维三次频率须环上分离面的指数小分裂
Exponentially small splitting of separatrices associated to 3D whiskered tori with cubic frequencies
摘要：
本文研究了一类近似可积哈密顿系统中具有三个频率的须状（双曲）环面的不变流形的分裂问题，该系统的双曲部分由摆给出。我们考虑一个具有快速频率向量$ω/\sqrt\varepsilon$的三维环面，其中$ω=(1,Ω,\widetildeΩ)$是一个三次无理数，它的两个共轭是复数，$ω$的分量生成场$\mathbb Q(Ω)$。一个典型的例子是三次黄金向量，由满足$Ω^3=1-Ω$和$\widetildeΩ=Ω^2$的（实数）$Ω$给出。对于这样的三维频率向量，连分数的标准理论不能应用，所以我们开发了一种方法来确定小因子$\langle k,ω\rangle$，$k\in{\mathbb Z}^3$的行为。应用庞加莱-Melnikov方法，这使我们能够进行一个仔细的研究的主谐波（这取决于$\varepsilon$）的Melnikov函数，获得一个渐近估计的最大分裂距离，这是指数小的$\varepsilon$，并有效的所有足够小的值~$\varepsilon$。这种估计的行为像$\exp\{-h_1(\varepsilon)/\varepsilon^{1/6}\}$，我们提供了，第一次在一个系统中的3个频率，一个准确的描述（正）功能$h_1(\varepsilon)$的指数的分子，表明它可以明确地从频率矢量$ω$的共振特性构建，并证明它是一个准周期函数（而不是周期性的）相对于$\ln\varepsilon$。通过这种方式，我们强调了频率的算术性质上的分裂的估计的强依赖性。
英文摘要：
We study the splitting of invariant manifolds of whiskered (hyperbolic) tori with three frequencies in a nearly-integrable Hamiltonian system, whose hyperbolic part is given by a pendulum. We consider a 3-dimensional torus with a fast frequency vector $ω/\sqrt\varepsilon$, with $ω=(1,Ω,\widetildeΩ)$ where $Ω$ is a cubic irrational number whose two conjugates are complex, and the components of $ω$ generate the field $\mathbb Q(Ω)$. A paradigmatic case is the cubic golden vector, given by the (real) number $Ω$ satisfying $Ω^3=1-Ω$, and $\widetildeΩ=Ω^2$. For such 3-dimensional frequency vectors, the standard theory of continued fractions cannot be applied, so we develop a methodology for determining the behavior of the small divisors $\langle k,ω\rangle$, $k\in{\mathbb Z}^3$. Applying the Poincaré-Melnikov method, this allows us to carry out a careful study of the dominant harmonic (which depends on $\varepsilon$) of the Melnikov function, obtaining an asymptotic estimate for the maximal splitting distance, which is exponentially small in $\varepsilon$, and valid for all sufficiently small values of~$\varepsilon$. This estimate behaves like $\exp\{-h_1(\varepsilon)/\varepsilon^{1/6}\}$ and we provide, for the first time in a system with 3 frequencies, an accurate description of the (positive) function $h_1(\varepsilon)$ in the numerator of the exponent, showing that it can be explicitly constructed from the resonance properties of the frequency vector $ω$, and proving that it is a quasiperiodic function (and not periodic) with respect to $\ln\varepsilon$. In this way, we emphasize the strong dependence of the estimates for the splitting on the arithmetic properties of the frequencies.
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