高维哈密顿系统的不稳定性：多重共振不阻碍扩散
Instability of high dimensional Hamiltonian Systems: Multiple resonances do not impede diffusion
摘要：
我们考虑由形式为$$H(I,ϕ,p,q,t;ε) = h(I) + \sum_{j = 1}^n \pm(\frac{1}{2} p_j^2 + V_j(q_j)) + εQ(I,ϕ,p,q,t;ε)$$的哈密尔顿算子给出的模型，其中$I,ϕ$是d维作用和角度，$p,q$是n维实共轭变量，$t$是角度。这些都是更高维的类似物，在中心和双曲方向，在以前的论文中研究的模型由athors。所有这些模型都存在大间隙问题。我们证明了，对于足够小的$ε$，在正则性和显式非退化条件下的模型，有轨道的作用变量$I$执行相当任意的范围内的大小为O（1）。该域包括共振线，因此在$d$维KAM环面之间存在大的间隙。证明方法严格遵循[DLS（= Delshams，Llave and Seara）2006]的策略。当中心方向的维数$d$大于1时，出现的主要新现象是多重共振的存在。我们表明，由于这些多个共振发生在余维大于一个在行动空间$I$的集合，他们可以轮廓。这对应于物理学文献中称为共振扩散的机制。然而，本文与[DLS 2006]有很大不同。在证明的技术细节上，我们利用了散射映射理论[DLS 2008]，这在撰写上述论文时是不可用的。我们已经更详细地分析了施加在共振上的条件。更准确地说，我们发现，有一个简单的条件Melnikov潜力，使我们能够得出结论，共振交叉。特别地，该条件不依赖于共振。因此，即使将结果应用于[DLS 2006]中的模型，结果也是新的。
英文摘要：
We consider models given by Hamiltonians of the form $$H(I,ϕ,p,q,t;ε) = h(I) + \sum_{j = 1}^n \pm(\frac{1}{2} p_j^2 + V_j(q_j)) + εQ(I,ϕ,p,q,t;ε)$$ where $I,ϕ$ are d-dimensional actions and angles, $p,q$ are n-dimensional real conjugated variables, and $t$ is an angle. These are higher dimensional analogues, both in the center and hyperbolic directions, of the models studied in previous papers by the athors. All these models present the large gap problem. We show that, for $ε$ small enough, under regularity and explicit non-degeneracy conditions on the model, there are orbits whose action variables $I$ perform rather arbitrary excursions in a domain of size O(1). This domain includes resonance lines and, hence, large gaps among $d$-dimensional KAM tori. The method of proof follows closely the strategy of [DLS(=Delshams, Llave and Seara) 2006]. The main new phenomenon that appears when the dimension $d$ of the center directions is larger than one, is the existence of multiple resonances. We show that, since these multiple resonances happen in sets of codimension greater than one in the space of actions $I$, they can be contoured. This corresponds to the mechanism called diffusion across resonances in the Physics literature. The present paper, however, differs substantially from [DLS 2006]. On the technical details of the proofs, we have taken advantage of the theory of the scattering map [DLS 2008], not available when the above papers were written. We have analyzed the conditions imposed on the resonances in more detail. More precisely, we have found that there is a simple condition on the Melnikov potential which allows us to conclude that the resonances are crossed. In particular, this condition does not depend on the resonances. So that the results are new even when applied to the models in [DLS 2006].
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