Exponentially Small Estimates for Separatrix Splittings
Exponentially Small Estimates for Separatrix Splittings
复制标题
分界线分裂的指数小估计
DOI：
10.1007/978-1-4757-0435-8_13
复制
发表时间：
1991
期刊：
影响因子：
--
通讯作者：
P. Holmes
中科院分区：
文献类型：
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作者：
J. Scheurle;J. Marsden;P. Holmes
本文回顾了我们以前的估计，并给出了一个新的现象的例子。在涉及摄动参数ε的所有阶以上的渐近性的问题中，一个常见的假设是所研究的量（如分界线分裂距离或角度、孤波失配等）可以通过εbe-c/ε的表达式“估计”为ε→ 0。这里，a是带通常数（b可以是负的，也可以是“尖的”，通常是从真实的轴到复平面极点的距离）。我们的例子的主要目的是表明这个假设可能是错误的。这个例子，它涉及的分裂的快速强制系统的异宿轨道表明，即使从上面的估计（使用尖锐的值c）是不正确的。我们认为这种情况并不是孤立或特殊的，而是普遍发生的。我们特别注意到，在涉及超过所有阶的渐近性的情况下，当假设εbe-c/ε的估计时，需要证明它是合理的。
This paper reviews our previous estimates and gives an example exhibiting a new phenomenon. In problems involving asymptotics beyond all orders in a perturbation parameterε, it is a commonassumptionthat the quantity being studied (such as a separatrix splitting distance or angle, a solitary wave mismatch, etc.) can be “estimated” by an expression of the formaεbe-c/εasε→ 0. Here,a, bandcare constants (wherebcan be negative andcis “sharp”, often the distance from the real axis to a pole in the complex plane). The main purpose of our example is to show that this assumption can be wrong. The example, which concerns the splitting of separatrices in a rapidly forced system with a heteroclinic orbit shows that the even the estimate from above (using the sharp value of c) is incorrect. We argue that this situation is not isolated or particular, but happens rather generally. We especially note that in situations involving asymptotics beyond all orders, when an estimate of the formaεbe-c/εis assumed, it needs to be justified.