论文标题
Melnikov理论，用于三维流中的二维流形
Melnikov theory for two-dimensional manifolds in three-dimensional flows
论文作者
论文摘要
我们提出了一种Melnikov方法，用于分析与三维非数量保存自主系统中与鞍点相关的二维稳定或不稳定的歧管。这种歧管的时变扰动位置是在非常笼统的，非数量保存和任意时间依赖性（扰动）下获得的。在具有二维杂层流形的不受干扰的情况下，我们适应理论以量化分裂成稳定且不稳定的流形，从而获得了表征这些歧管的横向相互切除的时间变化位置的Melnikov函数。通过墨尼科夫功能，获得了由这种相交产生的叶量的公式，以及跨碎屑歧管的瞬时通量。我们的理论在流体力学中具有特定的应用，其中流动在三个维度，流动分离器是二维稳定/不稳定的歧管。我们使用Hill球形涡流的经典版本和漩涡版本展示了我们的理论。
We present a Melnikov method to analyze two-dimensional stable or unstable manifolds associated with a saddle point in three-dimensional non-volume preserving autonomous systems. The time-varying perturbed locations of such manifolds is obtained under very general, non-volume preserving and with arbitrary time-dependence, perturbations. In unperturbed situations with a two-dimensional heteroclinic manifold, we adapt our theory to quantify the splitting into a stable and unstable manifold, and thereby obtain a Melnikov function characterizing the time-varying locations of transverse intersections of these manifolds. Formulas for lobe volumes arising from such intersections, as well as the instantaneous flux across the broken heteroclinic manifold, are obtained in terms of the Melnikov function. Our theory has specific application to transport in fluid mechanics, where the flow is in three dimensions and flow separators are two-dimensional stable/unstable manifolds. We demonstrate our theory using both the classical and the swirling versions of Hill's spherical vortex.
