Melnikov Theory for Two-Dimensional Manifolds in Three-Dimensional Flows

Abstract

We present a geometric Melnikov method to analyze a two-dimensional stable or unstable manifold associated with a saddle point in three-dimensional nonvolume preserving autonomous flows. The time-varying perturbed location of such a manifold is obtained under very general, nonvolume preserving and with arbitrary time-dependence, perturbations. We demonstrate the explicit computability of the leading-order spatio-temporal location of the manifold using our formulas. 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 an instantaneous flux quantification in terms of a Melnikov function. The time-varying instantaneous flux theory does not require any intersections between perturbed manifolds, nor rely on descriptions of lobe dynamics. Our theory has specific application to transport in fluid mechanics, where the flow is in three dimensions and flow separators in forward/backward time are two-dimensional stable/unstable manifolds. We demonstrate our theory using both the classical and swirling versions of Hill’s spherical vortex.