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

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