Melnikov theory for finite-time vector fields

Abstract

Melnikov theory provides a powerful tool for analysing time-dependent perturbations of autonomous vector fields that exhibit heteroclinic orbits. The standard theory requires that the perturbed vector field be defined, and bounded, for all times. In this paper, Melnikov theory is adapted so that it is applicable to vector fields that are defined over sufficiently large, but finite, time intervals. Such an extension is desirable when investigating Lagrangian trajectories in fluid flows under the effect of viscous perturbations; the resulting velocity field can only be guaranteed to be close to the unperturbed velocity field, corresponding to the inviscid limit, for finite times. Applications to transport in the viscous barotropic vorticity equation are given.