Melnikov theory forfinite-time vector fields

Abstract

Melnikov theory provides a powerful tool for analysing time-dependentperturbations of autonomous vector fields that exhibit heteroclinicorbits. The standard theory requires that the perturbed vector field bedefined, and bounded, for all times. In this paper, Melnikov theoryis adapted so that it is applicable to vector fields that are definedover sufficiently large, but finite, time intervals. Such an extensionis desirable when investigating Lagrangian trajectories in fluid flowsunder the effect of viscous perturbations; the resulting velocityfield can only be guaranteed to be close to the unperturbed velocityfield, corresponding to the inviscid limit, for finite times.Applications to transport in the viscous barotropic vorticity equationare given.