Direct Chaotic Flux Quantification in Perturbed Planar Flows: General Time-Periodicity

Abstract

Chaotic flux occurring across a heteroclinic upon perturbing an area-preserving planar flow is examined. The perturbation is assumed to have general periodicity, extending the harmonic requirement that is often used. Its spatial and temporal parts are moreover not required to be separable. This scenario, though well-understood phenomenologically, has until now had no computable formula for the quantification of the resulting chaotic flux. This article derives such a formula, by directly assessing the unequal lobe areas that are transported via a turnstile mechanism. The formula involves a bi-infinite summation of quantities related to Fourier coefficients of the associated Melnikov function. These are themselves directly obtainable using a Fourier transform process. An example is treated in detail, illustrating the relative ease in which the flux computation can be performed using this theory.