Impulsive perturbations to differential equations: stable/unstable pseudo-manifolds, heteroclinic connections, and flux

TL;DR

This paper develops a mathematical framework to analyze how impulsive perturbations affect the structure of invariant manifolds and heteroclinic connections in two-dimensional flows, with applications to physical systems.

Contribution

It introduces new definitions for pseudo-manifolds under impulsive effects and derives criteria for heteroclinic persistence and flux quantification in this context.

Findings

Impulsive perturbations destroy smooth invariant manifolds.

New criteria for heteroclinic trajectory persistence are established.

Flux across broken heteroclinic manifolds can be quantified.

Abstract

State-dependent time-impulsive perturbations to a two-dimensional autonomous flow with stable and unstable manifolds are analysed by posing in terms of an integral equation which is valid in both forwards- and backwards-time. The impulses destroy the smooth invariant manifolds, necessitating new definitions for stable and unstable pseudo-manifolds. Their time-evolution is characterised by solving a Volterra integral equation of the second kind with discontinuous inhomogeniety. A criteria for heteroclinic trajectory persistence in this impulsive context is developed, as is a quantification of an instantaneous flux across broken heteroclinic manifolds. Several examples, including a kicked Duffing oscillator and an underwater explosion in the vicinity of an eddy, are used to illustrate the theory.