Nonautonomous control of stable and unstable manifolds in two-dimensional flows

TL;DR

This paper presents a method to control the position of stable and unstable manifolds in two-dimensional flows by applying specific nonautonomous perturbations, with theoretical error bounds and numerical validation.

Contribution

It introduces a novel control strategy for moving manifolds in 2D flows using nonautonomous perturbations, including error analysis and numerical demonstration.

Findings

Effective control of manifold locations demonstrated

Theoretical bounds on manifold deviation established

Numerical example confirms practical applicability

Abstract

We outline a method for controlling the location of stable and unstable manifolds in the following sense. From a known location of the stable and unstable manifolds in a steady two-dimensional flow, the primary segments of the manifolds are to be moved to a user-specified time-varying location which is near the steady location. We determine the nonautonomous perturbation to the vector field required to achieve this control, and give a theoretical bound for the error in the manifolds resulting from applying this control. The efficacy of the control strategy is illustrated via a numerical example.