On the shifts of stable and unstable manifolds of a hyperbolic cycle under perturbation

TL;DR

This paper presents a method to efficiently compute the shifts of stable and unstable manifolds of hyperbolic cycles under perturbations, which is essential for understanding and controlling chaos in dynamical systems.

Contribution

It introduces a functional derivative approach to calculate manifold shifts using the entire system as an argument, simplifying the analysis of homoclinic and heteroclinic intersections.

Findings

Manifold shifts are computable with minimal effort.

Shifts of intersection points can be directly analyzed.

Method applicable to chaos control scenarios.

Abstract

Stable and unstable manifolds, originating from hyperbolic cycles, fundamentally characterize the behaviour of dynamical systems in chaotic regions. This letter demonstrates that their shifts under perturbation, crucial for chaos control, are computable with minimal effort using functional derivatives by considering the entire system as an argument. The shifts of homoclinic and heteroclinic orbits, as the intersections of these manifolds, are readily calculated by analyzing the movements of the intersection points.