Sketching 1-D stable manifolds of 2-D maps without the inverse

TL;DR

This paper introduces a new algorithm for sketching 1-D stable manifolds of 2-D maps, including non-invertible systems, overcoming limitations of existing methods.

Contribution

A novel algorithm for computing stable manifolds in 2-D systems that works even when the system is not invertible.

Findings

Algorithm successfully computes stable manifolds in various examples.

Method extends to non-invertible systems, broadening applicability.

Demonstrates effectiveness through multiple illustrative cases.

Abstract

Saddle fixed points are the centerpieces of complicated dynamics in a system. The one-dimensional stable and unstable manifolds of these saddle-points are crucial to understanding the dynamics of such systems. While the problem of sketching the unstable manifold is simple, plotting the stable manifold is not as easy. Several algorithms exist to compute the stable manifold of saddle-points, but they have their limitations, especially when the system is not invertible. In this paper, we present a new algorithm to compute the stable manifold of 2-dimensional systems which can also be used for non-invertible systems. After outlining the logic of the algorithm, we demonstrate the output of the algorithm on several examples.