Estimates on the domain of validity for Lyapunov-Schmidt reduction

TL;DR

This paper provides estimates for the domain where Lyapunov-Schmidt reduction accurately captures the behavior of high-dimensional nonlinear systems near singular points, using bounds from the Implicit Function Theorem.

Contribution

It introduces a method to quantify the validity region of Lyapunov-Schmidt reduction based on new bounds from the Implicit Function Theorem.

Findings

Derived explicit estimates for the reduction's validity domain.

Applied estimates to a 2D system with a pitchfork bifurcation.

Demonstrated the practical use of bounds in bifurcation analysis.

Abstract

Lyapunov-Schmidt reduction is a dimensionality reduction technique in nonlinear systems analysis that is commonly utilised in the study of bifurcation problems in high-dimensional systems. The method is a systematic procedure for reducing the dimensionality of systems of algebraic equations that have singular points, preserving essential features of their solution sets. In this article, we establish estimates for the region of validity of the reduction by leveraging recently derived bounds on the Implicit Function Theorem. We then apply these bounds to an illustrative example of a two-dimensional system with a pitchfork bifurcation.