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Estimates on the domain of validity for Lyapunov-Schmidt reduction

Pranav Gupta, Anastasia Bizyaeva, Ravi Banavar

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.

Estimates on the domain of validity for Lyapunov-Schmidt reduction

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.
Paper Structure (11 sections, 5 theorems, 30 equations, 4 figures)

This paper contains 11 sections, 5 theorems, 30 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation and definitions
  4. Implicit Function Theorem
  5. Bifurcation problems and Lyapunov-Schmidt reduction
  6. Bounds on Lyapunov-Schmidt reduction
  7. Example: recurrent neural network model
  8. System
  9. Computation of Bounds
  10. Completion of LS reduction
  11. Conclusions

Key Result

Theorem II.1

Let $U\subset\mathbb{R}^{n}$ and $V\subset\mathbb{R}^{m}$ be be open sets. Consider a $\mathcal{C}^{\nu}$ smooth map where $\nu\geq1$, and consider a point $(\mathbf{x}_0,\mathbf{y}_0)\in U\times V$ for which $\mathrm{D}_{\mathbf{y}}f(\mathbf{x}_0,\mathbf{y}_0):\mathbb{R}^{m}\to\mathbb{R}^{m}$ is an isomorphism. Then for any ${\mathcal{N}}(\mathbf{y}_0)\subset V$, there exists an ${\mathcal{N}}(\

Figures (4)

  • Figure 2.1: a schematic of the bounds on Implicit Function Theorem: $(x,w)\in\mathcal{B}_{}\left(x_0,r_x\right)\times\mathcal{B}_{}\left(w_0,\epsilon\right)\longmapsto y(x,w)\in\mathcal{B}_{}\left(y_0,r_y\right)$ where $w_0 := f(x_0,y_0)$ satisfying $f(x,y(x,w)) = w$; for our purposes we assert $\epsilon=0$ (Fig 2. from jindal2023fdbklin)
  • Figure 2.2: sketch of $\ker(J)\times\mathbb{R}^{}$ and the zero sets of $\boldsymbol{\Phi},P\boldsymbol{\Phi}$ and $\phi$ in for a 2-dimensional system that has a pitchfork bifurcation. Figure based on Golubitsky1985
  • Figure 3.1: sketch of map $\varphi:\mathcal{B}((\mathbf{x}^\parallel_0,\lambda_0),r^\parallel)\to\mathcal{B}(\mathbf{x}^\perp_0,r^\perp)$
  • Figure 4.1: Flow of \ref{['eq:example']} close to the pitchfork bifurcation

Theorems & Definitions (7)

  • Theorem II.1: Implicit Function Theorem
  • Theorem II.2
  • Theorem II.3
  • Lemma III.1
  • proof
  • Theorem III.2
  • proof