Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Bounds of Validity for Bifurcations of Equilibria in a Class of Networked Dynamical Systems

Pranav Gupta, Ravi Banavar, Anastasia Bizyaeva

TL;DR

This work provides explicit, computable bounds for the validity of Lyapunov--Schmidt reduction in analyzing equilibria bifurcations of two classes of networked continuous-time dynamical systems, namely Hopfield-like and Firing Rate-like models. By expressing the bounds in terms of interpretable network quantities (notably the adjacency spectrum and spectral gap), the authors quantify the neighborhood around bifurcation points in which reduced-order predictions remain topologically accurate. They derive simplified general LS-bounds and then specialize them to Hopfield and Firing Rate networks, obtaining computable radii that depend on $A$, $S$, and their derivatives. The theory is applied to consensus/bifurcation problems in nonlinear opinion dynamics on $k$-regular graphs, showing how spectral properties govern robustness of bifurcation phenomena and illustrating the direct impact of network structure on dynamical transitions.

Abstract

Local bifurcation analysis plays a central role in understanding qualitative transitions in networked nonlinear dynamical systems, including dynamic neural network and opinion dynamics models. In this article we establish explicit bounds of validity for the classification of bifurcation diagrams in two classes of continuous-time networked dynamical systems, analogous in structure to the Hopfield and the Firing Rate dynamic neural network models. Our approach leverages recent advances in computing the bounds for the validity of Lyapunov-Schmidt reduction, a reduction method widely employed in nonlinear systems analysis. Using these bounds we rigorously characterize neighborhoods around bifurcation points where predictions from reduced-order models remain reliable. We further demonstrate how these bounds can be applied to an illustrative family of nonlinear opinion dynamics on k-regular graphs, which emerges as a special case of the general framework. These results provide new analytical tools for quantifying the robustness of bifurcation phenomena in dynamics over networked systems and highlight the interplay between network structure and nonlinear dynamical behavior.

Bounds of Validity for Bifurcations of Equilibria in a Class of Networked Dynamical Systems

TL;DR

This work provides explicit, computable bounds for the validity of Lyapunov--Schmidt reduction in analyzing equilibria bifurcations of two classes of networked continuous-time dynamical systems, namely Hopfield-like and Firing Rate-like models. By expressing the bounds in terms of interpretable network quantities (notably the adjacency spectrum and spectral gap), the authors quantify the neighborhood around bifurcation points in which reduced-order predictions remain topologically accurate. They derive simplified general LS-bounds and then specialize them to Hopfield and Firing Rate networks, obtaining computable radii that depend on , , and their derivatives. The theory is applied to consensus/bifurcation problems in nonlinear opinion dynamics on -regular graphs, showing how spectral properties govern robustness of bifurcation phenomena and illustrating the direct impact of network structure on dynamical transitions.

Abstract

Local bifurcation analysis plays a central role in understanding qualitative transitions in networked nonlinear dynamical systems, including dynamic neural network and opinion dynamics models. In this article we establish explicit bounds of validity for the classification of bifurcation diagrams in two classes of continuous-time networked dynamical systems, analogous in structure to the Hopfield and the Firing Rate dynamic neural network models. Our approach leverages recent advances in computing the bounds for the validity of Lyapunov-Schmidt reduction, a reduction method widely employed in nonlinear systems analysis. Using these bounds we rigorously characterize neighborhoods around bifurcation points where predictions from reduced-order models remain reliable. We further demonstrate how these bounds can be applied to an illustrative family of nonlinear opinion dynamics on k-regular graphs, which emerges as a special case of the general framework. These results provide new analytical tools for quantifying the robustness of bifurcation phenomena in dynamics over networked systems and highlight the interplay between network structure and nonlinear dynamical behavior.

Paper Structure

This paper contains 13 sections, 14 theorems, 53 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Notations and Preliminaries
  4. Lyapunov--Schmidt reduction
  5. Bounds of Validity for Lyapunov--Schmidt reduction
  6. Simplified General LS Bounds
  7. Main Results: Network Bounds
  8. Hopfield models
  9. Firing Rate models
  10. Application to Consensus Bifurcations
  11. Hopfield-structured NOD
  12. Firing-Rate-structured NOD
  13. Discussion

Key Result

Lemma II.1

Let $U \in \mathbb{R}^{n \times k}$ be a matrix with orthonormal column vectors, and let $P = U U^{\top}$ denote the associated orthogonal projection onto the column space of $U$. Then, for any matrix $Y \in \mathbb{R}^{n \times m}$ we have $\left\lVert{U^{\top} Y}\right\rVert = \left\lVert{P Y}\rig

Figures (2)

  • Figure 1: Consensus pitchfork bifurcation in opinion dynamics over a regular graph at $(x^{\star},u^{\star})=\left(\mathbf{0}, \tfrac{d}{k}\right)$; emerging branches of bistable equilibria lie on a manifold tangent to the consensus subspace $\{(x\mathbf{1}_{n},u):x,u\in\mathbb{R}^{}\}$ at the bifurcation point.
  • Figure 2: Variation in bounds of validity for Lyapunov--Schmidt reduction for consensus bifurcation over balanced regular graphs. For each $(n,k)$ pair, 100 independent random graphs were generated using the NetworkX package in Python. The reported mean values and corresponding error bars represent the average and standard deviation of the computed bounds on $r_{\parallel}$ across the sampled graphs. (a) illustrates the bounds with fixed degree $k\in\{12,18,24,30,36\}$, varying graph size $n$; (b) illustrates the bounds with fixed graph size $n\in\{12,18,24,30,36\}$, varying degree $k$.

Theorems & Definitions (27)

  • Lemma II.1
  • proof
  • Lemma II.2
  • proof
  • Proposition II.3
  • Lemma III.1
  • proof
  • Lemma III.2
  • proof
  • Theorem III.3
  • ...and 17 more