Bounds of Validity for Bifurcations of Equilibria in a Class of Networked Dynamical Systems
This provides new analytical tools for quantifying the robustness of bifurcation phenomena in networked systems, which is incremental but useful for researchers in nonlinear dynamics and network science.
The authors established explicit bounds of validity for classifying bifurcation diagrams in networked dynamical systems, such as Hopfield and Firing Rate models, by applying Lyapunov-Schmidt reduction bounds to rigorously characterize reliable neighborhoods around bifurcation points, and demonstrated this on nonlinear opinion dynamics on k-regular graphs.
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 neighbourhoods around bifurcation points where predictions from reduced-order bifurcation equations 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 behaviour.