Scale-free patterns at a saddle-node bifurcation in a stochastic system

TL;DR

This paper shows that in a stochastic system near a saddle-node bifurcation, scale-free spatial patterns emerge transiently during relaxation, characterized by critical exponents and specific spatial configurations.

Contribution

It reveals the emergence of scale-free patterns at a saddle-node bifurcation in stochastic systems and characterizes their properties through numerical experiments.

Findings

Scale-free patterns appear only at a specific relaxation time.

Critical exponents are determined numerically.

Patterns are linked to the spatial configuration of exit times from a marginal saddle.

Abstract

We demonstrate that scale-free patterns are observed in a spatially extended stochastic system whose deterministic part undergoes a saddle-node bifurcation. Remarkably, the scale-free patterns appear only at a particular time in relaxation processes from a spatially homogeneous initial condition. We characterize the scale-free nature in terms of the spatial configuration of the exiting time from a marginal saddle where the pair annihilation of a saddle and a node occurs at the bifurcation point. Critical exponents associated with the scale-free patterns are determined by numerical experiments.