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

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.

Figure 1

Potential $v$ as a function of $\varphi$ for several values of $ϵ$.

Figure 2

Contour curves defined by ${\varphi }_i\left(t\right)=0.5$ for several values of $T$. $N=4096$.

Figure 3

Spatial pattern at time $t=828$ for the system with $T={10}^{-6}$. $N=4096$.

Figure 4

$\overline{\varphi }$ as a function of $t$. $N=1024$.

Figure 5

${\chi }_{\varphi }$ as a function of $t$. $N=1024$. Inset: ${\chi }_{\varphi }\left(t_{*}\right)$ as a function of $T$ in a log-log scale. The guide line represents ${\chi }_{\varphi }\left(t_{*}\right)\simeq T^{-1∕3}$.

Figure 6

$\chi (k_n;T)T$ as a function of $k_n{T}^{-1∕3}∕{(-\ln T)}^{1∕2}$ for several values of $T$. $N=4096$. The statistical error bar is about the symbol size. By comparing the systems with $N=512$, 1024, 2048, and 4096, it is found that the finite-size effects are fairly small.