Noise-Induced Macroscopic Bifurcations in Globally Coupled Chaotic Units

Large populations of globally coupled identical maps subjected to independent additive noise are shown to undergo qualitative changes as the features of the stochastic process are varied. We show that, for strong coupling, the collective dynamics can be described in terms of a few effective macroscopic degrees of freedom, whose deterministic equations of motion are systematically derived through an order parameter expansion.

Figure 1

System of $N=2^{20}$ globally coupled logistic maps in the chaotic regime [$f(x)=1-a{x}^2$, $a=1.57$] subjected to uniformly distributed noise of variance ${\sigma }^2$. (a)–(c) $K=0.7$: PDFs for the mean field (solid line) and one single element (dashed-dotted line; dotted line at even times, dashed line at odd times) for different noise intensities: (a) ${\sigma }^2=0.0025$, (b) ${\sigma }^2=0.0324$, and (c) ${\sigma }^2=0.09$. (d) Bifurcation diagram for $X$ as a function of ${\sigma }^2$ for $K=1$. (e) First period-doubling bifurcations for the mean field (○) and for the second-order reduced system (solid line, see text) in the $(\sigma ,K)$ plane. (f) Same as (d) but for the reduced system Eqs. (9) with $K=0.7$ (the diagram of the full system almost coincides). The dotted lines in (d) and (f) indicate the location of the bifurcations reported in (e).

Figure 2

Population of $N=2^{22}$ excitable maps [Eq. (2) with $\alpha =0.4$, $\beta =1$, $\gamma =8$, and $K=0.9$] subjected to Gaussian noise of variance ${\sigma }^2$. (a) Bifurcation diagram for $X$ vs $\sigma$. (b) Critical value ${\sigma }^{*}$ of the noise intensity above which the mean field displays large amplitude oscillations for the population ($K=1$, □; $K=0.8$, ○; $K=0.6$, $*$) and for the order parameter reduction (solid line).

Figure 3

Population of Fig. 1. (a) First return map of $X$ for the full system (dark dots), its zeroth-order approximation (solid line), and its second-order approximation (light dots) ($K=0.4$ and ${\sigma }^2=0.001$). (b) ${\Omega }_4$ versus ${\Omega }_2$ for the same parameters (dark dots, full system; zeroth order, triangle; second order, solid line; fourth-order, light dots). (c) and (d) Distances $d_x$ and $d_{\Omega }$ defined in text (circles for the full system, dashed lines for the second-order approximation, solid line for fourth-order system). [(c) ${\sigma }^2=0.03$; (d) $K=0.6$].