Noisy localized structures induced by large noise

We investigate the influence of large noise on the formation of localized patterns in the framework of the cubic-quintic complex Ginzburg-Landau equation. The interaction of localization and noise can lead to filling in or noisy localized structures for fixed noise strength. To focus on the interaction between noise and localization we cover a region in parameter space, in particular, subcriticality, for which stationary stable deterministic pulses do not exist. Possible experimental tests of the work presented for autocatalytic chemical reactions and bioinspired systems are outlined.

Figure 1

The amplitude averaged in time and space, $\langle A\rangle$ is plotted (black dots) as a function of the noise strength $\eta$ for $\mu =-1.40$ for an average over 50 realizations, time $T={10}^4$ for each realization and a cut-off value 1.5 when taking the average over the amplitude.

Figure 2

The modulus of $A,R\left(x\right)$, is plotted as a function of space for $\mu =-1.40$ in the asymptotic time regime: (a) snapshot of a noisy zero solution for $\eta =0.497$; (b) snapshot of a noisy localized state for $\eta =0.497$; (c) snapshot of a noisy localized solution demonstrating very succinctly the coexistence between the upper and lower noisy attractors for $\eta =0.53$; and (d) snapshot of a noisy filled-in finite amplitude solution for $\eta =0.53$.

Figure 3

Three typical examples for time series of noisy localized states are presented for $T=600$ in the asymptotic time regime. Here red (dark gray) and yellow (light gray) indicate values above and below a threshold value of $\left|A\right|=1.5$, respectively. (a) shows the time evolution of an intermittent localized state of finite width emerging and disappearing intermittently as a function of time for $\mu =-1.40$ and $\eta =0.497$, (b) shows the time evolution of a noisy localized solution for $\mu =-1.40$ and $\eta =0.514$, of the type shown as a snapshot in Fig. 2, while (c) shows the time evolution of a state filling the full width alternating intermittently with a state showing the coexistence between the upper and lower attractor for $\mu =-1.40$ and $\eta =0.53$.

Figure 4

(a) A snapshot of $R\left(x\right)$ is shown for $\mu =-1.60$ and $\eta =1.6$. (b) Plot of $\langle A\rangle$ for $\mu =-1.6$ as a function of $\eta$ for the range $0.5<\eta <2.0$ with 50 realizations for each noise amplitude and $T={10}^4$ for each realization.

Figure 5

$\text{log}(\langle A\rangle )$ is plotted as a function of $\text{log}\left(\eta \right)$. The power law $\text{log}(\langle A\rangle )\sim \text{log}\left(\eta \right)$ is obtained over more than two decades $\left(0.001<\eta <0.1\right)$. $\mu =-1.40$ and without cutoff for $\left|A\right|$.