Noise-induced standing waves in oscillatory systems with time-delayed feedback

In oscillatory reaction-diffusion systems, time-delay feedback can lead to the instability of uniform oscillations with respect to formation of standing waves. Here, we investigate how the presence of additive, Gaussian white noise can induce the appearance of standing waves. Combining analytical solutions of the model with spatiotemporal simulations, we find that noise can promote standing waves in regimes where the deterministic uniform oscillatory modes are stabilized. As the deterministic phase boundary is approached, the spatiotemporal correlations become stronger, such that even small noise can induce standing waves in this parameter regime. With larger noise strengths, standing waves could be induced at finite distances from the (deterministic) phase boundary. The overall dynamics is defined through the interplay of noisy forcing with the inherent reaction-diffusion dynamics.

Figure 1

Main solutions in the parameter space spanned by $\tau$ and $\mu$. The solid line defines the stability boundary of uniform oscillations in the deterministic system (above the curve). Below that curve, the diamond symbols indicate simulations displaying standing waves in the deterministic system (data from Fig. 8 of Ref. [13] and own simulations). The circles denote the parameter values chosen as defined in Fig. 2, while the left triangles indicate the parameter values as used for Figs. 3 and 3; the right triangles represent the parameter values as used for Figs. 3 and 3, and the down triangle stands for the parameter value for Fig. 5. The crosses represent the parameter values used in Fig. 6. Note that with the exception of Figs. 2–2(d), all simulations were performed in the deterministically stable regime characterized by uniform oscillations where standing waves do not exist. The other parameters are: $\alpha =-1.4, \beta =2, \omega =2\pi -\alpha , \xi =\pi /2$.

Figure 2

Main spatiotemporal solutions for different feedback magnitudes and noise strengths: (a) uniform oscillations, (b) standing waves without noise, (c) standing waves with small noise, (d) spatiotemporal chaos. Shown are space-time diagrams in gray scale for $\left|A\right|$ (top panels) and $\mathrm{Re}A$ (bottom panels) for a time interval of $t=25$ in the asymptotic regime and system size $L=256$. The delay time is $\tau =0.5$ and the values of $\mu$ are $\mu =0.50$ (a), $\mu =0.15$ (b), $\mu =0.15$ (c), $\mu =0$ (d). The noise magnitude is $D=0.05$ in (c) and zero otherwise. Black (white) denotes low (high) values of the respective quantity (rescaled for each simulation). For $\left|A\right|$, these values are $\left(\right|A{|}_{\text{min}},\left|A{|}_{\text{max}}\right)=(0.94,1.13)$ (b), $\left(\right|A{|}_{\text{min}},\left|A{|}_{\text{max}}\right)=(0.9,1.15)$ (c), $\left(\right|A{|}_{\text{min}},\left|A{|}_{\text{max}}\right)=(0.15,1.2)$ (d). For (a), the amplitude is constant $\left|A\right|=1.085$. The other parameters are as described in the caption of Fig. 1.

Figure 3

(a, b) Amplitudes of temporal (a) and spatial (b) correlation functions for $\tau =0.5$ for three different values of $\mu$ (see legend) and $D=1$. We observe that the closer $\mu$ to the critical ${\mu }_c=0.19848$, the larger the magnitude of the correlation functions. For illustration, we rescale (multiply) the temporal correlation functions with 1000 ($\mu =0.5$) and 50 ($\mu =0.25$) and the spatial correlation functions with 100 ($\mu =0.5$) and 50 ($\mu =0.25$). (c, d) Amplitudes of temporal (c) and spatial (c) correlation functions for $\mu =0.42$ for three different values of $\tau$ (see legend) and $D=1$. We observe that the closer $\tau$ to the critical ${\tau }_c=0.94244$, the larger the magnitude of the correlation functions. For illustration, we rescale (multiply) the correlation functions with 10 ($\tau =0.8$) and 5 ($\tau =0.9$). All other parameters are as described in the caption of Fig. 1.

Figure 4

Amplitudes of temporal and spatial correlation functions for $\mu =0.42$ as $\tau$ is varied (a) and for $\tau =0.5$ as $\mu$ is varied (b). This figure complements the data shown in Fig. 3. To quantify the oscillating amplitudes, we choose the average value of the amplitude in one period within the asymptotic regime for large $t^{\prime }$ and $r$. As $\mu$ is varied in (a), the amplitude of the correlation functions is low in the area of deterministically stable oscillations, but increases toward the limits of the stability region. As $\tau$ is varied in (b), we observe qualitatively similar behavior: here the uniform oscillations lose stability as $\tau$ is lowered. The other parameters are as described in the caption of Fig. 3.

Figure 5

Spatiotemporal simulations for different values of $D$ and fixed feedback strength. In space-time plots, we show $\left|A\right|$ (upper panels) and $\text{Re}A$ (lower panels) for $D=0.05$ (a), $D=0.2$ (b), and $D=0.5$ (c). The feedback magnitude is $\mu =0.25$, the delay time $\tau =0.5$, and the other parameters are as described in the caption of Fig. 1. For low $D$, we see that although $\left|A\right|$ shows some intermittent spatially periodic patches, their amplitude is actually quite small and the pattern actually is indistinguishable from uniform oscillations. For intermediate $D$, intermittent spatially periodic patches are seen in the pattern, reminiscent of standing waves. For large $D$, the noise is too strong to induce standing waves and the pattern corresponds to noisy uniform oscillations.

Figure 6

Spatiotemporal simulations for different values of $\mu$ and fixed noise. In space-time plots, we show $\left|A\right|$ (upper panels) and $\text{Re}A$ (lower panels) for $\mu =0.2$ (a), $\mu =0.35$ (b), and $\mu =0.5$ (c). The noise strength is $D=0.05$, the delay time $\tau =0.5$, and the other parameters are as described in the caption of Fig. 1. For low $\mu$, we observe clearly spatially periodic patterns that correspond to noise-induced standing waves. For larger $\mu$ (and hence further from the deterministic onset of standing waves), standing waves are weaker. For high $\mu$, the noise is not enough to induce standing waves.

Figure 7

Amplitude of noise-induced standing waves. In (a), this amplitude is shown as a function of the noise strength $D$ for a fixed set of $\tau =0.5$ and $\mu =0.25$. As qualitatively seen in Fig. 5, for small $D$ the amplitude is very small and not perceivable, for intermediate $D$ the amplitude is large and visible, while for large noise strengths, the overall pattern becomes too irregular to actually identify standing waves. In panel (b), the amplitude of standing waves is shown as a function of the feedback strength $\mu$ for a fixed set of $\tau =0.5$ and noise strength $D=0.05$. The leftmost point ($\mu =0.15$) corresponds to deterministically stable standing waves with relatively large amplitude. Starting from $\mu =0.2$, we enter the regime where standing waves do not exist as deterministic solutions, and we see that the amplitude diminishes as we move away from the stability boundary of uniform oscillations. The other parameters are as described in the caption of Fig. 1.