Inverse stochastic resonance in electroconvection by multiplicative colored noise

A kind of inverse stochastic resonance (ISR) observed in ac-driven electroconvection (EC) in a nematic liquid crystal is presented. In successive pattern evolutions by increasing noise intensity $V_{\mathrm{N}}$, a typical EC (with a normalized amplitude ${\mathit{A}}_0=1$ at $V_{\mathrm{N}}=0$) disappears $({\mathit{A}}_0\to 0)$, and then the rest state (${\mathit{A}}_0=0$) reenters into the EC (${\mathit{A}}_0=1$); eventually, it develops into complicated $\mathrm{EC}({\mathit{A}}_0>1)$. The reversed bell-shaped behavior of ${\mathit{A}}_0\left(V_{\mathrm{N}}\right)$ is evidence of ISR. The present ISR may be explained by taking into account colored noise characterized by its intensity and correlation time.

Figure 1

Pattern evolution with increasing noise intensity $V_{\mathrm{N}}(f_{\mathrm{c}}=500\mathrm{Hz})=0$ (a); 7.8 V (b); 12.7 V (c); 13.5 V (d); 17.4 V (e). The amplitude $A$ of Williams domain (WD, at a fixed $V=7.1\mathrm{V}$ and $f=30\mathrm{Hz}$) is varied by $V_{\mathrm{N}}$; $A$ first decreases to zero [$\mathit{A}\sim 0$ at (b)], and then recovers (c) with increasing $V_{\mathrm{N}}$. The size of each frame is $X\times Y=640\mathrm{pixels}\times 480\mathrm{pixels}(=0.421\mathrm{mm}\times 0.316\mathrm{mm})$.

Figure 2

Noise intensity $V_{\mathrm{N}}$-dependent threshold voltage $V_{\mathrm{c}}$ for WD with respect to the cutoff frequency $f_{\mathrm{c}}$ of noise. The solid lines were calculated by the typical relationship between $V_{\mathrm{c}}$ and $V_{\mathrm{N}}$: $V_{\mathrm{c}}^2=V_{\mathrm{c}0}^2+b\left(f_{\mathrm{c}}\right)V_{\mathrm{N}}^2$ [24, 25]. Here, $V_{\mathrm{c}0}$ and $b$ indicate a threshold voltage for WD ($V_{\mathrm{N}}=0$) and a sensitivity of WD to noise, respectively. In the case of $f_{\mathrm{c}}=500\mathrm{Hz}$, the linear relationship is inapplicable to the complicated behavior. See also Fig. 1.

Figure 3

Spatiotemporal plot for the pattern evolution (i.e., the amplitude $A$ of one-dimensional images of patterns) with increasing or decreasing noise intensity $V_{\mathrm{N}}(f_{\mathrm{c}}=500\mathrm{Hz})$. An arbitrarily selected one-dimensional line ($X$) in WD [at a fixed $V=7.1\mathrm{V}$ and $f=30\mathrm{Hz}$, Fig. 1] was successively placed in order of time $T$ (with step increase of $V_{\mathrm{N}}$). As seen in Fig. 1, WD $(\mathit{A}\ne 0)$ disappears ($\mathit{A}\sim 0$ around $V_{\mathrm{N}}=7.8\mathrm{V}$) and reappears $(\mathit{A}\ne 0)$ near $V_{\mathrm{N}}=12.0\mathrm{V}$) with increasing $V_{\mathrm{N}}(0\to 17.4\mathrm{V})$; the reverse variation of $A$ is also obtained with decreasing $V_{\mathrm{N}}(17.4\mathrm{V}\to 0)$. For each value of $V_{\mathrm{N}}$, the measurement time is 150 s (i.e., $X\times T=640\mathrm{pixels}\times 30\mathrm{pixels}$); total $X\times T=640\mathrm{pixels}\times 600\mathrm{pixels}(=0.421\mathrm{mm}\times 3000\mathrm{s})$.

Figure 4

Spatiotemporal plots for the pattern evolution (i.e., $XT$ plots) at a fixed $V=7.1\mathrm{V},f=30\mathrm{Hz}$, and $V_{\mathrm{N}}(f_{\mathrm{c}}=500\mathrm{Hz})=0$ (a); 13.5 V (b); 15.5 V (c); 17.4 V (d). The size of each plot is $X\times T=640\mathrm{pixels}\times 900\mathrm{pixels}(=0.421\mathrm{mm}\times 900\mathrm{s})$.

Figure 5

A normalized brightness-intensity $\left({\mathit{I}}_0\right)$ variation with increasing noise intensity $V_{\mathrm{N}}$ with respect to the cutoff frequency $f_{\mathrm{c}}$ (at a fixed $V=7.1\mathrm{V}>V_{\mathrm{c}}=6.6\mathrm{V}$, and $f=30\mathrm{Hz}$). ${\mathit{I}}_0$ was measured at fixed points on $X$ (a one-dimensional image; see Figs. 3 and 4) which indicate the peak intensity $I_{\max }$ [see Fig. 7]; it was measured for 30 s at each fixed $V_{\mathrm{N}}$ [with steplike increase $(0\to 10.6\mathrm{V})$]. Notice that there exists ${\mathit{I}}_0\sim 0$ in the reversed bell-shaped variation for $f_{\mathrm{c}}=500\mathrm{Hz}$; it corresponds to Fig. 1. For $f_{\mathrm{c}}=200\mathrm{Hz},{\mathit{I}}_0(T>120\mathrm{s})$ should be neglected because it behaves like that because of appearance of defects [i.e., change of the points $X\left(I_{\max }\right)]$ at larger $V_{\mathrm{N}}$.

Figure 6

(a) A normalized brightness-intensity $\left({\mathit{I}}_0\right)$ variation with steplike increase of $V_{\mathrm{N}}$ with respect to the cutoff frequency $f_{\mathrm{c}}$ at a PL cell and (b) its comparison for three different cells (TW: twist cell, HO: homeotropic cell, PL: planar cell) at an identical $f_{\mathrm{c}}=500\mathrm{Hz}$. Notice that there exists ${\mathit{I}}_0\sim 0$ at a certain $V_{\mathrm{N}}^{*}$ for each different cell.

Figure 7

(a) Brightness intensity $\mathit{I}$($X$) in a one-dimensional image ($X=640$ pixels) at various noise intensities $V_{\mathrm{N}}(f_{\mathrm{c}}=500\mathrm{Hz})$. (b) Amplitude variation with increasing and decreasing $V_{\mathrm{N}}$. A normalized amplitude ${\mathit{A}}_0$ was calculated for some peaks after averaging the intensity difference $(I_{\max }-I_{\min })$ at each peak point. $I$($X$) for $V_{\mathrm{N}}=3.9$ and 5.8 V corresponds to the rest state (i.e., no EC). ${\mathit{A}}_0$ shows a reversed bell-shaped variation in $V_{\mathrm{N}}$ (with a background error ${\mathit{A}}_0=0.06$); this coincides with the pattern evolution (Fig. 1) and the spatiotemporal plot (Fig. 3).

Figure 8

Schematic diagram for the inverse stochastic resonance (ISR). If noise-induced amplitude can be supposed in a linear relation $\mathit{A}(V+V_{\mathrm{N}})={\mathit{A}}_{\mathrm{S}}\left(V\right)+{\mathit{A}}_{\mathrm{N}}\left(V_{\mathrm{N}}\right)$, the present ISR is possible (top panel). In the region of noise intensity $V_{\mathrm{N}}$ indicating $|{\mathit{A}}_{\mathrm{N}}\left(V_{\mathrm{N}}\right)|>{\mathit{A}}_{\mathrm{S}}\left(V\right)$, the rest state [i.e., no EC, Fig. 1] can be realized. The ♦ symbols (a)–(e) mean $A$ for Figs. 1–1, respectively; ${\mathit{A}}_{\mathrm{S}}\left(V_1\right)$ represents a constant $A$ for a fixed $V_1$ (e.g.,$=7.1\mathrm{V}$ in Fig. 1). Compare ${\mathit{A}}_{\mathrm{N}}\left(V_{\mathrm{N}}\right)$ with respect to $f_{\mathrm{c}}$ (bottom panel). No ISR is expected for extremely colored noises ($f_{\mathrm{c}}<f_{\mathrm{c}}^{†}$) and whitelike noises ($f_{\mathrm{c}}>f_{\mathrm{c}}^{†}$); it will be found that $\mathit{A}(V+V_{\mathrm{N}})$ monotonically decreases ($f_{\mathrm{c}}>f_{\mathrm{c}}^{†}$) or increases ($f_{\mathrm{c}}<f_{\mathrm{c}}^{†}$) with increasing $V_{\mathrm{N}}$.