Responses of spatiotemporal chaos to oscillating forces

The responses of soft-mode turbulence, a kind of spatiotemporal chaos seen in electroconvection of a nematic liquid crystal, to alternating-current magnetic fields is investigated to uncover the dynamical properties of spatiotemporal chaos. The dynamical responses can be measured by an order parameter, $M_p\left(t\right)$, which indicates ordering in the convective roll pattern induced by the magnetic field. Determined by properties of the liquid crystal in a magnetic field, $M_p\left(t\right)$ oscillates in accordance with the square of the magnetic field. The relaxation time of the system was obtained by fitting the frequency dependence of the complex susceptibility for the pattern obtained from the oscillation of $M_p\left(t\right)$ to the Debye-type relaxation spectra. However, for the high-frequency regime, the susceptibility deviates from the spectra because slow and large fluctuations of $M_p\left(t\right)$ contribute to the oscillation. The properties of this type of fluctuation were investigated by introducing a dynamic ordering parameter defined as the period average of $M_p\left(t\right)$.

Figure 1

Electroconvective patterns in a homeotropic nematic system and their 2D power spectra. (a) $H=0,\varepsilon =0.1$. This pattern corresponds to the SMT. (b) $H=400\text{G},\varepsilon =0.1$. (c, d) 2D power spectra obtained from (a) and (b), respectively, by fast Fourier transformation.

Figure 2

$M_p\left(t\right)$ for $f_{\mathrm{eff}}=0.2$ mHz (a) and $f_{\mathrm{eff}}=2.8$ mHz (b). Filled (blue) circles represent $M_p\left(t\right)$, and solid (red) lines show $H^2\left(t\right)$.

Figure 3

Dependence of $M_{p0}^2$ on $f_{\mathrm{eff}}$ plotted on a logarithmic scale.

Figure 4

(a) Dependences of ${\chi }_p^{\prime }$ [(red) circles] and ${\chi }_p^{\prime \prime }$ [(blue) triangles] on $f_{\mathrm{eff}}$. $f_{\mathrm{eff}}$ is represented on a logarithmic scale. Solid lines show fittings to Eqs. (8) and (9). ${\chi }_p^{\prime }\left(0\right)=3.4\times {10}^{-6}{\mathrm{G}}^{-2}$ was obtained by the fitting. (b) Relation between ${\chi }_p^{\prime }\left(f_{\mathrm{eff}}\right)$ and ${\chi }_p^{\prime \prime }\left(f_{\mathrm{eff}}\right)$ plotted using $f_{\mathrm{eff}}$ as a common parameter (Cole-Cole plot).

Figure 5

Typical $M_p\left(t\right)$ [solid (red) line] shown in the high-$f_{\mathrm{eff}}$ regime and $M_i$ [(blue) circles] obtained from Eq. (10). $f_{\mathrm{eff}}=40$ mHz. $M_i$ data are plotted at $t=i{T}_{\mathrm{eff}}$.

Figure 6

Dependence of the variance $S^2$ of $M_i$ on $f_{\mathrm{eff}}$.