Electric-field-induced weakly chaotic transients in ferroelectric liquid crystals

Nonlinear dynamics induced in surface stabilized ferroelectric liquid crystals by strong alternating external electric fields is studied both theoretically and experimentally. As has already been shown, molecular reorientations induced by sufficiently strong fields of high-enough frequencies can reveal a long transient behavior that has a weakly chaotic character. The resulting complex dynamics of ferroelectric liquid crystals can be considered not only as a consequence of irregular motions of particular molecules but also as a repercussion of a surface-enforced partial decorrelation of nonlinear molecular motions within smectic layers. To achieve more insight into the nature of this phenomenon and to show that the underlying complex field-induced behavior of smectic liquid crystals is not exceptional, ranges of system parameters for which the chaotic behavior occurs are determined. It is proved that there exists a large enough set of initial phase trajectory points, for which weakly chaotic long-time transitory phenomena occur, and, thereby, it is demonstrated that such a chaotic behavior can be regarded as being typical for strongly field-driven thin liquid crystal systems. Additionally, the influence of low-amplitude random noise on the duration of the transient processes is numerically studied. The strongly nonlinear contribution to the electro-optic response, experimentally determined for liquid crystal samples at frequencies lower than the actual field frequency, is also analyzed for long-time signal sequences. Using a statistical approach to distinguish numerically response signals of samples from noise generated by measuring devices, it is shown that the distribution of sample signals distinctly differs from the device noise. This evidently corroborates the occurrence of the nonlinear low-frequency effect, found earlier for different surface stabilized liquid crystal samples.

Figure 1

Geometry of the studied liquid crystal system. Smectic layers are aligned perpendicular to plates of a cell of thickness $d$. The local polarization ${\vec{P}}_S$ is perpendicular to the molecular director $\vec{n}$ and lies parallel to the layer plane. The orientation of the director is determined by tilt and azimuthal angles, $\theta$ and $\varphi$, respectively. The external alternating electric field $E\left(t\right)$ is applied perpendicular to the boundary plates.

Figure 2

Trajectory, determined by Eqs. (6, 7, 8) for $U=20\mathrm{V}$, $f=4\mathrm{kHz}$, and for the initial conditions $a_0^{\left(0\right)}=-0.01$, $a_1^{\left(0\right)}=0.3$, and $a_2^{\left(0\right)}=-0.003$. Projections of the trajectory onto the ($a_0,a_2$) plane (a) and onto the ($a_2,a_1$) plane (b) are shown for 185 000 initial iteration steps (blue points) and for $6\times {10}^6$ iterations steps (red points) after ${10}^7$ skipped trajectory points.

Figure 3

Sets of initial values of coefficients $a_0$ and $a_2$ for which trajectories exhibit long weakly chaotic transients (like those illustrated in Figs. 2 and 3). Each set has been obtained for $U_0=20\mathrm{V}$ and $f=4\mathrm{kHz}$ but for different initial values of $a_1^{\left(0\right)}$: $a_1^{\left(0\right)}=0.3$ (a), $a_1^{\left(0\right)}=0.2$ (b), $a_1^{\left(0\right)}=0.15$ (c), $a_1^{\left(0\right)}=0.12$ (d), $a_1^{\left(0\right)}=0.1$ (e), and $a_1^{\left(0\right)}=0.05$ (f). The sets (shown as black regions) represent respective sections of the basin of chaotic transients in the plane ($a_0,a_2)$.

Figure 4

Sets of initial values of coefficients $a_1$ and $a_2$ leading to weakly chaotic transients of the type shown in Figs. 2 and 3 for different values of the remaining parameter: $a_0^{\left(0\right)}=-0.01$ (a), $a_0^{\left(0\right)}=-0.05$ (b), $a_0^{\left(0\right)}=-0.07$ (c), $a_0^{\left(0\right)}=-0.08$ (d), $a_0^{\left(0\right)}=-0.09$ (e), $a_0^{\left(0\right)}=-0.1$ (f), $a_0^{\left(0\right)}=-1.05$ (g), and $a_0^{\left(0\right)}=-0.11$ (h). The resulting sections of basin of chaotic transients have been obtained at $U_0=20\mathrm{V}$ and $f=4\mathrm{kHz}$.

Figure 5

Sets of initial values of coefficients $a_0$ and $a_1$ that yield weakly chaotic transients for different initial values of $a_2^{\left(0\right)}$: $a_2^{\left(0\right)}=-0.001$ (a), $a_2^{\left(0\right)}=0.001$ (b), $a_2^{\left(0\right)}=-0.003$ (c), $a_2^{\left(0\right)}=0.003$ (d), as determined at $U_0=20\mathrm{V}$ and $f=4\mathrm{kHz}$.

Figure 6

Molecular azimuthal angle as a function of the number of points of trajectory determined for $U_0=60\mathrm{V}$, $f=4\mathrm{kHz}$, and $\tilde{x}=0.01$ (1), $\tilde{x}=0.5\left(2\right)$, $\tilde{x}=1.0$ (3). Starting trajectory points were taken to be $a_0^{\left(0\right)}=-0.001$, $a_1^{\left(0\right)}=0.3$, and $a_2^{\left(0\right)}=0.01$. The plot (a) shows the azimuthal angle for $N=2\times {10}^4$ trajectory points, while the diagram (b) displays continuation of the plot (a) for iteration steps ${10}^6\le n\le {10}^6+2\times {10}^4$.

Figure 7

Plots of the time dependence of the molecular azimuthal angle, analogous to drawings shown in Figs. 6 and 6, respectively, but obtained in the presence of random noise of the level $\mu =0.2$ (see the text).

Figure 8

Exemplary oscilloscope image of time variations of the output voltage of the selective voltmeter, obtained for a time step $\mathrm{\Delta }t=5\times {10}^{-4}\mathrm{s}$. The voltmeter was set to the frequency $f_s=176\mathrm{Hz}$ while the voltage frequency was fixed at $f=4\mathrm{kHz}$. The voltage amplitude was $U_0=20\mathrm{V}$ (a), $U_0=2\mathrm{V}$ (b), and $U_0=0\mathrm{V}$ (c).

Figure 9

The function $G\left(q\right)$ plotted for discrete time sequences of the output voltage as measured for the applied voltage of amplitudes $U_0=0,2\mathrm{V}$, and $20\mathrm{V}$. All experimental data were obtained for $\mathrm{\Delta }t=5\times {10}^{-4}\mathrm{s}$ and $N=5\times {10}^4$.