Spatiotemporal Chaotic Localized State in Liquid Crystal Light Valve Experiments with Optical Feedback

The existence, stability properties, and dynamical evolution of localized spatiotemporal chaos are studied. We provide evidence of spatiotemporal chaotic localized structures in a liquid crystal light valve experiment with optical feedback. The observations are supported by numerical simulations of the Lifshitz model describing the system. This model exhibits coexistence between a uniform state and a spatiotemporal chaotic pattern, which emerge as the necessary ingredients to obtain localized spatiotemporal chaos. In addition, we have derived a simplified model that allows us to unveil the front interaction mechanism at the origin of the localized spatiotemporal chaotic structures.

Figure 1

Chaoticon structure observed in the LCLV experiment. (a) Temporal evolution of the light intensity isosurface; the camera acquisition rate is 30 fps. (b) Instantaneous snapshot. (c) Plot of the light intensity at a given time.

Figure 2

Bifurcation diagram of states observed in the LCLV experiment; the average pattern intensity is plotted versus the input laser intensity $I_{\mathrm{in}}$; the upper and lower branches correspond to the spatiotemporal chaotic pattern and uniform state, respectively. Insets display example pictures of observations in the respective regions. The shaded area represents the coexistence region. An enlarged view of the chaoticon is shown in the upper central inset.

Figure 3

Bifurcation diagram of the Lifshitz model, Eq. (1), $\mu =-0.09$, $\nu =-1$, $b=-3.5$, $c=10$. The upper and lower branches correspond to the spatiotemporal chaotic pattern and uniform state, respectively. The insets display pattern profiles obtained in the respective regions. The shaded area represents the coexistence region.

Figure 4

Spatiotemporal diagrams of localized structures of model (1), $\mu =-0.09$, $\nu =-1$, $b=-3.5$, $\eta =0.04$, $c=10$, for (a) 2, (b) 5, (c) 8, and (d) 24 cells using 512 points in space ($dx=0.5$) and a temporal step $dt=0.01$. Temporal evolutions of $u$ at two given points for (e) 2 and (f) 24 cells. (g) The largest Lyapunov exponent versus the number of cells.

Figure 5

Numerical observations of chaoticons and respective spatiotemporal diagrams in (a) the Lifshitz model, Eq. (1), $\mu =-0.09$, $\nu =-1$, $b=-3.87$, $\eta =0.04$, $c=10$ and (b) the phenomenological Nagumo-Kuramoto model, Eqs. (2, 3), $\alpha =0.5$, $\beta =0.3$, $k=11$, $\gamma =0.3$.

Figure 6

Fronts interaction law accounting for the equilibrium chaoticon widths derived from Eq. (4). The closed (empty) circles represents stable (unstable) chaoticons.