Quantitative and qualitative characterization of zigzag spatiotemporal chaos in a system of amplitude equations for nematic electroconvection

Abstract

It has been suggested by experimentalists that a weakly nonlinear analysis of the recently introduced equations of motion for the nematic electroconvection by M. Treiber and L. Kramer [Phys. Rev. E 58, 1973 (1998)] has the potential to reproduce the dynamics of the zigzag-type extended spatiotemporal chaos and localized solutions observed near onset in experiments [M. Dennin, D. S. Cannell, and G. Ahlers, Phys. Rev. E 57, 638 (1998); J. T. Gleeson (private communication)]. In this paper, we study a complex spatiotemporal pattern, identified as spatiotemporal chaos, that bifurcates at the onset from a spatially uniform solution of a system of globally coupled complex Ginzburg-Landau equations governing the weakly nonlinear evolution of four traveling wave envelopes. The Ginzburg-Landau system can be derived directly from the weak electrolyte model for electroconvection in nematic liquid crystals when the primary instability is a Hopf bifurcation to oblique traveling rolls. The chaotic nature of the pattern and the resemblance to the observed experimental spatiotemporal chaos in the electroconvection of nematic liquid crystals are confirmed through a combination of techniques including the Karhunen-Loeve decomposition, time-series analysis of the amplitudes of the dominant modes, statistical descriptions, and normal form theory, showing good agreement between theory and experiments.

Complex dynamics near a Hopf bifurcation with symmetry: A parameter study

“…An important question is which of the observed behaviour is found in a spatiotemporal setting, when the normal form is extended to the system of globally coupled Ginzburg Landau equations introduced in [32] and investigated to some extent in [22,33,36]. A preliminary study for fixed values of the parameters in region A of Figure 2(c) [22] showed that the Ginzburg Landau dynamics depends on the initial conditions.…”

“…As is apparent from (2), z 1 and z 3 are amplitudes of waves travelling in the directions À(p c , q c ) and (p c , q c ), and z 2 and z 4 are amplitudes of waves travelling in the directions (p c , Àq c ) and À(p c , Àq c ), respectively. The two wave pairs are sometimes referred to as 'zig' and 'zag' waves [35,36]. When the spatial periodicity constraint is removed, the representation (2), with the z j considered time and space dependent, leads to a globally coupled system of PDEs of the Ginzburg Landau type, whose spatially uniform solutions are governed by (1) [32].…”

“…When the spatial periodicity constraint is removed, the representation (2), with the z j considered time and space dependent, leads to a globally coupled system of PDEs of the Ginzburg Landau type, whose spatially uniform solutions are governed by (1) [32]. Analytical and numerical studies of these spatiotemporal equations are presented in [22,33,36]. In this article we do not consider the Ginzburg Landau system, but point out that the dynamics of (1) is also present in this spatiotemporal system, although the stability of the attractors of (1) in the spatiotemporal setting requires further investigation.…”

A period doubling route to spatiotemporal chaos in a system of Ginzburg-Landau equations for nematic electroconvection

“…Overall, the temporal dynamics resulting from Simulation 1 is a low-dimensional temporal chaos, reminiscent of the chaotic attractor of the normal form for the same values of the coefficients, but with a few additional modes being activated. We note that a similar type of low-dimensional spatiotemporal chaotic dynamics shown by (7), referred to as "zigzag-chaos", has been identified and analyzed in [44].…”

Spatiotemporal chaotic dynamics in a transmission line model

“…We use the method of Karhunen-Loeve (KL) spatial modes decomposition combined with time-series analysis of the resulting mode amplitudes [5] . The KL modes are orthogonal and represent statistically independent parts of the data and are ranked according to the amount of data correpsonding to them.…”