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

Iuliana Oprea; Gerhard Dangelmayr

doi:10.3934/dcdsb.2018095

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2019, Volume 24, Issue 1: 273-296. Doi: 10.3934/dcdsb.2018095

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

Iuliana Oprea^, and
Gerhard Dangelmayr

Department of Mathematics, Colorado State University, Fort Collins, CO 80523-1874, USA

* Corresponding author: Iuliana Oprea
* Corresponding author: Iuliana Oprea

Received: November 30, 2016

Revised: September 30, 2017

Early access: March 2018

Published: January 2019

Work supported by NSF grant DMS-1615909.

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Abstract

In this paper we investigate the transition from periodic solutions to spatiotemporal chaos in a system of four globally coupled Ginzburg Landau equations describing the dynamics of instabilities in the electroconvection of nematic liquid crystals, in the weakly nonlinear regime. If spatial variations are ignored, these equations reduce to the normal form for a Hopf bifurcation with $O(2) × O(2)$ symmetry. Both the amplitude system and the normal form are studied theoretically and numerically for values of the parameters including experimentally measured values of the nematic liquid crystal Merck I52. Coexistence of low dimensional and extensive spatiotemporal chaotic patterns, as well as a temporal period doubling route to spatiotemporal chaos, corresponding to a period doubling cascade towards a chaotic attractor in the normal form, and a kind of spatiotemporal intermittency that is characteristic for anisotropic systems are identified and characterized. A low-dimensional model for the intermittent dynamics is obtained by perturbing the eight-dimensional normal form by imperfection terms that break a continuous translation symmetry.

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Mathematics Subject Classification: Primary: 35B36; Secondary: 35B32, 76W05.

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Figure 1. Time series $|A_j(T)|$, $1\leq j\leq 4$, showing the period doubling sequence, obtained from numerical simulations of (10) for (a) $\alpha = 0.024$, (b) $\alpha = 0.025$, (c) $\alpha = 0.02515$, and (d) $\alpha = 0.0252$. In (c) the upper time series in the two plots are for $|A_1|$, $|A_4|$, and the lower time series for $|A_2|$, $|A_3|$.

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Figure 2. Phase space plots of the quasiperiodic solutions and the chaotic attractor, for (a) $\alpha = 0.025$, (b) $\alpha = 0.025025$, and (c) $\alpha = 0.02515$ (see text). Left plots: $|A_4|$ versus $|A_1|$, right plots: $|A_3|$ versus $|A_2|$.

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Figure 3. Pattern snapshot $U$, equation (9), for Simulation 1.

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Figure 4. Time series and phase space plots of dominant and secondary mode amplitudes plots for Simulation 1.

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Figure 5. Averages of the mode amplitudes $|a_j(m,n,T)|$ for Simulation 2. (a): Time averages. (b) Time series of the $(m,n)$-averages for $j = 2,3$.

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Figure 6. Snapshots of (a): $|A_j|$ ($1\leq j \leq 4$) and (b): pattern $U$, equation (9), for Simulation 2.

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Figure 7. Time series of dominant and secondary Ginzburg Landau mode amplitudes for Simulation 3.

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Figure 8. Time series of the amplitudes $|z_j(T)|$, equation (14).

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Figure 9. Real parts of snapshots of simulations of the CML (15) after transients have died out for $\lambda = 2.3$, $c = 0.25$, $d = 0.05$, $\alpha = -\beta = 0.5$ and $M = N = 64$. Initial condition for (a) is a small random perturbation of a uniform state, while for (b) it is fully random.

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Table 1. The six basic periodic solutions of (10).

Name (Shortcut)	(A₁, A₂, A₃, A₄)	Pattern
Travelling Waves (TW)	(A, 0, 0, 0)	\|A\|cos$(\omega t+p_cx+q_cy)$
Travelling Rectangles-$x$ (TR$_x$)	(A, 0, 0, A)	2\|A\|cos$(\omega t+p_cx)\cos(q_cy)$
Travelling Rectangles-$y$(TR$_y$)	(A, A, 0, 0)	2\|A\|cos $(\omega t+q_cy)\cos(p_cx)$
Standing Waves (SW)	(A, 0, A, 0)	2\|A\|cos$(\omega t)\cos(p_cx+q_cy)$
Standing Rectangles (SR)	(A, A, A, A)	4\|A\|cos$(\omega t)\cos(p_cx)\cos(q_cy)$
Alternating Waves (AW)	(A, iA, A, iA)	2\|A\|$[\cos(\omega t)\cos(p_cx+q_cy)$ $-\sin(\omega t)\cos(p_cx-q_cy)]$

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