Steady state Hopf mode interaction in anisotropic systems
Date
2013
Authors
Maple, Jennifer, author
Oprea, Iuliana, advisor
Dangelmayr, Gerhard, advisor
Shipman, Patrick, committee member
Fassnacht, Steven, committee member
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Abstract
A paradigm example of pattern formation in anisotropic extended systems is the electroconvection of nematic liquid crystals, due to its easily accessible control parameters and the variety of patterns near onset. Some of the patterns observed are oblique and normal rolls which can be stationary or traveling, and more complex structures such as worms, defects and spatiotemporal complexity, including spatiotemporal intermittency and chaos, can occur, see e.g., Dennin et al, Science 272, 1996. During electroconvection experiments on the nematic liquid crystal mixture Phase V, a mode interaction between oblique stationary rolls and normal traveling rolls has been observed by Acharya et al, Int. J. Mol. Sci. 12, 448, 2011; a system of four globally coupled Ginzburg Landau equations for slowly varying spatiotemporal amplitudes of ideal roll patterns governing the dynamics of anisotropic systems close to the experimentally observed codimension-two point has been set up, two equations for the steady oblique rolls and two for the normal traveling rolls. This dissertation pursues a theoretical and numerical study of the patterns predicted by this system of globally coupled Ginzburg Landau equations. Acharya et al presented a bifurcation analysis of the normal form that follows from the Ginzburg Landau system by ignoring slow variations. The basic solutions of the normal form are two types of pure mode solutions corresponding to ideal oblique stationary and normal traveling rolls, respectively, and superpositions of pure mode solutions, which are referred to as mixed mode solutions. Acharya et al distinguished two cases for the bifurcations of these solutions. In one case the mixed mode solution is stable and a continuous transition between the steady oblique rolls and the normal traveling rolls is predicted. For the other case, the mixed mode solution is unstable and bistability occurs between the steady oblique rolls and the normal traveling rolls. In the present work, a numerical code was developed to simulate the spatiotemporal system of globally-coupled, complex Ginzburg-Landau equations using a pseudo-spectral method. The simulations of the system resulted in patterns that were consistent with the normal form analysis. Steady oblique and normal traveling rolls were found numerically. A region of bistability of the steady oblique rolls and normal traveling rolls was found numerically, and a continuous transformation between the two primary branches via a stable mixed mode branch has been observed when the main bifurcation parameter is varied. Mixed mode solutions have been found that involved either amplitudes of steady rolls aligned in two different ("zig" and "zag") directions, or amplitudes of two counter-propagating normal traveling rolls, for parameter values near the primary instabilities and when the initial conditions favored their appearance, and a bifurcation diagram showing the occurrence of steady state, steady oblique rolls, normal traveling rolls, mixed mode solutions, as well as bistability of the steady oblique rolls and normal traveling rolls has been obtained numerically.
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Subject
anisotropic systems
pseudo spectral method
mode interaction