Electrohydrodynamic convection in liquid crystals driven by multiplicative noise: Sample stability

We study the stochastic stability of a system described by two coupled ordinary differential equations parameterically driven by dichotomous noise with finite correlation time. For a given realization of the driving noise (a sample), the long time behavior is described by an infinite product of random matrices. The transfer matrix formalism leads to a Frobenius-Perron equation, which seems not solvable. We use an alternative method to calculate the largest Lyapunov exponent in terms of generalized hypergeometric functions. At the threshold, where the largest Lyapunov exponent is zero, we have an exact analytical expression also for the second Lyapunov exponent. The characteristic times of the system correspond to the inverse of the Lyapunov exponents. At the threshold the first characteristic time diverges and is thus well separated from the correlation time of the noise. The second time, however, depending on control parameters, may reach the order of the correlation time. We compare the corresponding threshold with a threshold from a simple mean-field decoupling and with the threshold describing stability of moments. The different stability criteria give similar results if the characteristic times of the system and the noise are well separated, the results may differ drastically if these times become of similar order. Digital simulation strongly confirms the criterion of sample stability. The stochastic differential equations describe in the frame of a simple one-dimensional model and a more realistic two-dimensional model the appearance of normal rolls in nematic liquid crystals. The superposition of a deterministic field with a ''fast'' stochastic field may lead to stable region that extends beyond the threshold values for deterministic or stochastic excitation alone, forming thus a stable tongue in the space of control parameters. For a certain measuring procedure the threshold curve may appear discontinuous as observed previously in experiment. For a different set of material parameters the stable tongue is absent.