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 dr iving noise (a sample), the long time behavior is described by an infi nite product of random matrices. The transfer matrix formalism leads t o a Frobenius-Perron equation, which seems not solvable. We use an alt ernative method to calculate the largest Lyapunov exponent in terms of generalized hypergeometric functions. At the threshold, where the lar gest Lyapunov exponent is zero, we have an exact analytical expression also for the second Lyapunov exponent. The characteristic times of th e system correspond to the inverse of the Lyapunov exponents. At the t hreshold the first characteristic time diverges and is thus well separ ated from the correlation time of the noise. The second time, however, depending on control parameters, may reach the order of the correlati on time. We compare the corresponding threshold with a threshold from a simple mean-field decoupling and with the threshold describing stabi lity of moments. The different stability criteria give similar results if the characteristic times of the system and the noise are well sepa rated, the results may differ drastically if these times become of sim ilar order. Digital simulation strongly confirms the criterion of samp le stability. The stochastic differential equations describe in the fr ame of a simple one-dimensional model and a more realistic two-dimensi onal 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 valu es for deterministic or stochastic excitation alone, forming thus a st able tongue in the space of control parameters. For a certain measurin g procedure the threshold curve may appear discontinuous as observed p reviously in experiment. For a different set of material parameters th e stable tongue is absent.