Abstract
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 behaviour 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 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 which 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.
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