The Rapid Communications section is intended for the accelerated publication of important new results. Since manuscripts submitted to this section are given priority treatment both in the editorial once and in production, authors should explain in their submittal letter why the work justifies this special handling. A Rapid Communication should be no longer than 4 printed pages and must be accompanied by an abstract. Page proofs are sent to authors We consider a system of globally coupled bistable systems under the influence of noise and periodic modulations.

We investigate a system of harmonically coupled identical nonlinear constituents subject to noise in different spatial arrangements. For global coupling, we find for infinitely many constituents the coexistence of several ergodic components and a bifurcation behavior like in first-order phase transitions. These results are compared with simulations for finite systems both for global coupling and for nearest-neighbor coupling on two-and three-dimensional cubic lattices. The mean-field-type results for global coupling provide a better understanding of the more complex behavior in the latter case.

We study a stochastic process X(t) which is a particular case of the Rayleigh process and whose square is the Bessel process, with various applications in physics, chemistry, biology, economics, finance, and other fields. The stochastic differential equation is dX(t)=(nD/X(t))dt+√(2D)dW(t), where W(t) is the Wiener process. The drift term can arise from a logarithmic potential or from taking X(t) as the norm of a multidimensional random walk. Due to the singularity of the drift term for X(t)=0, different natures of boundary at the origin arise depending on the real parameter n: entrance, exit, and regular. For each of them we calculate analytically and numerically the probability density functions of first-passage times or first-exit times. Nontrivial behavior is observed in the case of a regular boundary.

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 sample͒, 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.

We report on-off intermittency in electroconvection of nematic liquid crystals driven by a dichotomous stochastic electric voltage. With increasing voltage amplitude we observe laminar phases of undistorted director state interrupted by shorter bursts of spatially regular stripes. Near a critical value of the amplitude the distribution of the duration of laminar phases is governed over several decades by a power law with exponent −3/2. The experimental findings agree with simulations of the linearized electrohydrodynamic equations near the sample stability threshold.PACS numbers: 05.40.+j, 47.20.-k, 47.54.+r, 61.30-v Systems at a threshold of stability driven by a stochastic or chaotic process coupling multiplicatively to the system variables may exhibit on-off intermittency characterized by specific statistical properties of the intermittent signal. Quiescent (or laminar) periods (off-states) are interrupted by bursts of large variation (on-states); the duration of laminar periods is governed by power laws with exponents universal over a broad class of different systems. Early theoretical studies considered systems with few degrees of freedom modeled by differential equations [1] and mappings [2]. There is increasing interest in systems with many degrees of freedom [3], described by random map lattices [4], larger systems of coupled nonlinear elements [5], and partial differential equations [5,6]. Experimental results are available mainly for nonlinear electric circuits [7]; on-off intermittency was also observed in a spin wave experiment [8], in optical feedback [9], and a gas discharge plasma system [10]. Here we first report about on-off intermittency in a spatially extended dissipative system, viz. electroconvection (EC) in nematic liquid crystals driven by a stochastic voltage.EC in planarly aligned nematics is a standard system for pattern formation, for recent reviews see, e.g. [11]. In the presence of an electric field E a spontaneous fluctuation of the director leads due to the anisotropic conductivity to a formation of space charges which tend to destabilize the homogeneously ordered state. With increasing strength of the driving field one observes a hierarchy of convection patterns of increasing complexity. The patterns depend on external parameters such as amplitude, frequency and wave form of the driving voltage which are conveniently adjustable in the experiment. The hydrodynamic flow induces a modulation of the director field and thus of the effective indices of refraction which leads to transmission patterns easily observed with a microscope.In previous experiments, the superposition of a deterministic AC field with a stochastic field, E = E det (t) + E stoch (t), was studied. A variety of noise induced phenomena including stabilization or destabilization of the homogeneous state, and a change from continuous to discontinuous behaviour of the threshold as a function of the noise strength was observed [12][13][14][15][16] and has stimulated theoretical work [17,18].As long as the characteristi...