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The time-dependent process of directional crystallization in the presence of a mushy layer is considered with allowance for arbitrary fluctuations in the atmospheric temperature and friction velocity. A nonlinear set of mushy layer equations and boundary conditions is solved analytically when the heat and mass fluxes at the boundary between the mushy layer and liquid phase are induced by turbulent motion in the liquid and, as a result, have the corresponding convective form. Namely, the 'solid phase-mushy layer' and 'mushy layer-liquid phase' phase transition boundaries as well as the solid fraction, temperature and concentration (salinity) distributions are found. If the atmospheric temperature and friction velocity are constant, the analytical solution takes a parametric form. In the more common case when they represent arbitrary functions of time, the analytical solution is given by means of the standard Cauchy problem. The deterministic and stochastic behaviour of the phase transition process is analysed on the basis of the obtained analytical solutions. In the case of stochastic fluctuations in the atmospheric temperature and friction velocity, the phase transition interfaces (mushy layer boundaries) move faster than in the deterministic case. A cumulative effect of these noise contributions is revealed as well. In other words, when the atmospheric temperature and friction velocity fluctuate simultaneously due to the influence of different external processes and phenomena, the phase transition boundaries move even faster. This article is part of the theme issue 'From atomistic interfaces to dendritic patterns'.This article is part of the theme issue 'From atomistic interfaces to dendritic patterns'.

We study excitability phenomena for the stochastically forced FitzHugh-Nagumo system modeling a neural activity. Noise-induced changes in the dynamics of this model can be explained by the high stochastic sensitivity of its attractors. Computational methods based on the stochastic sensitivity functions technique are suggested for the analysis of these attractors. Our method allows us to construct confidence ellipses and estimate a threshold value of a noise intensity corresponding to the neuron excitement. On the basis of the proposed technique, a supersensitive limit cycle is found for the FitzHugh-Nagumo model.

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We study a stochastically forced predator-prey model with Allee effect. In the deterministic case, this model exhibits non-trivial stable equilibrium or limit cycle corresponding to the coexistence of both species. Computational methods based on the stochastic sensitivity functions technique are suggested for the analysis of the dispersion of random states in stochastic attractors. Our method allows to construct confidence domains and estimate the threshold value of the intensity for noise generating a transition from the coexistence to the extinction.

We study the stochastic dynamics of a Hodgkin-Huxley neuron model in a regime of coexistent stable equilibrium and a limit cycle. In this regime, noise may suppress periodic firing by switching the neuron randomly to a quiescent state. We show that at a critical value of the injected current, the mean firing rate depends weakly on noise intensity, while the neuron exhibits giant variability of the interspike intervals and spike count. To reveal the dynamical origin of this noise-induced effect, we develop the stochastic sensitivity analysis and use the Mahalanobis metric for this four-dimensional stochastic dynamical system. We show that the critical point of giant variability corresponds to the matching of the Mahalanobis distances from attractors (stable equilibrium and limit cycle) to a three-dimensional surface separating their basins of attraction.

The effects of stochastic perturbations in a nonlinear alpha Omega-dynamo model are investigated. By using transformation of variables we identify a "slow" variable that determines the global evolution of the non-normal alpha Omega-dynamo system in the subcritical case. We apply an adiabatic elimination procedure to derive a closed stochastic differential equation for the slow variable for which the dynamics is determined along one of the eigenvectors of the full system. We derive the corresponding Fokker-Planck equation and show that the generation of a large scale magnetic field can be regarded as a first-order phase transition. We show that the an advantage of the reduced system is that we have explicit expressions for both the stochastic and deterministic potentials. We also obtain the stationary solution of the Fokker-Planck equation and show that an increase in the intensity of the multiplicative noise leads to qualitative changes in the stationary probability density function. The latter can be interpreted as a noise-induced phase transition. By a numerical simulation of the stochastic galactic dynamo model, we show that the qualitative behavior of the "empirical" stationary pdf of the slow variable is accurately predicted by the stationary pdf of the reduced system.