Lev Ryashko

Sea Ice Dynamics Induced by External Stochastic Fluctuations

The influence of stochastic fluctuations in the atmosphere and in the ocean caused by different occasional phenomena (noises) on dynamic processes of sea ice growth with a mushy layer is studied. It is shown that atmospheric temperature variances substantially increase the sea ice thickness, whereas dispersion variations of turbulent flows in the ocean to a great extent decrease the ice content produced by false bottom evolution.

Stochastic sensitivity analysis of noise-induced excitement in a prey–predator plankton system

In this paper, the excitable Truscott–Brindley dynamical model of the interacting populations under environmental noise is considered. We study a probabilistic mechanism of the noise-induced outbreaks in a zone of stable deterministic equilibria. The stochastic sensitivity functions technique and method of confidence ellipses are applied for the description of the spatial arrangement of random states near the deterministic attractor. The effectiveness of our method for the constructive analysis of the stochastic excitement in Truscott–Brindley model is demonstrated. A critical value for the intensity of noise-generating noise-induced outbreaks is estimated.

How environmental noise can contract and destroy a persistence zone in population models with Allee effect

A problem of the analysis of the noise-induced extinction in population models with Allee effect is considered. To clarify mechanisms of the extinction, we suggest a new technique combining an analysis of the geometry of attractors and their stochastic sensitivity. For the conceptual one-dimensional discrete Ricker-type model, on the base of the bifurcation analysis, deterministic persistence zones are constructed in the space of initial states and biological parameters. It is shown that the random environmental noise can contract, and even destroy these persistence zones. A parametric analysis of the probabilistic mechanism of the noise-induced extinction in regular and chaotic zones is carried out with the help of the unified approach based on the sensitivity functions technique and confidence domains method.

Sensitivity analysis of the noise-induced oscillatory multistability in Higgins model of glycolysis

A phenomenon of the noise-induced oscillatory multistability in glycolysis is studied. As a basic deterministic skeleton, we consider the two-dimensional Higgins model. The noise-induced generation of mixed-mode stochastic oscillations is studied in various parametric zones. Probabilistic mechanisms of the stochastic excitability of equilibria and noise-induced splitting of randomly forced cycles are analysed by the stochastic sensitivity function technique. A parametric zone of supersensitive Canard-type cycles is localized and studied in detail. It is shown that the generation of mixed-mode stochastic oscillations is accompanied by the noise-induced transitions from order to chaos.

Controlling the equilibria of nonlinear stochastic systems based on noisy data

Abstract For controlling an equilibrium of a nonlinear stochastic system, the problem of stabilization and synthesis with a required dispersion is studied. This problem is solved for the case where the feedback regulator uses noisy data. The new approach is based on an extension of the stochastic sensitivity synthesis method. Technically, this problem is reduced to the analysis of some quadratic matrix equations. A solution to the problem of minimizing the stochastic sensitivity is given. Details of such analysis are discussed for 2D and 3D nonlinear stochastic oscillators.

Stochastic variability and noise-induced generation of chaos in a climate feedback system including the carbon dioxide dynamics

In this work, a non-linear dynamics of a simple three-dimensional climate model in the presence of stochastic forcing is studied. We demonstrate that a dynamic scenario of mixed-mode stochastic oscillations of the climate system near its equilibrium can be formed. As this takes place, a growth of noise intensity increases the frequency of interspike intervals responsible for the abrupt climate changes. In addition, a certain enhancement of stochastic forcing abruptly increases the atmospheric carbon dioxide and decreases the Earths ice mass. A transition from order to chaos occurring at a critical noise is shown.

Nonlinear dynamics of mushy layers induced by external stochastic fluctuations

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’.

Methods of Stochastic Analysis of Complex Regimes in the 3D Hindmarsh–Rose Neuron Model

A problem of the stochastic nonlinear analysis of neuronal activity is studied by the example of the Hindmarsh–Rose (HR) model. For the parametric region of tonic spiking oscillations, it is shown that random noise transforms the spiking dynamic regime into the bursting one. This stochastic phenomenon is specified by qualitative changes in distributions of random trajectories and interspike intervals (ISIs). For a quantitative analysis of the noise-induced bursting, we suggest a constructive semi-analytical approach based on the stochastic sensitivity function (SSF) technique and the method of confidence domains that allows us to describe geometrically a distribution of random states around the deterministic attractors. Using this approach, we develop a new algorithm for estimation of critical values for the noise intensity corresponding to the qualitative changes in stochastic dynamics. We show that the obtained estimations are in good agreement with the numerical results. An interplay between noise-indu...

Noise-induced quasi-periodic oscillations in Hindmarsh-Rose neuron model

We study the phenomenon of noise-induced quasi-periodic oscillations in the stochastic Hindmarsh-Rose neuron model. We show that with the increase of the noise intensity the quiescent regime in this model transforms into the quasi-periodic (bursting) one with the formation of the stochastic torus. This phenomenon is confirmed by changes in the probability distribution of random trajectories and by the interspike intervals statistics. We show that the emergence of the torus bursting oscillations is related to the peculiarities of the geometrical arrangement of deterministic trajectories near the equilibrium and its stochastic sensitivity.

Mean square stabilisation of complex oscillatory regimes in nonlinear stochastic systems

ABSTRACT A problem of stabilisation of the randomly forced periodic and quasiperiodic modes for nonlinear dynamic systems is considered. For this problem solution, we propose a new theoretical approach to consider these modes as invariant manifolds of the stochastic differential equations with control. The aim of the control is to provide the exponential mean square (EMS) stability for these manifolds. A general method of the stabilisation based on the algebraic criterion of the EMS-stability is elaborated. A constructive technique for the design of the feedback regulators stabilising various types of oscillatory regimes is proposed. A detailed parametric analysis of the problem of the stabilisation for stochastically forced periodic and quasiperiodic modes is given. An illustrative example of stochastic Hopf system is included to demonstrate the effectiveness of the proposed technique.