Irina Bashkirtseva

Stochastic sensitivity of 3D-cycles

The limit cycles of nonlinear systems under the small stochastic disturbances are considered. The random trajectories of forced system leave the deterministic cycle and form some stochastic bundle around it. The probabilistic description of this bundle near cycle based on stochastic sensitivity function (SSF) is suggested. The SSF is a covariance matrix for periodic solution of linear stochastic first approximation system. This matrix is a solution of the boundary problem for linear matrix differential equation. For 3D-cycles this matrix differential equation on the basis of singular expansion is reduced to the system of three scalar equations only. The possibilities of SSF to describe some peculiarities of stochastically forced Roessler model are demonstrated.

Sensitivity analysis of the stochastically and periodically forced Brusselator

The problem of sensitivity of nonlinear system limit cycle with respect to small stochastic and periodic disturbances is considered. Sensitivity analysis on the basis of quasipotential function is performed. The quasipotential is used widely in statistical physics (for instance by Graham for analysis of nonequilibrium thermodynamics problem). We consider an application of quasipotential technique to sensitivity problem. For the plane orbit case an approximation of quasipotential is expressed by some scalar function. This function (sensitivity function) is introduced as a base tool of a quantitative description for a system response on the external disturbances. New cycle characteristics (sensitivity factor, parameter of stiffness) are considered. The analysis of the forced Brusselator based on sensitivity function is shown. From this analysis the critical value of Brusselator parameter is found. The dynamics of forced Brusselator for this critical value is investigated. For small stochastic disturbances the burst of response amplitude is shown. For small periodic disturbances the period doubling regime of the transition to chaos scenario is demonstrated.

Confidence tori in the analysis of stochastic 3D-cycles

We present a new computer approach to the spatial analysis of stochastically forced 3D-cycles in nonlinear dynamic systems. This approach is based on a stochastic sensitivity analysis and uses the construction of confidence tori. A confidence torus as a simple 3D-model of the stochastic cycle adequately describes its main probabilistic features. We suggest an effective algorithm for construction of the confidence tori using a discrete set of confidence ellipses. The ability of these tori to visualize thin effects observed for the period-doubling bifurcations zone in the stochastic Roessler model are shown. For this zone, the geometrical growth of stochastic sensitivity of the forced cycles under transition to chaos is presented.

Analysis of noise-induced transitions from regular to chaotic oscillations in the Chen system

The stochastically perturbed Chen system is studied within the parameter region which permits both regular and chaotic oscillations. As noise intensity increases and passes some threshold value, noise-induced hopping between close portions of the stochastic cycle can be observed. Through these transitions, the stochastic cycle is deformed to be a stochastic attractor that looks like chaotic. In this paper for investigation of these transitions, a constructive method based on the stochastic sensitivity function technique with confidence ellipses is suggested and discussed in detail. Analyzing a mutual arrangement of these ellipses, we estimate the threshold noise intensity corresponding to chaotization of the stochastic attractor. Capabilities of this geometric method for detailed analysis of the noise-induced hopping which generates chaos are demonstrated on the stochastic Chen system.

NOISE-INDUCED BACKWARD BIFURCATIONS OF STOCHASTIC 3D-CYCLES

We study stochastically forced multiple limit cycles of nonlinear dynamical systems in a period-doubling bifurcation zone. Noise-induced transitions between separate parts of the cycle are considered. A phenomenon of a decreasing of the stochastic cycle multiplicity with a noise intensity growth is investigated. We call it by a backward stochastic bifurcation (BSB). In this paper, for the BSB analysis we suggest a stochastic sensitivity function technique. As a result, a method for the estimation of critical values of noise intensity corresponding to BSB is proposed. The constructive possibilities of this general method for the detailed BSB analysis of the multiple stochastic cycles of the forced Roessler system are demonstrated.

On control of stochastic sensitivity

Consideration was given to generation of the desired probabilistic characteristics of the attractors of the nonlinear stochastic control systems. In the case of small random perturbations, the scatter of trajectories in the neighborhood of stable equilibrium is describable in terms of the stochastic sensitivity matrix. Problem was posed of generating the desired stochastic sensitivity matrix. The corresponding notions of reachability and full controllability were examined. A criterion for full controllability was obtained, and a constructive description of the reachability set was given. Efficiency of the proposed constructions was demonstrated by way of example of control of the stationary distribution of the random states of the perturbed stochastic Van der Pol’s equation.

Noise-induced chaos and backward stochastic bifurcations in the lorenz model

We study the phenomena of stochastic D- and P-bifurcations of randomly forced limit cycles for the Lorenz model. As noise intensity increases, regular multiple limit cycles of this model in a period-doubling bifurcations zone are deformed to be stochastic attractors that look chaotic (D-bifurcation) and their multiplicity is reduced (P-bifurcation). In this paper for the comparative investigation of these bifurcations, the analysis of Lyapunov exponents and stochastic sensitivity function technique are used. A probabilistic mechanism of backward stochastic bifurcations for cycles of high multiplicity is analyzed in detail. We show that for a limit cycle with multiplicity two and higher, a threshold value of the noise intensity which marks the onset of chaos agrees with the first backward stochastic bifurcation.

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.

ANALYSIS OF STOCHASTIC CYCLES IN THE CHEN SYSTEM

We study the stochastically forced Chen system in its parameter zone under the transition to chaos via period-doubling bifurcations. We suggest a stochastic sensitivity function technique for the analysis of stochastic cycles. We show that this approach allows to construct the dispersion

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.