Stochastic sensitivity and noise-induced bifurcations of limit cycles

Deformations of limit cycles for dynamical systems forced by random disturbances are studied. A mathematical tool based on stochastic sensitivity analysis is shortly presented. A phenomenon of noise-induced bifurcation of supersensitive limit cycle of randomly forced Brusselator is discussed

Stochastic sensitivity analysis of noise-induced oscillations in Adler model

We consider Adler model with the saddle-node bifurcation on the invariant circle. Near this bifurcation, even weak random disturbances can generate noise-induced spike oscillations. Corresponding random phase trajectories form stochastic bundle. A dispersion of random trajectories in this bundle is non-uniform. To approximate this dispersion, we propose a new constructive approach based on the stochastic sensitivity analysis and method of "freezing" of the phase variable. A mathematical description of this approach is given. An extension of this theory for complex systems with so-called sequential dynamics is discussed. © 2019 Author(s).Russian Science Foundation, RSF: N 16-11-10098The work was supported by Russian Science Foundation (N 16-11-10098)

Analysis of the stochastically forced invariant manifolds of dynamic systems

We consider the randomly forced invariant manifolds of nonlinear dynamic systems. To study the dispersion of random states near the general deterministic attractors, we discuss two approaches. The first approach is based on the approximation of the quasipotential, and the second one uses the linear extension systems. A new semi-analytical method based on the stochastic sensitivity functions is suggested. The corresponding mathematical theory is shortly presented. Constructive applications of this theory to the analysis of equilibria and oscillatory regimes are given. © 2017 Author(s).The work was supported by Russian Science Foundation (No 16-11-10098)

Attainability analysis in the problem of stochastic equilibria synthesis for nonlinear discrete systems

A nonlinear discrete-time control system forced by stochastic disturbances is considered. We study the problem of synthesis of the regulator which stabilizes an equilibrium of the deterministic system and provides required scattering of random states near this equilibrium for the corresponding stochastic system. Our approach is based on the stochastic sensitivity functions technique. The necessary and important part of the examined control problem is an analysis of attainability. For 2D systems, a detailed investigation of attainability domains is given. A parametrical description of the attainability domains for various types of control inputs in a stochastic Henon model is presented. Application of this technique for suppression of noise-induced chaos is demonstrated

Stochastic sensitivity analysis of noise-induced intermittency and transition to chaos in one-dimensional discrete-time systems

We study a phenomenon of noise-induced intermittency for the stochastically forced one-dimensional discrete-time system near tangent bifurcation. In a subcritical zone, where the deterministic system has a single stable equilibrium, even small noises generate large-amplitude chaotic oscillations and intermittency. We show that this phenomenon can be explained by a high stochastic sensitivity of this equilibrium. For the analysis of this system, we suggest a constructive method based on stochastic sensitivity functions and confidence intervals technique. An explicit formula for the value of the noise intensity threshold corresponding to the onset of noise-induced intermittency is found. On the basis of our approach, a parametrical diagram of different stochastic regimes of intermittency and asymptotics are given. © 2012 Elsevier B.V. All rights reserved

Analysis+of+stochastic+sensitivity+of+Turing+patterns+in+distributed+reaction-diffusion+systems

In this paper, a distributed stochastic Brusselator model with diffusion is studied. We show that a variety of stable spatially heterogeneous patterns is generated in the Turing instability zone. The effect of random noise on the stochastic dynamics near these patterns is analysed by direct numerical simulation. Noise-induced transitions between coexisting patterns are studied. A stochastic sensitivity of the pattern is quantified as the mean-square deviation from the initial unforced pattern. We show that the stochastic sensitivity is spatially non-homogeneous and significantly differs for coexisting patterns. A dependence of the stochastic sensitivity on the variation of diffusion coefficients and intensity of noise is discussed

Stabilizing stochastically-forced oscillation generators with hard excitement: A confidence-domain control approach

In this paper, noise-induced destruction of self-sustained oscillations is studied for a stochastically-forced generator with hard excitement. The problem is to design a feedback regulator that can stabilize a limit cycle of the closed-loop system and to provide a required dispersion of the generated oscillations. The approach is based on the stochastic sensitivity function (SSF) technique and confidence domain method. A theory about the synthesis of assigned SSF is developed. For the case when this control problem is ill-posed, a regularization method is constructed. The effectiveness of the new method of confidence domain is demonstrated by stabilizing auto-oscillations in a randomly-forced generator with hard excitement.© EDP Sciences Società Italiana di Fisica Springer-Verlag 2013

Noise-induced shifts in the ecological model with delay

A Hassell-type mathematical model of population dynamics with delay and stochastic disturbances is considered. In this bistable model, one of the attractors corresponds to the extinction, and the other one describes non-trivial stable modes of dynamics. These modes can be both regular and chaotic. Structural stability zones are separated by local and global bifurcations. We study how noise shifts these bifurcation points and contracts the persistence zone. Abilities of the theoretical analysis of these phenomena with the help of the stochastic sensitivity function technique is discussed. © 2019 Author(s).Russian Science Foundation, RSF: N 16-11-10098The work was supported by Russian Science Foundation (N 16-11-10098)

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-induced bursting and transitions from order to chaos is discussed. © 2018 World Scientific Publishing Company.The work was supported by Russian Science Foundation (N 16-11-10098)

Modality analysis of patterns in reaction-diffusion systems with random perturbations

In this paper, a distributed Brusselator model with diffusion is investigated. It is well known that this model undergoes both Andronov–Hopf and Turing bifurcations. It is shown that in the parametric zone of diffusion instability the model generates a variety of stable spatially nonhomogeneous structures (patterns). This system exhibits a phenomenon of the multistability with the diversity of stable spatial structures. At the same time, each pattern has its unique parametric range, on which it may be observed. The focus is on analysis of stochastic phenomena of pattern formation and transitions induced by small random perturbations. Stochastic effects are studied by the spatial modality analysis. It is shown that the structures possess different degrees of stochastic sensitivity. © 2019 Udmurt State University. All right reserved.Russian Science Foundation, RSF: 16–11–10098Funding. This research was supported by the Russian Science Foundation (project no. 16–11–10098)