Stochastic phenomena in the dynamical tumor-immune system

A nonlinear dynamical model describing the interaction of tumor and immune cells is considered. Regimes of "dormant tumor" and "tumor explosion" are studied by methods of the bifurcation theory and stochastic simulation. Noise-induced transitions from "dormant tumor" to "tumor explosion" and back are investigated parametrically in dependence of the noise intensity and parameter of the inactivation of immune cells by tumor ones. Three scenarios of the noise-induced transitions are discussed: (i) from "dormant tumor" to "tumor explosion", (ii) from "tumor explosion" to "dormant tumor", (iii) oscillations between these two regimes. © 2019 Author(s).Russian Science Foundation, RSF: N 16-11-10098The work was supported by Russian Science Foundation (N 16-11-10098)

Approximation of stochastic equilibria for dynamical systems with parametrical noise

A problem of the approximation for stochastic equilibria for the system with parametrical noise is considered. Our approach is based on first approximation stochastic systems technique. For these systems, we use a spectral theory of positive operators for the analysis of exponential mean square stability. Using this method we approximate a dispersion of random states in stochastic equilibrium of nonlinear dynamical system with parametrical noise

Noise-induced excitability of the complex liquid flows

We study an excitability for the stochastically forced system modeling a dynamics of the complex liquid flows. A phenomenon of noiseinduced generation of large-amplitude oscillations in a zone of stable equilibria is studied. © 2013 Irina Bashkirtseva

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 phenomena in Ricker-type population model with delay

A phenomenon of the noiseinduced extinction is studied on the base of the conceptual Rickertype model with the delay and Allee effect. This nonlinear discrete population model exhibits the persistence with the different form of attractors, both regular and chaotic. For this model, the persistence zones are defined by points of the crisis bifurcations. The phenomenon of the noiseinduced extinction is investigated with the help of direct numerical simulations and by the semianalytical new method based on the stochastic sensitivity functions. In the stochastic analysis, a geometrical approach taking into account a mutual arrangement of the confidence domains and basins of attraction is used. © 2017 Author(s).The work was supported by Russian Science Foundation (grant No 16-11-10098)

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

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)

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. © 2013 World Scientific Publishing Company

Controlling Stochastic Sensitivity by Feedback Regulators in Nonlinear Dynamical Systems with Incomplete Information

The problem of synthesis of stochastic sensitivity for equilibrium modes in nonlinear randomly forced dynamical systems with incomplete information is considered. We construct a feedback regulator that uses noisy data on some system state coordinates. For parameters of the regulator providing assigned stochastic sensitivity, a quadratic matrix equation is derived. Attainability of the assigned stochastic sensitivity is reduced to the solvability of this equation. We suggest a constructive algorithm for solving this quadratic matrix equation. These general theoretical results are used to solve the problem of stabilizing equilibrium modes of nonlinear stochastic oscillators under conditions of incomplete information. Details of our approach are illustrated on the example of a van der Pol oscillator. © 2021 by the author. Licensee MDPI, Basel, Switzerland.Funding: The work was supported by Russian Foundation for Basic Research (20-01-00165)