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In this paper, we address the stabilization problem for nonlinear systems in a critical case. Namely, we study the class of canonical nonlinear systems. Canonical nonlinear systems or chain of power integrators is an important subject of research. Studying such systems is complicated by the fact that they cannot be mapped onto linear systems. Moreover, they have the uncontrollable first approximation. Previous results on smooth stabilization of such systems were obtained under the assumption that the powers in the right-hand side are strictly decreasing. In this work, we consider a case of non-increasing powers in the right-hand side for a three-dimensional system. A popular approach for studying such systems is the backstepping method, which is a method of step-wise stabilization. This method requires a sequential investigation of lower-dimensional subsystems. Backstepping enables the study of a wide range of nonlinear triangular systems but requires technically complex and cumbersome computations. Therefore, a natural question arises about constructing stabilizing controls of a simple form. Polynomial controls can serve as an example of such controls. In the paper, we demonstrate that linear controls can be considered as stabilizing controls. We derive sufficient conditions for the coefficients of the linear control that ensure the asymptotic stability of the zero equilibrium point of the corresponding closed-loop system. The asymptotic stability is proven using the Lyapunov function method, which is found as the sum of squares. The negative definiteness of the Lyapunov function derivative in a neighborhood of the origin guarantees asymptotic stability. In contrast to the case of strictly decreasing powers, additional conditions on the control coefficients, apart from their negativity, emerge. The obtained result extends to a broader class of nonlinear systems through stabilization by nonlinear approximation. This allows the consideration of systems with higher-order terms in the right-hand side. The effectiveness of the applied approach is illustrated by several model examples. The method used in this work to investigate the case of non-increasing powers can be applied to systems of higher dimensions.

The purpose of this study is to investigate the phenomenon of self-organization of the dynamics of charge carriers in semiconductor structures. Investigate the basic model, give a numerical estimate for given parameters and propose its modification. Determine the dependence of the evaluation results on the control parameter. Consider the dynamics when the control parameter depends on time. Carry out theoretical analysis, numerical simulation and build graphs. Methods. In this paper, we investigate the possibilities and limitations of the basic model of generation-recombination dynamics in semiconductors, proposed earlier by other researchers. A modification of this model was proposed and considered. Results. Various types of charge carrier concentration dynamics are demonstrated.Theoretical analysis of the model was carried out. Numerical simulation has shown that for certain values of the control parameter, stable states are observed. Numerical estimates of the control parameter were obtained, phase portraits of the nonlinear equation were constructed, and the behavior of the dynamical system was considered when the control parameter is periodic. The extended model showed a qualitatively new behavior in comparison with the basic one. Conclusion. It is shown that charge dynamics in semiconductor structures can exhibit different behaviors. The patterns and estimates obtained are consistent with those obtained earlier. The results obtained can be verified experimentally and will be useful in the development of photo- and beta-voltaic devices.

Aim of this study is to compare different experimental paradigms and to determine parameters suitable for conducting a neurophysiological experiment with visual stimuli to assess foreign language proficiency and providing further time series analysis of electrical brain activity to reveal specific biomarkers. Methods. This paper explores the possibilities and limitations of various experimental studies using the metaanalysis paradigm. Statistical approaches are used to determine significance of the results. Results. We review the current state of research in the field of experimental works related to visual stimulus presentation and verbal performance acquisition. Generalizations and analytical estimates of the experimental parameters used in the studies are carried out to provide recommendations for future experimental research. Conclusion. In this area of applied research, we have developed experimental design and algorithms for working with multiple data sources. In addition, experimental encephalographic studies have been carried out, that allowed the optimal temporal structure selection.

In paper the problem of construction of exact solutions for nonlinear system of two equations of elliptic type is studied. Nonlinear systems of the equations of elliptic type are applied as mathematical models in the theory warm and a mass transfer of the reacting systems, in the theory of chemical reactors, the theory of burning and mathematical biology. In a one-dimensional case can carry the model of magnetic isolation of the vacuum diode described by the ordinary differential equations to the same class of the equations. Finding of exact solutions for nonlinear elliptic systems plays an important role as for development of the theory and establishment of properties of all set of solutions, and for applications. Exact solutions can be used for testing and verification of numerical methods of the solution of boundary value problems. In this paper the system of two equations of elliptic type with one nonlinearity depending on a difference of squares of required functions is considered. Conditions on nonlinearity under which the system is reduced to one equation are found. It is shown that in this case the system is reduced to the semi-linear elliptic equation of a special kind, only one component different from Helmholtz’s equation. The case of the system which isn’t reduced is separately studied at any nonlinearity to one equation. For this case the integro-differential equation to which have to satisfy radially symmetric solutions is obtained. Cases when this equation is reduced to the ordinary differential equation are specified and is integrated in an explicit form. A number of examples of construction of the exact solutions set by elementary functions for systems with the two-dimensional and three-dimensional operator of Laplace is given.

The purpose of this work is to consider the possibility of nonlinear influence of social media algorithms to the users opinions. A social media inherent algorithm of information ranging interacts with the user inherent bias and that increases the positive feedback loop. The result of this interaction is receiving by the user the only one side of an opinion and the user looses the very possibility to receive the opposite information. The conditions for the society polarization by means of a social media are investigated. Methods. In this paper, a model of users opinions dynamics was studied. There are two types of user’s strategy was considered: strategy 1, when a user puts "like" on information with proximity to his own view, but differed in any direction; strategy 2, when a user puts "like" on information along his own view, but more strict. Results. It was shown that for strategy 1 the society comes to a consensus, but for strategy 2 the society polarizes to the two opposite views. Considering the mixed society, where both strategies are used, it was found that the bifurcation to the society polarization appears when there are more than 40% of people using strategy 2. Conclusion. The inherent algorithms of social media, which are created to adapt in coming information to the user’s interests, creates or amplifies the bias of the user’s opinion and locks the user in an information chamber of only one type. That effect is substantially created by the social media algorithm itself. Thus interaction of users within a social media may increase the polarization of a society more than if they would communicate offline.

We prove a small-gain sufficient condition for (global) finite-time input-to-state stability (FTISS) of infinite networks. The network under consideration is composed of a countable set of finite-dimensional subsystems of ordinary differential equations, each of which is interconnected with a finite number of its “neighbors” only and is affected by some external disturbances. We assume that each node (subsystem) of our network is finite-time input-to-state stable (FTISS) with respect to its finite-dimensional inputs produced by this finite set of the neighbors and with respect to the corresponding external disturbance. As an application we obtain a new theorem on decentralized finite-time input-to-state stabilization with respect to external disturbances for infinite networks composed of a countable set of strict-feedback form systems of ordinary differential equations. For this we combine our small-gain theorem proposed in the current work with the controllers design developed by S. Pavlichkov and C. K. Pang (NOLCOS-2016) for the gain assignment of the strict-feedback form systems in the case of finite networks. The current results address the finite-time input-to-state stability and decentralized finite-time input-to-state stabilization and redesign the technique proposed in recent work S. Dashkovskiy and S. Pavlichkov, Stability conditions for infinite networks of nonlinear systems and their application for stabilization, Automatica. – 2020. – 112. – 108643, in which the case of $\ell_{\infty}$-ISS of infinite networks was investigated. The current paper extends and generalizes its conference predecessor to the case of finite-time ISS stability and decentralized stabilization in presence of external disturbance inputs and with respect to these disturbance inputs. In the special case when all these external disturbances are zeroes (i.e. are abscent), we just obtain finite-time stability and finite-time decentralized stabilization of infinite networks accordingly.

The paper considers methods for estimating stability using Lyapunov functions, which are used for nonlinear polynomial control systems. The apparatus of the Gro¨bner basis method is used to assess the stability of a dynamical system. A description of the Gro¨bner basis method is given. To apply the method, the canonical relations of the nonlinear system are approximated by polynomials of the components of the state and control vectors. To calculate the Gro¨bner basis, the Buchberger algorithm is used, which is implemented in symbolic computation programs for solving systems of nonlinear polynomial equations. The use of the Gro¨bner basis for finding solutions of a nonlinear system of polynomial equations is considered, similar to the application of the Gauss method for solving a system of linear equations. The equilibrium states of a nonlinear polynomial system are determined as solutions of a nonlinear system of polynomial equations. An example of determining the equilibrium states of a nonlinear polynomial system using the Gro¨bner basis method is given. An example of finding the critical points of a nonlinear polynomial system using the Gro¨bner basis method and the Wolfram Mathematica application software is given. The Wolfram Mathematica program uses the function of determining the reduced Gro¨bner basis. The application of the Gro¨bner basis method for estimating the attraction domain of a nonlinear dynamic system with respect to the equilibrium point is considered. To determine the scalar potential, the vector field of the dynamic system is decomposed into gradient and vortex components. For the gradient component, the scalar potential and the Lyapunov function in polynomial form are determined by applying the homotopy operator. The use of Gro¨bner bases in the gradient method for finding the Lyapunov function of a nonlinear dynamical system is considered. The coordination of input-output signals of the system based on the construction of Gro¨bner bases is considered.

The purpose of this work is to determine the ability of the partial directed coherence method to identify the directed interaction between nonlinear systems correctly in presence of nonlinear couplings between systems, as well as in the case when the measured signals are generated by objects of high dimension. The another purpose is to determine the dependence of the coupling estimation results on the parameters: series length, sampling rate, model dimension and the coupling architecture. Methods. In this paper, the possibilities and limitations of the frequency-resolved approach (partial directed coherence) to describe the couplings between high-dimensional time series are investigated. Surrogate time series constructed by permutation of realization are used to determine the significance of the results. Results. Coupling architecture in ensembles of small-dimensional oscillators can be correctly identified for linear and nonlinear systems connected in case of both linear and nonlinear coupling. For complex composite signals, when each measured time series is the sum of the signals of many individual oscillators, the technique is not specific enough, revealing non-existent connections, and it is not sensitive enough, missing the existing ones. Conclusion. The criteria for applying the partial directed coherence method to different signals are formulated. The measure does not show indirect couplings at sufficient series length, sampling rate and model dimension in contrast to the pairwise methods of Granger causality and transfer entropy. The measure works well for noisy time series. The method allows to study the connectivity in an ensemble of an arbitrary number of oscillators. The method allows to determine at what frequencies the interaction occurs. The partial directed coherence method gives acceptable results for series of length 80 and more characteristic periods in comparison with the Granger causality method, for which the efficiency is declared already at 4–16 characteristic periods.

The paper deals with an output control approach for structural uncertain MIMO systems under parametrical uncertainties, cross- reactions and external disturbances. The proposed technique ensures high quality of transients and high robustness in the steady state toward the disturbances without using high-gain components in the controller. A model transformation with one linear filter for control signal is used to obtain a structural determined form of the plant (“straight-feedback form”). The dynamical order of the filer is equal to the relative degree of MIMO plant. The procedure of relative dynamic degree estimation for nonlinear MIMO plant with cross-couplings is considered. The control algorithm combines an auxiliary loop method and backstepping method. The first one is a robust approach to unknown bounded disturbances evaluation and compensation. Backstepping is a well-known iterative procedure of control law synthesis by consecutive analysis of the plant state equations. The considered combination allows estimating undesired dynamics in each state equation and compensates it by creation of virtual control laws with equal magnitude toward disturbances and reverse sign. Experimental verification of the considered control algorithm is given with the use of a laboratory platform called “Twin Rotor MIMO System”. The platform can be controlled in two angle positions and represents the simplified helicopter dynamics.

Subject of Research. The paper presents an output control approach for a class of nonlinear MIMO (Multiple Input Multiple Output) systems. Quadrocopter is chosen to illustrate proposed adaptive control approach based on the high-gain principle, the so-called, “consecutive compensator”. Basic Concepts. The dynamic properties of quadrocopter can be described by a non-linear multi-channel system. Accordingly, the first step is accomplished by its decomposition, whereby the model is divided into two parts: static conversion and several dynamical channels with one input and one output). After the decomposition, a derivation of the simplified mathematical model of the quadrocopter is carried out; and, in addition to the absolute coordinate system, the local system rigidly connected with the robot is introduced. The appropriate coordinate transformation is taken into account. Interim Results. Decomposition of the mathematical model gives the possibility to create control system in two steps. Firstly, the law is applied; in accordance with it the so-called virtual control actions for each model channel of the quadrocopter are generated. A well-known method of robust control, "consecutive compensator", is used here, successfully well-accepted as the most easily implemented one from an engineering point of view. This method gives the possibility to control an object output under its uncertainty quantification. Secondly, the inverse transformation is performed, resulting in real control actions distributed between the system actuators. We assume that wind disturbance affects each channel of the object. What is more, the values and directions of these disturbances are changed negligibly slow. Thus, we can treat them as unknown constants, subject to compensation. Main Result. We have developed a multi-channel control system for the quadrocopter by different methods (decomposition of the mathematical model and the output of its simplified analogue, coordinate transformation, "consecutive compensator"). The obtained closed-loop system has been successfully simulated and corresponding results have been received.

The paper deals with a mathematical model of dynamical system with single degree of freedom, presented in the form of ordinary differential equations with nonlinear parts in the form of polynomials with constant and periodic coefficients. A modified method for the study of self-oscillations of nonlinear mechanical systems is presented. A refined method of transformation and integration of the equation, based on Poincare-Dulac normalization method has been developed. Refinement of the method lies in consideration of higher order nonlinear terms by Chebyshev economization technique that improves the accuracy of the calculations. Approximation of the higher order remainder terms by homogeneous forms of lower orders is performed; in the present case, it is done by cubic forms. An application of the modified method for the Van-der-Pol equation is considered as an example; the expressions for the amplitude and the phase of the oscillations are obtained in an analytical form. The comparison of the solution of the Van-der-Pol equation obtained by the developed method and the exact solution is performed. The error of the solution obtained by the modified method equals to 1%, which shows applicability of the developed method for analysis of self-oscillations of nonlinear dynamic systems with constant and periodic parameters.

In plasma modeling, partial differential equations and equation systems are usually applied, such as Boltzmann or Vlasov equations. Their solutions must meet initial and boundary conditions which presents a stubborn problem. Thus, the task is commonly reduced to a simpler one, e.g., to solving ordinary differential equations. This is the basis for model of magnetic electron isolation in vacuum diode proposed by a group of French mathematicians. The model is described by a system of two nonlinear ordinary second-order differential equations, and the problem of finding all exact solutions, i.e. full integration is concerned. In this paper, the whole concept is further developed into a class of elliptic equation systems with multidimensional Laplace operator, including both generalization of the above vacuum diode model and other systems applied in chemical technology, mathematical biology, etc. It is established that only solutions of Helmholtz linear equation can be solutions of the elliptic systems considered, and the properties of the former solutions can be inherited by the latter ones. Method of finding radially symmetric exact solutions is offered. A series of example control systems are observed, for which parametrical families of exact solutions (including those anisotropic by spatial variables) described by elementary or harmonious functions are found. Examples of global solutions defined on entire space are specified. The explicit expressions of exact solutions obtained have both theoretical and applied value as they can be used for testing, development and adaptation of numerical methods and algorithms of finding approximate solutions for boundary problems within the generalized model of magnetic isolation.