Vol 33, No 4 (2025)
Editorial
431-434
Bifurcation in dynamical systems. Deterministic chaos. Quantum chaos
A method for constructing a complete bifurcation picture of a boundary value problem for nonlinear partial differential equations: application of the Kolmogorov-Arnold theorem
Abstract
The purpose of this study is to develop a numerical method for bifurcation analysis of nonlinear partial differential equations, based on the reduction of partial differential equations to ordinary ones, using the Kolmogorov-Arnold theorem. Methods. This paper describes a method for reducing partial differential equations to ordinary ones using the Kolmogorov-Arnold theorem, as well as methods for the bifurcation analysis of nonlinear boundary value problems for ordinary differential equations. Results. The paper presents a new method for solving and bifurcation analysis of nonlinear boundary value problems for partial differential equations, which allow variational formulation. The method was applied to a nonlinear two-dimensional Bratu problem with Dirichlettype boundary conditions. Conclusion. A new method of bifurcation analysis for nonlinear partial differential equations has been developed. Specifically, a method has been proposed for reducing partial different equations to ordinary equations, which allows the use of the developed apparatus of bifurcation analysis for boundary value problems of ordinary differential equations. This method allows conducting bifurcation analysis for arbitrary nonlinear partial differential equations.
Izvestiya VUZ. Applied Nonlinear Dynamics. 2025;33(4):435-465
435-465
Applied problems of nonlinear oscillation and wave theory
Nonlinear dynamics for cylindrical resonator of wave solid-state gyroscope with different number of electrostatic control sensors
Abstract
The purpose of this work is to determine differences in the nonlinearities of mathematical models of dynamics and nonlinear effects of dynamics for the cylindrical resonator of wave solid–state gyroscope using a different number of electrostatic control sensors. Methods. In this article resonator oscillations nonlinearity caused by finite ratio of the small deflection of the resonator to the small gap of the electrostatic sensor is considered. To construct approximate mathematical models Tikhonov’s theorem on the passage to the limit is used and a small parameter singularly included in the system of differential equations is also taken into account. The equations of resonator dynamics are averaged by using the Krylov–Bogolyubov method. Results. The difference between the nonlinear terms in the equations of resonator dynamics with eight and sixteen control sensors is determined. It is found that nonlinear effects are more pronounced in the case of the gyroscope with sixteen control sensors. The angular drift velocity and the displacement of the resonant peak of the amplitude-frequency response are greater than in the case of eight control sensors. It is shown that in the case of eight control sensors, the angular drift velocity has a variable value and also contains a small uncompensated component. Conclusion. Mathematical models of the dynamics for the cylindrical resonator of wave solid-state gyroscope taking into account the nonlinearities caused by the excitation of oscillations by eight and sixteen electrostatic control sensors are deduced. The difference between the nonlinear effects of the resonator dynamics for wave solid-state gyroscope with different number of control sensors is shown. The angular drift velocity and the displacement of the resonant peak of the amplitude-frequency response are obtained. Conclusions about the applicability of gyroscope with eight control sensors are discussed.
Izvestiya VUZ. Applied Nonlinear Dynamics. 2025;33(4):466-484
466-484
Free linear vibrations of a viscoelastic spherical shell with filler
Abstract
Purpose. Thin multilayered shells are widely used in aviation, shipbuilding, and mechanical engineering. Recently, interest in the dynamic analysis of shell structures under various load effects has increased. This study examines the effect of moving normal internal pressure on a viscoelastic cylindrical shell. Methods. The viscoelastic medium filling the spherical shell has a significantly lower instantaneous modulus of elasticity than the shell itself. The solution is presented for the free vibrations of the viscoelastic system "shell–filler."An analytical frequency equation in the form of a transcendental equation is derived and solved numerically using the M¨uller method. Results. It has been found that, at certain values of viscoelastic and density parameters, low-frequency natural vibrations occur. These vibrations represent an aperiodic motion since the imaginary part of the natural frequency is large. For viscoelastic mechanical systems, the dependence of damping coefficients on physical and mechanical parameters has been identified. Conclusion. A theory and methods for calculating the complex natural frequencies of vibrations of an elastic spherical inhomogeneity in an elastic medium have been developed. A classification of such vibrations into radial, torsional, and spheroidal modes has been carried out. The problem is reduced to finding those frequencies at which the system of motion equations has nonzero solutions in the class of infinitely differentiable functions.
Izvestiya VUZ. Applied Nonlinear Dynamics. 2025;33(4):485-496
485-496
Fiber optic gyroscope with output signal noise reduction system: effect of system parameters on noise reduction
Abstract
The purpose of this work is to determine the optimal parameters for matching the signals of the measuring channels of a fiber-optic gyroscope with an output signal noise reduction system. Methods. The paper analyzes the circuit of a fiber-optic gyroscope with an output signal noise reduction system based on subtracting the signal of the reference measuring channel containing the intensity noise of the optical radiation source from the signal of the main measuring channel containing the useful information signal and the noise component. Four signal matching conditions are defined, failure to meet which can lead to a decrease in the noise suppression efficiency or an increase in the noise level. For each of the conditions under consideration, the parameters of the noise reduction system are determined and mathematical expressions are derived for the dependence of their influence on the level of spectral density of noise in the output signal of the gyroscope, mathematical modeling is carried out in a wide range of variation of the parameters under consideration. To confirm the obtained results, a computer simulation of the operation of a digital fiber-optic gyroscope with a closed feedback loop for measuring angular velocity and an output signal noise reduction system was carried out under conditions of varying the parameters of the noise reduction system. Results. For each parameter under consideration, the shape and degree of influence on the noise level were obtained, and the optimal value of each parameter was determined, at which the minimum value of the spectral density of noise in the output signal was observed. Conclusion. The conducted study confirms the correctness of the derivation of analytical expressions describing the formation of the noise component of the output signal of a fiber-optic gyroscope with a noise reduction system under conditions of incomplete matching of the signals of the main and reference measuring channels. A quantitative assessment of the requirements for the accuracy of this matching was obtained for such parameters as: time matching, matching by signal intensity, by bandwidth and by the state of polarization of optical radiation.
Izvestiya VUZ. Applied Nonlinear Dynamics. 2025;33(4):497-512
497-512
Dynamic model of economic cycles origin
Abstract
The purpose of the work is to research the phenomenon of the self-oscillation of the economic cycles for the model of the network with different types of element couplings (cooperative and competitive). As the element of the production network there are considered the typical system of automatic control of the added value,which determines the profitability of production. Methods. The research methods of the theory of nonlinear dynamic systems. Results. Modeling of nonlinear dynamics of small ensembles of connected production elements indicates the significant role of connections, in particular, the antagonistic (competitive) relationships lead to the emergence of regular and irregular economic fluctuations (cycles).
Izvestiya VUZ. Applied Nonlinear Dynamics. 2025;33(4):513-530
513-530
Innovations in applied physics
On the structure of a viscous liquid flow under periodic influences which have no predominant direction in space
Abstract
Purpose of the work is the revealing and the researching of peculiarities of the dynamics of a viscous liquid which is undergoing by periodic in time influences possessing no predominant direction in space underpossible simplest hydro-mechanical conditions (which are able to provide non-trivial behavior of the liquid). Methods. The analytic investigational methods for boundary problems for Navier–Stokes and continuity equations are used that are the method of perturbations, the method of Fourier. Results. A new problem on the flow of a viscous liquid is formulated and solved. The hydro-mechanical system consists of an incompressible viscous liquid and a moving absolutely solid wall which creates periodic influences to the liquid. A new peculiarity of the viscous liquid dynamics is revealed consisting in that in simplest hydro-mechanical conditions the liquid in the background of oscillatios (in average of time) performs a motion of a new type — a stationary fading with the distance from the wall motion which is characterized by the presence of a laminate structure. A physical interpretation of the formulation of the problem under consideration is given. Conclusion. The obtained results can be used in particular in a scientific search of ways to control hydro-mechanical systems, under a developing methods of a creation of prescribed flows of liquid media.
Izvestiya VUZ. Applied Nonlinear Dynamics. 2025;33(4):531-544
531-544
Nonlinear dynamics and neuroscience
Investigation of wave processes and rhythmic activity of the human brain using the Walsh orthogonal function system
Abstract
The purpose of this work is to study the wave processes and rhythmic activity of the brain based on multiscale parametric maps of electroencephalograms obtained as a result of algorithmic application of a system of discrete functions. Methods. For visualization, a previously developed multi-scale method for constructing parametric mappings of molecular genetic information was used, in which a set of four nucleotides is considered as a system of orthogonal Walsh functions. Results. The article proposes a new method of visualization of electroencephalography data for the study of rhythmic and wave processes of bioelectric activity of the brain. To analyze the electroencephalography data, the stage of transcoding the recorded amplitudes was previously carried out by one-to-one conversion of the EEG signal into a symbolic sequence, the alphabet of which consisted of four characters. Based on this method, the EEG signals of the subject were compared at rest and under mental stress. The study analyzed the readings of electrodes registering biopotentials of the frontal lobes of the brain. Conclusion. New methods have made it possible to identify various configurations of clusters in the frequency space of visualization, which can be used for comparative analysis of encephalograms and identification of features of recorded EEG signals. Specialized software has been developed as a tool for studying the rhythmic activity of the brain by constructing parametric displays of electroencephalograms.
Izvestiya VUZ. Applied Nonlinear Dynamics. 2025;33(4):545-556
545-556
Comparison of ensemble and correlation graphs in the task of classifying brain states based on fMRI data
Abstract
The study of functional brain networks that support cognitive processes is one of the central goals of modern neuroscience. Functional magnetic resonance imaging (fMRI) is widely used to obtain data on brain activity. However, the high dimensionality and dynamic nature of fMRI data makes their processing challenging. Network-based methods of data representation offer a promising approach to describe the brain as a network, where nodes correspond to brain regions and edges correspond to functional connections between them. This allows us to further explore the topology of brain networks and their role in cognitive states. The purpose of this paper is to compare ensemble and correlation graphs in a brain state classification task based on functional magnetic resonance imaging (fMRI) data. Methods. This paper presents a novel method for representing fMRI data in graph form based on ensemble learning. To demonstrate the effectiveness of the data representation method, we compared it with correlated graphs by applying a graph neural network to classify brain states. Results and Conclusion. Our results showed that ensemble graphs lead to significantly more accurate and stable classification. The better classification performance suggests that using this method we are more efficient in identifying functional connections between brain regions during cognitive tasks.
Izvestiya VUZ. Applied Nonlinear Dynamics. 2025;33(4):557-566
557-566
Interaction and synchronization of rhythms in a model of the brain limbic system
Abstract
The purpose of this study is to build a mathematical network model for interaction of rhythms in the limbic system of the brain, when pathological epileptiform activity is spreading from the focus. Methods. Based on well-known anatomical rules and biophysical laws, networks of model neurons are constructed for the hippocampus, the entorhinal and frontal cortex of both hemispheres. The task of chaotic dynamics synchronization in a relatively large network of heterogeneous neurooscillators by the signal of a ring generator of regular activity (a model of epileptic focus) was considered. Results. The regular pulse activity was shown to be able partly synchronize of suppress own activity in the limbic system model despite significant differences between the model equations of different cell types and the presence of natural fundamental oscillation frequencies in the beta and gamma ranges, with some spectral peaks become shifted. This effect stays valid not for a single model, but for a class of models that differ by the connectivity matrices. Conclusion. The mechanism of pathological rhythmic activity propagation from the epileptic focus to the whole limbic system is modelled for the first time, with using biologically relevant models of the limbic system of both brain hemispheres.
Izvestiya VUZ. Applied Nonlinear Dynamics. 2025;33(4):567-589
567-589
Recurrent neural network consisting of FitzHugh–Nagumo systems: characteristics required for training
Abstract
The purpose of this study is to determine the feasibility and features of training a recurrent neural network consisting of FitzHugh-Nagumo systems with delayed feedback to predict an impulse (spike signal). Methods. The network under consideration consisted of N=60 FitzHugh–Nagumo systems with different lag times. During training, the problem of which neuron should be activated and with what strength of lagged feedback was solved. The network was trained using gradient descent from different initial conditions. In the process of research, it was found that the use of standard recurrent network training characteristics such as mean squared error or mean absolute error was not applicable to this task, so an alternative method for computing the loss function was proposed. Results. The proposed combined loss function is the sum of MSE error and interspike interval error, and therefore has the following advantages: 1 – includes the information about spike periodicity and interspike intervals, 2 – responds adequately to the absence of a network output signal, 3 – takes into account small amplitude fluctuations in addition to impulse dynamics, which allows predicting complex quasi-periodic signals. It has been shown that gradient descent can be used for the problem, but several initial conditions must be used because of the nonlinearity of the loss function. The more initial conditions, the more accurate the result. Conclusion. The problem of predicting a pulse (spike) signal using a self-closed recurrent neural network consisting of FitzHughNagumo systems with delayed feedback has been successfully solved. It was clearly shown what features should be taken into account during loss function calculation and how the gradient descent should be realized.
Izvestiya VUZ. Applied Nonlinear Dynamics. 2025;33(4):590-604
590-604