Izvestiya VUZ. Applied Nonlinear Dynamics
ISSN (print): 0869-6632, ISSN (online): 2542-1905
Founder: Saratov State University
Editor-in-Chief: Yu.V. Gulyaev, Member of the RAS, Ph.D., Professor
Frequency / Access: 6 issues per year / open
Included in: White List (2nd level), Higher Attestation Commission List, RISC, WoS, Scopus
The founder and the publisher of the journal is Saratov State University.
Active since 1993, 6 issues (1 volume) per year.
The journal subscription index is 73498. The subscription is available in online catalogue Ural-Press Group of Companies. The price is not fixed.
Registered by the Federal Service for Supervision of Communications, Information Technology, and Mass Media. Certificate of mass media registration No 1492 from 19.12.1991, re-registration in 24.08.1998, re-registration in 20.03.2020.
The journal is published in Russian (English articles are also acceptable, with the possibility of publishing selected articles in other languages by agreement with the editors), the articles data as well as abstracts, keywords, figure captions and references are consistently translated into English.
Scientific and technical journal "Izvestiya VUZ. Applied Nonlinear Dynamics" is an original interdisciplinary publication of wide focus. The journal is the oldest Russian specialized periodical on nonlinear dynamics (synergetics), chaos theory and their applications.
The journal publishes original research in the following areas (headings):
- Nonlinear Waves. Solitons. Autowaves. Self-Organization.
- Bifurcation in Dynamical Systems. Deterministic Chaos. Quantum Chaos.
- Applied Problems of Nonlinear Oscillation and Wave Theory.
- Modeling of Global Processes. Nonlinear Dynamics and Humanities.
- Innovations in Applied Physics.
- Nonlinear Dynamics and Neuroscience.
- Science for Education. Methodical Papers. History of Nonlinear Dynamics. Personalia.
Current Issue
Vol 33, No 2 (2025)
Editorial
On the 130th Anniversary of the Korteweg-de Vries Solitary Wave and the 60th Anniversary of the Word "Soliton"
Abstract
In 2025, it will be 130 years since the publication of the article by D. Korteweg and G. de Vries, which explored the famous nonlinear partial differential equation describing waves on water under the assumption that the water depth is much less than the wavelength but much greater than the amplitude, now known by the names of these authors (although it was recorded in another form earlier by J. Boussinesq).
Izvestiya VUZ. Applied Nonlinear Dynamics. 2025;33(2):145-152
145-152
Bifurcation in dynamical systems. Deterministic chaos. Quantum chaos
On the probabilistic description of the asynchronous phases occurrence in intermittent generalized synchronization regime of one-dimensional maps
Abstract
The purpose of the present study is to explain and describe (with the help of the probabilistic model) the process of breaking the stage of synchronous behavior and the emergence of a section of asynchronous dynamics in the regime of intermittent generalized chaotic synchronization in one-dimensional dynamical systems with discrete time. Methods. In this paper, a probabilistic model is used to quantitatively describe the observed characteristics of the behavior of two unidirectionally coupled systems being near the onset of the synchronous regime. Results. An analytical expression for the probability to observe the destruction of the synchronous phase on an interval of fixed duration under the assumption of uniformly distributed variable, as well as the form of the probability density function of the system state for the destruction intervals of synchronous dynamics are obtained. Conclusion. The paper presents quantitative estimates of the process of destruction of synchronous behavior in the regime of intermittent generalized chaotic synchronization for one-dimensional dynamical systems with discrete time. The generality of processes near the boundary of the synchronous motion for generalized chaotic synchronization and noise-induced synchronization is shown.
Izvestiya VUZ. Applied Nonlinear Dynamics. 2025;33(2):153-164
153-164
Applied problems of nonlinear oscillation and wave theory
Degenerate cases in discrete Lotka – Volterra dynamical systems
Abstract
The purpose of the work is to study the asymptotic behavior of trajectories of interior points of discrete Lotka–Volterra dynamical systems with degenerate skew-symmetric matrices operating in two-dimensional and three-dimensional simplexes. It turned out that in a number of applied problems, the Lotka–Volterra mappings of this type arise and the simplex points in this case are considered as the state of the system under study. In this case, the mapping preserving the simplex determines the discrete law of evolution of this system. For an arbitrary starting point, we can construct a sequence — an orbit that determines its evolution. And if in this case the mapping in question is an automorphism, we can define both a positive and a negative orbit for the point in question. At the same time, the limiting sets of positive and negative orbits are of particular interest. Methods. It is known that for Lotka–Volterra mappings it is possible to define limit sets, which in the case of non-degenerate mappings consist of a single point. In this paper, we define these sets for degenerate Lotka–Volterra mappings by constructing the Lyapunov function and applying Jacobian spectrum analysis. It should be noted that these sets allow us to describe the dynamics of the systems under consideration. Results. Taking into account that the considered mappings are automorphisms, using the Lyapunov functions and applying the analysis of the Jacobian spectrum, sets of limit points of both positive and negative trajectories are constructed and it is proved that in the degenerate case they are infinite. It is also shown that partially oriented graphs can be constructed for degenerate mappings. Conclusion. Degenerate cases of Lotka–Volterra mappings have not been considered by other authors before us. These mappings are interesting because they can be considered as discrete models of epidemiological situations, in particular, for studying the course of airborne viral infections. The results obtained in this work provide a detailed description of the dynamics of the trajectories of Lotka–Volterra mappings with degenerate matrices. In addition, partially oriented graphs were constructed for the systems under consideration in order to visually represent the dynamics of epidemiological situations.
Izvestiya VUZ. Applied Nonlinear Dynamics. 2025;33(2):165-183
165-183
Dynamics of interacting SIRS+V models of infectious disease spread
Abstract
The purpose of this work is study of processes of spread of infectious diseases in metapopulations interacted through spontaneous migration. The method is based on theoretical examination of the structure of the phase space of a system of coupled ODEs and numerical study of the transient processes in dependence on the coupling between subsystems. Results. A model of interacting populations in the form of two identical SIRS+V systems with mutual diffusion coupling is proposed and investigated. It was found that the long-term dynamics of the metapopulation does not differ from the behavior of an individual population; however, its transitional dynamics may be different and significantly depends on the values of the migration coefficients of infected and healthy individuals. In particular, under certain conditions, a complete suppression of infection waves can be observed in a secondarily infected population. Discussion. Despite the extreme simplicity of the model and the observed regimes, the results may be interesting from the point of view of practical recommendations for planning a strategy to combat transmission between communities, since they reveal the influence of the intensity of migrations of sick and healthy individuals on the spread of the epidemic in metapopulations.
Izvestiya VUZ. Applied Nonlinear Dynamics. 2025;33(2):184-198
184-198
Modeling of global processes. Nonlinear dynamics and humanities
Periodic regimes in a hybrid dynamical predator-prey system with migration and intraspecific competition
Abstract
The goal of the paper is to construct and analyze a hybrid model describing patch biocommunity dynamics with variable structure interspecific interactions. Species interaction structure variations are implied by predator’s population migration from a patch caused by food resources lack and patch colonization in a case of its sufficient amount. Methods. The model is presented by a three dimensional nonlinear hybrid system consisting of three dynamical subsystems. Switchings between subsystems are regulated by a patch food attractivity value the notion of which was introduced by one of the authors. Due to a food attractivity usage the system possesses a memory because of which variations of interactions’ structure obtain inertia typical for ecological processes. Results.The following regimes of patch biocommunity are introduced: interspecific interaction, predator’s migration and prey’s dynamics in the absence of predators. Symbolic dynamics corresponding to patch regimes variations is investigated. Results delivering conditions of existence of periodic trajectories in a hybrid system, as well as periodic symbolic regime sequences, are obtained. A bifurcation value of a parameter characterized predator’s resource requirements is determined. Numerical example is given. Conclusion. On the basis of obtained results concerning periodic symbolic regime sequences, expressed via system parameters relations, it is possible to predict a predator population migration from a patch and its recolonization. Moreover, appears a possibility to estimate time periods of recolonization processes which is an important practical problem in ecology.
Izvestiya VUZ. Applied Nonlinear Dynamics. 2025;33(2):199-218
199-218
Innovations in applied physics
Effects of a rotational motion of a liquid between curvilinear walls
Abstract
The purpose of the work is the revealing and the researching of peculiarities of an average in time rotational motion of a viscous liquid which is bordering with solid bodies (curvilinear walls) under periodic in time influences which are characterized by the presence or the absence of a predominant direction in space. 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 a decomposition by degrees of a small parameter), the method of Fourier, an averaging. Results. A new problem on the motion of a viscous liquid is formulated and solved. New hydro-mechanical effects are revealed. Conclusion. The fulfilled investigation is a continuation of previous investigations of non-trivial dynamics of hydro-mechanical systems under periodic influences. In particular the work is directed to the determination of the range of possibilities to create quality changes of a hydro-mechanical systems dynamics by periodic influences. The obtained results can be used in a scientific researching of perspective approaches to the solving actual applied and fundamental problems.
Izvestiya VUZ. Applied Nonlinear Dynamics. 2025;33(2):219-232
219-232
Nonlinear dynamics and neuroscience
Astrocyte-induced synchronization of neural network activity
Abstract
The purpose of this work is to study the role of mechanisms of astrocytic regulation of synaptic transmission in the processes of synchronization formation in signaling of neural networks by mathematical modeling methods. Methods. The paper presents a model of a small neuron-astrocyte ensemble. The Hodgkin-Huxley model is used as a model of the membrane potential dynamics of a neuron. The case of an ordered topology of connections (“all-to-all”) in a neural network is considered. The astrocyte network is modeled as a network of diffusion-coupled calcium oscillators with an ordered topology (in which the matrix of connections is structured in a certain way, interaction with the nearest neighbors). A biophysical model of calcium dynamics is used as an astrocyte model. The effect of astrocytes on neurons is taken into account as a slow modulation of synaptic connections weights in the neural network, proportional to calcium signals in nearby astrocytes. In other words, at the network level, the possibility of adaptive restructuring of oscillatory wave patterns due to astrocyte-induced regulation of synaptic transmission is being studied. The synchronization of neuronal activity is estimated by calculating the coherence of the neural network signaling. Results. The influence of astrocytes on the dynamics of the neural network consists in the excitation of time-correlated patterns of neural activity caused by an astrocyte-dependent increase in synaptic interaction between neurons on the time scales of astrocytic dynamics. It has been shown that synchronized calcium signaling of the astrocytic network leads to coordinated burst (bundle) activity of the neural network, which occurs against the background of uncorrelated spontaneous impulse activity induced by external noise stimulation. The influence of specific biophysical mechanisms of astrocytic modulation of synaptic transmission on the dynamic properties of local synchronization structures in neural ensembles has been investigated. The characteristics of the coordinated bundle activity of a neural network are studied depending on the properties of external noise stimulation, the strength of astrocytic regulation of synaptic transmission, as well as the degree of neurons influence on astrocytes.
Izvestiya VUZ. Applied Nonlinear Dynamics. 2025;33(2):233-248
233-248
Dynamics of recurrent neural networks with piecewise linear activation function in the context-dependent decision-making task
Abstract
Purpose. This paper aims to elucidate the dynamic mechanism underlying context-dependent two-alternative decision-making task solved by recurrent neural networks through reinforcement learning. Additionally, it seeks to develop a methodology for analyzing such models based on dynamical systems theory. Methods. An ensemble of neural networks with piecewise linear activation functions was constructed. These models were optimized using the proximal policy optimization method. The trial structure, featuring constant stimuli over extended periods, allowed us to treat inputs as system parameters and consider the system as autonomous during finite time intervals. Results. The dynamic mechanism of two-alternative decision-making was uncovered and described in terms of attractors of autonomous systems. The possible types of attractors in the model were characterized, and their distribution within the ensemble of models relative to the cognitive task parameters was studied. A stable division into functional populations was observed in the ensemble of models, and the evolution of these populations’ composition was examined. Conclusion. The proposed approach enables a qualitative description of the problem-solving mechanism in terms of attractors, facilitating the study of functional model dynamics and identification of populations underlying dynamic objects. This methodology allows for tracking the evolution of system attractors and corresponding populations during the learning process. Furthermore, based on this understanding, a two-dimensional network was developed to solve a simplified context-free two-alternative decision problem.
Izvestiya VUZ. Applied Nonlinear Dynamics. 2025;33(2):249-265
249-265
Nonlinear waves. Solitons. Autowaves. Self-organization
Influence of coupling topology and noise on the possibility of frequency tuning in ensembles of FitzHugh–Nagumo oscillators
Abstract
Purpose. The study focuses on analyzing spike activity and synchronization in ensembles of FitzHugh–Nagumo neurons, both with and without external noise excitation. In these networks oscillations at different frequencies can be induced depending on the excitability parameter of individual elements and the coupling strength between them. Additionally, variations in these parameters can lead to synchronization among the elements. The research investigates the dynamics of both a single-layer network, which includes a common element, and a three-layer network with an intermediate neuron-hub layer. Methods. To analyze the dynamics of the networks under investigation, we calculate the time-averaged spike frequencies of all elements. These frequencies are then averaged for each outer layer and compared with the frequency of the central element, as well as with each other in the case of a multilayer network. In order to assess the impact of coupling strength on the spike activity and synchronization of the network elements, we construct frequency distributions and frequency difference distributions in a plane of coupling strength coefficients. Results. It has been shown that small single-layer and three-layer networks of identical oscillators (FitzHugh–Nagumo neurons) with simple coupling topologies can exhibit different spike activity in different parts of the system. In this case, the neurons transition to a self-oscillatory mode due to repulsive coupling between the elements. The research has established that in a single-layer network, a ring of elements can synchronize in frequency with the central element within a specific range of coupling strength values. In a three-layer system, layer synchronization can also be observed. Weak noise has minimal impact on the synchronization boundaries of all three layers, in terms of coupling parameters. However, as the noise intensity increases, synchronization area decreases. At the same time, the noise leads to the emergence of a new synchronization region in which relay synchronization of the layers is observed in the absence of synchronization with the hub. Conclusion. The study explored the potential of exciting oscillations and achieving synchronization in single-layer and three-layer networks of coupled FitzHugh–Nagumo oscillators. The coupling strength between the elements varied in order to investigate its impact. Although the study only provided a broad understanding of the spike activity of excitable neurons in the two network models examined, it adequately demonstrated the crucial role of coupling in the spiking activity of these neurons.
Izvestiya VUZ. Applied Nonlinear Dynamics. 2025;33(2):266-282
266-282