Volume 33, Nº 3 (2025)
Editorial
287-288
Applied problems of nonlinear oscillation and wave theory
Influence of Gaussian noise and Levy noise on the phase dynamics of the ensemble of Kuramoto-like oscillators of first and second order
Resumo
The purpose of this study is to determine the stability threshold of the dynamic modes of the ensemble of phase Kuramoto-like oscillators, describing the behavior of a simple power grid model with a ring topology, under the external influence of Gaussian noise and Levy noise, to evaluate the results and determine the threshold values of noise at which the considered dynamic model is the most sensitive to noise and demonstrates a change of the steady state. Methods. In this paper, two ensembles of Kuramoto-like phase oscillators with the same topology but different number of oscillators are investigated. The ensembles consist of second and first order phase oscillators modeling the dynamics of generators and consumers in the power grid, respectively. In this work, mode maps are computed and used, from which regions with different synchronous dynamics are selected. In the selected regions, a set of initial conditions is fixed and the ensemble under study is modeled in the presence of noise of different types and intensities. The obtained result is evaluated with the help of calculated spatio-temporal diagrams, values of the Kuramoto parameter and statistical characteristics estimated from the realizations of oscillations in time. Results. It has been shown that a power grid model consisting of Kuramoto-like phase oscillators exhibits different robustness to external noise disturbances depending on the type of noise disturbance and the steady-state dynamic regime. It was demonstrated that the frequency synchronization mode of all oscillators, independent of the initial conditions, is insensitive to the influence of white noise of high intensity, both Gaussian and Levy noise. Whereas, in the region of coexistence of synchronous and asynchronous behavior, depending on the initial conditions, a change of phase dynamics under the influence of different noise is observed. Numerical experiment has shown that the power grid model is more susceptible to Levy noise due to the noise features associated with random emissions, which in turn can be interpreted as random impulses. Conclusion. In a power grid model represented by two ensembles consisting of different numbers of Kuramoto-like phase oscillators of second and first order, different modes of frequency and phase dynamics of the oscillators are established. A numerical experiment with the influence of Gaussian noise and Levy noise is carried out for the obtained modes. It is shown that the model under study is more sensitive to Levy noise, the influence of which leads to a change of the dynamic mode due to the influence of strong random pulses.
Izvestiya VUZ. Applied Nonlinear Dynamics. 2025;33(3):289-306
289-306
Influence of the topology of coupled antiferromagnetic oscillators on their mutual synchronization
Resumo
The purpose the purpose of this study is to mathematically describe an ensemble of oscillators coupled by a common dipole field, to model and study the synchronization of the three topologies under consideration: a ring, a grid, and a special case of a lattice-chain, with subsequent identification of a priority topology that would ensure synchronization over a larger range of initial conditions of the ensemble. Methods. To simplify the numerical modeling and study the synchronization of systems of antiferromagnetic oscillators coupled by a common dipole field, the Kuramoto model for coupled oscillators, as well as the order parameter and its average value, were used. Results. A mathematical model for antiferromagnetic oscillators coupled by a common dipole field was obtained for three topologies: ring, grid, and chain. Using the Kuramoto model of coupled oscillators and the order parameter and its average value, the synchronization of the considered arrays was studied, and it was found that the lattice is the preferred topology for the same ensemble parameters. Conclusion. The paper studies the problem of synchronization of ensembles of antiferromagnetic spin-Hall oscillators (AFM SCHO) combined into different topologies: ring, grid and chain. It is shown that the grid is the most preferable topology for achieving synchronization at lower values of the coupling strength constant between the moscillators. The ring and chain require higher values of the coupling strength constant, which is their disadvantage. Nevertheless, with a sufficiently strong coupling, synchronization is also possible in these topologies.
Izvestiya VUZ. Applied Nonlinear Dynamics. 2025;33(3):307-321
307-321
Coherent resonance in the microwave chaos generators
Resumo
The purpose of this study is the observation of coherent resonance phenomenon in two microwave single-mode chaos generators with a delayed feedback under external noise influence. The first generator is a vacuum chaos generator based on a traveling wave tube and a multi-resonator drift klystron. The second generator is a solid-state chaos generator based on a transistor amplifier and a spin-wave transmission line supporting a magnetostatic surface wave (MSSW) propagation. The chaotic dynamics of the vacuum generator is caused by the presence of a falling section on the amplitude characteristic of the drift klystron. In the solid-state generator, the development of chaotic dynamics is associated with a nonlinear parametric three-wave decay of the MSSW into short-wave spin waves. Methods. To observe the coherent resonance phenomenon in single chaotic oscillators (the microwave chaos generators), the time filtering method under the noise influence with a limited frequency band is used. The method is based on the effect of forced chaos synchronization (through its suppression) by external noise. The proposed method has the greatest efficiency in frequency separation of the power spectra of chaotic and noise signals (the power spectra of both signals should not overlap). Results. In the studied microwave chaos generators of various physical natures, the existence of the “on-off” intermittency mode, in that the coherent resonance phenomenon is observed, has been experimentally established. It has been shown that the autocorrelation time of the envelope of a chaotic MW signal has a maximum value at a certain power level of the noise MW signal, and this maximum value depends on the noise MW signal bandwidth. Conclusion. The developed method for the coherent resonance observation can also be applied to the multimode (broadband) ring chaos generators with the delayed feedback.
Izvestiya VUZ. Applied Nonlinear Dynamics. 2025;33(3):322-340
322-340
Scenarios of passive particle transport in the velocity field of a vortex pair in shear flow
Resumo
The purpose of the work is to analyze the transport of passive particles in the velocity field of a two-vortex configuration on a plane with a possible presence of a shear flow. We model the system using two point vortices and a shear flow, where the velocity components depend linearly on one coordinate. Scenarios of particle transport and mixing are studied depending on the intensity of one vortex (in the region of [-1, 1]∖{0}) and various shear flows with fixed initial positions of the vortices and an intensity of the second equal to unity. In the investigation, we mainly use numerical methods of dynamical systems analysis. We apply 8th-order of accuracy integrators to solve the Cauchy problem for a system of ordinary differential equations. The study also involved constructing Poincare sections and fields of local Lyapunov exponents, as well as studying transformations of marker circles (fluid contours) on a plane. Results. Depending on the signs of the vortex intensities and the direction of the shear flow, the following scenarios were found: mixing of particles near the vortex structure; movement of a vortex pair along closed orbits with the transfer of particles from its vicinity and mixing near the orbits; mixing of particles in a large area on the plane; movement of a vortex pair to infinity with the transfer of particles from the vicinity of its initial position over long distances; disintegration of the pair and movement of vortices in different directions to infinity with the transfer of particles from the vicinity of their initial positions. In the presence of a shear flow, stochastic scattering of passive particles is typical, which is because of their chaotic dynamics. Conclusion. We show that depending on the signs of intensities and parameters of the shear flow, a vortex pair can be a ’carrier’ moving particles from the vicinity of its initial position over long distances, a ’mixer’ of particles in a limited area of the plane, a ’scatterer’ of particles from a certain area along its path to infinity. The results of the article can be useful in explaining the complexity of transfer processes in fluids and gas flows when vortex pairs arise in them.
Izvestiya VUZ. Applied Nonlinear Dynamics. 2025;33(3):341-360
341-360
Modeling of global processes. Nonlinear dynamics and humanities
Simplified models for describing multitasking modes in living recognition systems
Resumo
The purposeof this study is to explore the possibilities and create versions of simplified languages that describe the variety of dynamic modes during the alternating execution of multiple tasks. These languages can be used for an adequate description of experimentally recorded variants of behavior in living systems under such conditions. Methods. The study was conducted using a physical methodology based on qualitative analysis of possible solutions and confirmed by the results of computational experiments. To qualitatively describe the versions of possible mechanisms for switching between dynamic modes of operation of living systems in situations presented by the environment for these systems (expressed through changes in emotional or energy states), the simplest basic model was used. This model includes two balance equations corresponding to either the first or the second solvable problem. For this system, a two-dimensional phase space is constructed, allowing for tracking characteristic changes in the trajectories of the representing points in the ’null-isocline’ system, which depend on the control parameters. Various trajectories of representing points are considered depending on the initial conditions, visually demonstrating the main modes of transitional processes in the developing system. Results and discussion. Classification of dynamic modes in the system has been conducted, depending on control parameters. Such dynamic modes form the basis for simplified language descriptions. The proposed simplified mathematical model allowed for the examination of a wide range of states and various types of its evolutionary changes in full accordance with known examples of behavior modes in living systems.
Izvestiya VUZ. Applied Nonlinear Dynamics. 2025;33(3):361-380
361-380
Nonlinear dynamics and neuroscience
Application of phase dynamics modeling and recurrence methods to assess the characteristics of the relationship between physiological rhythms
Resumo
The purpose of this work is to apply two methods of nonlinear dynamics to assess the characteristics of the relationship between time series extracted from physiological rhythms. The analyzed time series were respiratory rhythm fluctuations, arterial pressure variability curves, and variability of neuronal activity intervals in the medulla oblongata of rats before and during pain exposure. Methods. To solve the problem of identifying the relationship and assessing the asymmetry and direction of the relationship, a method for modeling the phase dynamics of weakly coupled and weakly noisy systems and a method for calculating averaged conditional probabilities of recurrences of time series generated by interacting systems were used. As characteristics of the relationship between systems, estimates of the intensity of the influence of one system on another and estimates in the differences of the averaged conditional probabilities of recurrences were used. Results. To verify the robustness of the applied methods to noise, an analysis of a well-studied model of unidirectionally coupled Van der Pol oscillators was performed. The correct determination of the direction of coupling by both methods with weak noise and a decrease in the possibility of identifying the direction by the phase modeling method with increasing noise, and the preservation of the possibility of correctly determining the direction by the recurrence method were confirmed. For experimentally obtained and weakly noisy biological time series, an asymmetry of the coupling with a predominant influence of the respiratory rhythm on the variability of neuronal activity and arterial pressure, and the influence of arterial pressure variability on the neuronal activity of the reticular formation of the medulla oblongata was found in most of the analyzed data. Conclusion. The application of two methods for assessing the characteristics of the relationship between weakly noisy time series, both model and experimental, showed quite consistent results in the predominant influence of one system on the other.
Izvestiya VUZ. Applied Nonlinear Dynamics. 2025;33(3):381-398
381-398
Markers of patients’ condition after orthodontic treatment: application of recurrent analysis to EEG data obtained during cognitive tests
Resumo
The purpose of this study — is to study the differences in recurrent indicators based on electroencephalography signals of patients after orthodontic treatment during cognitive tests. Depending on the type of impact (installation of braces or aligners), identify markers in the canals, which can be used to further determine the strength of stress from orthodontic intervention for subsequent correction of treatment. Methods. Recurrence analysis was used to study electroencephalography data. In particular, recurrent indicators were constructed for each channel of each patient. Results. The channels in which changes in recurrent indicators with different types of orthodontic influence are the greatest are demonstrated. For these channels, the dynamics of recurrent indicators in them is described to highlight some markers of stress and pain experienced by the patient. Conclusion. In the course of the study, recurrent indices were constructed based on the electroencephalography data of patients after orthodontic treatment. It was shown that the highest differences in patients of different groups were demonstrated by the temporal and occipital canals (O1, O2, T3, T4, T5, T6). Thus, the value of recurrent indices of this group of indices should be used as a marker of the patient’s condition.
Izvestiya VUZ. Applied Nonlinear Dynamics. 2025;33(3):399-411
399-411
Nonlinear waves. Solitons. Autowaves. Self-organization
Kink dynamics in the φ4 model with two extended impurities
Resumo
The purpose of this study is to use numerical methods to consider the problem of nonlinear kink dynamics for the φ4 equation in a model with two identical extended «impurities» (or spatial inhomogeneity of the potential). Methods. The φ4 model with inhomogeneities was numerically solved using the method of lines for partial differential equations. The kink was launched in the direction of the inhomogeneities with different initial velocities. The distance between the two impurities was also varied. The kink trajectory after interaction with the impurities was studied. The discrete Fourier transform was used to find the oscillation frequencies of the kink after interaction with spatial inhomogeneities. Results. The interaction between the kink and two identical extended impurities described by rectangular functions is described. Possible scenarios of kink dynamics are determined, taking into account resonance effects, depending on the magnitude of the system parameters and initial conditions. Critical and resonant velocities of the kink motion are found depending on the impurity parameters and the distance between them. Significant differences are observed in the kink dynamics when interacting with repulsive and attractive impurities. It is established that among the found scenarios of kink dynamics for the case of extended rectangular impurities, there are scenarios of resonant kink dynamics obtained earlier for the case of one extended impurity, for example, quasi-tunneling and repulsion from an attractive potential. Conclusion. An analysis of the influence of system parameters and initial conditions on possible scenarios of kink dynamics is carried out. Critical and resonant kink velocities are found as functions of the impurity parameters and the distance between them.
Izvestiya VUZ. Applied Nonlinear Dynamics. 2025;33(3):412-425
412-425