Vol 33, No 6 (2025)
Bifurcation in dynamical systems. Deterministic chaos. Quantum chaos
On the interaction of a system with multifrequency oscillations with a chaotic generator
Abstract
The purpose of the work: to study the influence of the dynamics of a chaotic system on a system with multi-frequency quasi-periodicity and the Landau-Hopf scenario. The Kislov-Dmitriev chaotic system and an ensemble of van der Pol oscillators with non-identical excitation parameters are chosen as the object of study. Methods. The analysis was carried out using graphs of Lyapunov exponents and the criterion for identifying types of quasiperiodic bifurcations based on them. Results. Scenarios of the changing of the regime’s types are presented as the coupling parameter between the subsystems decreased. They may have certain features. Thus, the transition from a three-frequency to a four-frequency regime occurs not through a quasiperiodic Hopf bifurcation, but through a chaos window. The latter is characterized by three or four zero Lyapunov exponents. Inside this chaotic window, a peculiar bifurcation is possible. It is corresponding to an increase in the number of zero Lyapunov exponents according to the type of a saddle-node Hopf bifurcation. Chaos with a different number of zero exponents is observed as the coupling parameter of van der Pol oscillators varied. In this case, a cascade of points corresponding to a step-by-step increase in the number of zero exponents occurs according to a different scenario. It is to a certain extent similar to a quasiperiodic Hopf bifurcation. When the control parameter When the control parameter of the Kislov-Dmitriev system increases, hyperchaos with three zero Lyapunov exponents may appear in the combined system. An inverted order of changing modes is also possible. In this case, for example, a three-frequency regime turns into a four-frequency regime through a chaotic window. Conclusion. The obtained results expand conception about high-dimensional chaos with several zero Lyapunov exponents and its transformations with parameter changes.
Izvestiya VUZ. Applied Nonlinear Dynamics. 2025;33(6):785-803
785-803
Applied problems of nonlinear oscillation and wave theory
Nonlinear vibrations of a high-rise structure with a dynamic vibration damper
Abstract
Objective: To study the dynamic behavior of high-rise structures with dynamic oscillation dampers under various kinematic influences, taking into account the nonlinear elastic, viscoelastic, and elastoplastic properties of the structure’s material and the viscoelastic properties of the oscillation damper. Dynamic oscillation dampers can be used both at the stage of design, development and construction of structures, as well as in cases where structural quality deficiencies are identified during operation. Their adjustment allows for a simple way to achieve the desired effect of reducing oscillation levels. Methods. A mathematical model, methodology, and algorithm are proposed for evaluating the dynamic behavior of high-rise structures equipped with a dynamic vibration absorber, taking into account the nonlinear properties of materials under actual operating conditions. To account for internal dissipation in the structure’s material, a nonlinear hereditary Boltzmann-Volterra viscoelasticity model is employed, along with elastic-plastic properties based on a bilinear diagram. This model is characterized by a hysteresis loop, which describes the relationship between the structure’s response and deformation, represented in the form of a parallelogram. Results. The forced vibrations of high-rise structures near the resonance mode were investigated taking into account the linear, nonlinear elastic, viscoelastic and elastic-plastic properties of the structure material with a dynamic vibration damper under various kinematic effects in the base. The reliability of the method was verified by a test example considering the reaction of an elastic-plastic frame as a system with one degree of freedom under a given load. The effect of vibration damping of a high-rise structure was revealed taking into account the nonlinear viscoelastic and elastic-plastic properties of the structure material together with a viscoelastic dynamic vibration damper. Conclusion. The influence of the material’s dissipative properties on the structure’s oscillations has been established. Recommendations for optimizing the structure’s performance, taking into account the dynamic oscillation damper, have been proposed. The effectiveness of damping oscillations in a high-rise structure has been demonstrated, considering the nonlinear viscoelastic and elastoplastic properties of the structure’s material in conjunction with a viscoelastic dynamic oscillation damper.
Izvestiya VUZ. Applied Nonlinear Dynamics. 2025;33(6):804-822
804-822
Modulation instability and soliton formation under interaction of an electromagnetic wave with a beam of unexcited non-isochronous electron–oscillators
Abstract
This paper develops the theory of modulation instability (MI) in the interaction of an electromagnetic wave with a counterpropagating beam of unexcited electron-oscillators under the cyclotron resonance conditions. The purpose of this study is to establish the pattern of possible wave propagation regimes in such a system. Methods. The theoretical analysis is based on the nonlinear Schrodinger equation, which enables to determine the conditions for occurrence of MI and obtain a simple analytical expression for the boundary between the absolute and convective MI on the wave frequency – wave amplitude parameter plane. The theoretical conclusions about possible regimes of wave propagation are verified by direct 3-D particle-in-cell (PIC) simulation of the electronwave interaction. The obtained results show that above the boundary of cyclotron absorption band non-stationary self-modulation regimes occur. These regimes are caused by absolute MI and can lead to the formation of solitonlike pulse trains. As the frequency of the input signal increases, self-modulation is replaced by a stationary single-frequency regime of wave propagation. This transition is due to the change of MI character from absolute to convective. The results of 3-D PIC simulation are consistent with the theoretical analysis of the averaged equations, and the same sequence of transitions between different dynamic regimes occurs as the input frequency increases. Conclusion. 3-D PIC simulation provided an opportunity to study a model that approximates the conditions of a potential experiment. The possibility of converting the 241.3-GHz signal into a close-to-periodic train of nanosecond pulses was demonstrated. Such an effect is useful for the generation of microwave frequency combs.
Izvestiya VUZ. Applied Nonlinear Dynamics. 2025;33(6):823-842
823-842
Modeling of global processes. Nonlinear dynamics and humanities
Spatiotemporal multistability scenarios for system of three competing species
Abstract
The aim of this work is to determine the conditions under which multistability is possible in system of three competing species described by reaction–diffusion–advection equations. Methods. Using the theory of cosymmetry and the concept of ideal free distribution, relations are established for the coefficients of local interaction, diffusion and directed migration, under which continuous families of solutions are possible. Compact scheme of the finite difference method is used to discretize the problem of species distribution on one-dimensional spatial area with periodicity conditions. Results. Conditions for parameters are found, under which stationary solutions proportional to the resource are obtained, corresponding to the ideal free distribution (IFD). The conditions under which two-parameter families of stationary distributions exist are studied. For parameters corresponding to IFD, family of periodic regimes is obtained in computational experiment. Conclusion. The obtained results demonstrate variants of multistability of species in resource-heterogeneous area and will further serve as a basis for the analysis of systems of interacting populations.
Izvestiya VUZ. Applied Nonlinear Dynamics. 2025;33(6):843-859
843-859
Innovations in applied physics
On planar models of resistive wall amplifier (theory and simulations)
Abstract
The purpose of this paper is to investigate a metamaterial-based planar resistive wall amplifier and to demonstrate the applicability of simplified models for preliminary evaluations. Methods. Two two-dimensional models are considered. The first model constitutes of an infinitely wide sheet beam immersed in a strong longitudinal magnetic field propagating between two identical layers of metamaterial, with the symmetry plane placed along the beam center; the layers of metamaterial and an envelope of parallel metal plates are separated by vacuum gaps. The second model is a periodic structure of thin sheet beams immersed in a strong longitudinal magnetic field propagating through drift channels in an infinite slab of metamaterial. In both cases, the frequency properties of the metamaterial are accounted by the Drude model. The dispersion equations for these models are derived. The transition to one-dimensional linear theory is demonstrated and discussed. The results of linear theory and numerical simulations in CST Particle Studio for each model are compared and analyzed. In numerical simulation, the initial beam density modulation is utilized. In the linear regime, the gain is evaluated by the ratio of the maximum amplitudes of the Fourier transform of the collector current to the emission current. Results and conclusion. The obtained theoretical results show the sensitivity of the metamaterial-based planar resistive wall amplifier performance to geometrical dimensions and properties of the medium. It is shown that by using metamaterial it is possible to obtain a significant increase of the initial beam modulation. A qualitative correspondence between the results of planar linear theory and numerical simulation for both models is shown. The hierarchy of models is formulated.
Izvestiya VUZ. Applied Nonlinear Dynamics. 2025;33(6):860-872
860-872
Nonlinear dynamics and neuroscience
Nonlinear properties of the brain theta rhythm
Abstract
The purpose of this study is to identify nonlinearity of local brain field potential signals during theta rhythm. Methods. The nonlinearity in the signal is manifested in the synchronous behavior of the fundamental frequency and its higher harmonics; to detect this, skeletons in different frequency ranges were used, the phase synchronization index was calculated and cross-spectral analysis was performed. Results. The fundamental frequency and the second harmonic in the spectrum during the theta rhythm were identified from multi-hour recordings in 14 animals. Based on the skeletons constructed from these recordings, frequency synchronization between the fundamental frequency and the second harmonic in the ratio of 2:1 was diagnosed at intervals of up to 10 s in both symmetrical channels. Phase synchronization was also diagnosed in a number of recordings, but at shorter intervals (about 2–4 s). Conclusion. The study proved that the spectral component observed at the theta-rhythm in healthy laboratory animals in the signals of local brain field potentials at the range containing twice fundamental frequency of the rhythm is indeed the second harmonic of the fundamental frequency. Thus, the theta rhythm is often a significantly nonlinear signal.
Izvestiya VUZ. Applied Nonlinear Dynamics. 2025;33(6):873-897
873-897
The impact of internal noise on the performance of convolutional neural network
Abstract
Purpose. This study aims to establish the characteristics of noise propagation and accumulation in convolutional neural networks. The article investigates how the accuracy of a trained convolutional network varies depending on the type and intensity of noise exposure. Methods. White Gaussian noise sources were used as the basis for noise exposure. Two types of noise exposure were applied to artificial neurons: additive and multiplicative. Additionally, the effects of correlated and uncorrelated noise on the layers of neurons were examined. Results. The findings indicate that additive noise (both correlated and uncorrelated) accumulates more significantly in networks with convolutional layers compared to those without. The relationship between network accuracy and the intensity of multiplicative correlated noise is similar for both types of networks. However, the impact of multiplicative uncorrelated noise is more favorable for networks with convolutional layers. The study also considered pooling layers, specifically MaxPooling and MeanPooling, which significantly enhance accuracy in the presence of additive noise within the convolutional layer. The decline in accuracy due to increasing intensity of multiplicative correlated noise is nearly identical for networks with and without pooling layers. Conversely, networks employing MaxPooling demonstrate reduced resilience to uncorrelated multiplicative noise. Conclusion. The study demonstrates that additive noise severely degrades network performance when a convolutional layer is present, though this negative effect can be mitigated by including a pooling layer immediately following the convolutional layer. In contrast, the effects of multiplicative noise are less clear-cut. In most cases, its impact remains consistent regardless of the presence of convolution and pooling layers. However, the use of MaxPooling in the pooling layer may compromise the network’s robustness against multiplicative uncorrelated noise.
Izvestiya VUZ. Applied Nonlinear Dynamics. 2025;33(6):898-916
898-916
Modified FitzHugh-Nagumo oscillator with spiking activity dependent on the duration of external impulse action
Abstract
The purpose of the study is to develop and investigate a modified FitzHugh-Nagumo oscillator, the spiking activity of which is determined not only by the amplitude, but also by the duration of the external impulse signal applied to the input of the oscillator. Methods. We have added an equation to the system of known equations describing the dynamics of the FitzHugh-Nagumo oscillator with a constant threshold parameter value. This additional equation describes the change in the threshold parameter over time under the influence of external impulse signals. For various values of the parameters of external impulses, a numerical study of the dynamics of the proposed oscillator, which is in a state of equilibrium in the absence of external influence, is carried out. Results. It is shown that, unlike the classical FitzHugh-Nagumo oscillator, the modified oscillator is capable of demonstrating a sequence of several spikes in response to a single external impulse action, and the oscillator dynamics depends on both the amplitude and the duration of external impulses. In addition, the proposed oscillator can be excited by a sequence of impulses with an amplitude below the threshold. Conclusion. The proposed modified FitzHughNagumo oscillator can be used to construct spiking neural networks. Learning of such networks can be implemented by changing synaptic connections by adjusting the synapse weights corresponding to the duration of external impulse signals. The proposed modification of the FitzHugh-Nagumo oscillator can be implemented quite simply in a radio physical experiment using analog electronic elements and digital circuits regulating the duration of input impulses.
Izvestiya VUZ. Applied Nonlinear Dynamics. 2025;33(6):917-928
917-928
Nonlinear waves. Solitons. Autowaves. Self-organization
Power series reversion and exact solutions of nonlinear mathematical physics equations
Abstract
Purpose. Develop a new method for finding exact solutions to equations of nonlinear mathematical physics. Methods. The partial sum of a perturbation series, written for the original nonlinear equation, is represented as a power series in powers of the exponential function, which is the solution of the linearized equation. The rational generating function of the sequence of coefficients of the power series represents the exact solution of the original equation. The method is based on the property that inverted power series for soliton-like solutions terminates at powers at least one greater than the order of the pole of the solution. Results. The effectiveness of the method is demonstrated in constructing exact localized solutions of the nonintegrable Korteweg–de Vries–Burgers equation, as well as nonlinear integrable differential-difference equations. Conclusion. The proposed method is applicable to solving integrable and non-integrable differential equations with constant coefficients, as well as integrable differential-difference equations.
Izvestiya VUZ. Applied Nonlinear Dynamics. 2025;33(6):929-942
929-942