Volume 31, Nº 5 (2023)
Articles
547-548
Self-oscillating systems with controlled phase of external force
Resumo
The purpose of this work is to study self-oscillatory systems under adaptive external action. This refers to the situation when the phase of the external action additionally depends on the dynamical variable of the oscillator. In a review plan, the results are presented for the case of a linear damped oscillator. Two cases of self-oscillatory systems are studied: the van der Pol oscillator and an autonomous quasi-periodic generator with three-dimensional phase space. Methods. Methods of charts of dynamical regimes and charts of Lyapunov exponents are used, as well as the construction of phase portraits and stroboscopic sections. Results. In a review plan, the results are presented for the case of a linear damped oscillator. Two cases of self-oscillatory systems are studied: the van der Pol oscillator and an autonomous quasi-periodic generator with a three-dimensional phase space. The pictures of characteristic dynamical regimes are described. Scenarios for the development of multidimensional chaos are described. Illustrations are given of the influence of the control parameter, which is responsible for the degree of dependence of the phase on the oscillator variable, on the dynamics of the system at different frequencies of action. Conclusion. The taling into account of the dependence of the phase on a dynamical variable leads to an extension of the tongues of subharmonic resonances, which are weakly expressed in the classical van der Pol oscillator. This is especially noticeable for even resonances of periods 2 and 4. For the generator of quasi-periodic oscillations in the non-autonomous case, three-frequency tori are observed, their regions begin to dominate with an increase in the adaptivity parameter, displacing the tongues of resonant two-frequency tori. A variety of multidimensional chaos characterized by an additional Lyapunov exponent close to zero is discovered, the possibility of developing hyperchaos as a result of destruction is shown.
Izvestiya VUZ. Applied Nonlinear Dynamics. 2023;31(5):549-565
549-565
Multistability near the boundary of noise-induced synchronization in ensembles of uncoupled chaotic systems
Resumo
The aim of this work is to study the possibility of the existence of multistability near the boundary of noise-induced synchronization in chaotic continuous and discrete systems. Ensembles of uncoupled Lorenz systems and logistic maps being under influence of a common source of white noise have been chosen as an object under study. Methods. The noise-induced synchronization regime detection has been performed by means of direct comparison of the system states being under influence of the common noise source and by calculation of the synchronization error. To determine the presence of multistability near the boundary of this regime, the multistability measure has been calculated and its dependence on the noise intensity has been obtained. In addition, for fixed moments of time, the basins of attraction of the synchronous and asynchronous regimes have been received for one of the systems driven by noise for fixed initial conditions of the other system. The result of the work is a proof of the presence of multistability near the boundary of noise-induced synchronization. Conclusion. It is shown that the regime of intermittent noise-induced synchronization, as well as the regime of intermittent generalized synchronization, is characterized by multistability, which manifests itself in this case as the existence in the same time interval of the synchronous behavior in one pair of systems being under influence of a common noise source, whereas in the other pair the asynchronous behavior is observed. The found effect is typical for both flow systems and discrete maps being under influence of a common noise source. It can find an application in the information and telecommunication systems for improvement the methods for secure information transmission based on the phenomenon of chaotic synchronization.
Izvestiya VUZ. Applied Nonlinear Dynamics. 2023;31(5):566-574
566-574
Analytical method of optical wave behavior studying in nonlinear medium with periodically arranged conducting nanofilms
Resumo
The purpose of this work is to build the analytical model of the behavior of a harmonic wave in a nonlinear optical medium with periodically arranged nanofilms. Methods. The modernized method is presented of non-smooth transformation of the argument to eliminate the Dirac functions on the right side of the non-linear inhomogeneous differential equation describing linear polarized wave behavior within a non-linear optical medium with periodically arranged conducting nanofilms. Small parameter methods, in particular, the averaging method, is also used to find an approximate analytical solution. Results. The fully analytical model of the behavior of a linear polarized harmonic wave within a nonlinear optical medium with periodically arranged conducting nanofilms is constructed. Conclusion. For the case of propagation of a linearly polarized harmonic wave in a nonlinear optical medium with periodically arranged conducting nanofilms, the mathematical model based on the non-smooth argument transformation method is constructed. The model is fully analytical, all expressions are obtained directly from Maxwell’s equations by identical transformations. The limits of its applicability are determined by the limits of application of the wave theory of light.
Izvestiya VUZ. Applied Nonlinear Dynamics. 2023;31(5):575-585
575-585
Mathematical model of the photoplethysmogram for testing methods of biological signals analysis
Resumo
The purpose of this study was to develop a mathematical model of the photoplethysmogram, which can be used to test methods that introduce the instantaneous phases of the modulating signals. The model must reproduce statistical and spectral characteristics of the real photoplethysmogram, and explicitly incorporate the instantaneous phases of the modulating signals, so they can be used as a reference during testing. Methods. Anacrotic and catacrotic phases of the photoplethysmogram pulse wave were modeled as a sum of two density distributions for the skew normal distribution. The modulating signals were introduced as harmonic functions taken from the experimental instantaneous phases of the VLF (0.015...0.04 Hz), LF (0.04...0.15 Hz) and HF (0.15...0.4 Hz) oscillations in the real photoplethysmogram. The spectral power in the VLF, LF, and HF frequency ranges was calculated to compare the model and experimental data. Results. The model qualitatively reproduces the shape of the experimental photoplethysmogram pulse wave and shows less than 1% error when simulating the spectral properties of the signal. Conclusion. The proposed mathematical model can be used to test the methods for introduction of the instantaneous phases of the modulating signals in photoplethysmogram time-series.
Izvestiya VUZ. Applied Nonlinear Dynamics. 2023;31(5):586-596
586-596
Study of character of modulation instability in cyclotron resonance interaction of an electromagnetic wave with a counterpropagating rectilinear electron beam
Resumo
In this paper, the interaction of a monochromatic electromagnetic wave with a counterpropagating electron beam moving in an axial magnetic field is considered. The purpose of this study is to investigate the conditions for occurrence of modulation instability (MI) in such a system and to determine at which parameters of the incident wave the MI is absolute or convective. Methods. Theoretical analysis of the MI character is carried out by studying the asymptotic form of unstable perturbations using the saddle-point analysis. The analytical results are verified by numerical simulations. Results. Theoretically, the boundary of change in the character of MI on the plane of input signal parameters (amplitude and detuning of the frequency from the cyclotron resonance) is determined. Numerical simulations confirm that as the signal frequency increases, the regime of self-modulation, which corresponds to the absolute MI, is replaced by the stationary single-frequency transmission corresponding to the convective MI. The numerical results coincide with the analytical ones for the system, which is matched at the end. The matching is implemented by smooth increasing of the guiding magnetic field in the region of electron beam injection. Conclusion. Determining the analytical conditions for the implementation of the absolute MI is of practical interest, since the emerging self-modulation can lead to the generation of trains of pulses with the spectrum in the form of frequency combs.
Izvestiya VUZ. Applied Nonlinear Dynamics. 2023;31(5):597-609
597-609
Mutual synchronization of oscillations in a system of coupled evolutionary games
Resumo
The purpose of this study is to investigate collective dynamics of coupled communities that evolve according to the population game «Battle of the Sexes». A separate community includes two interacting populations of players of opposite sex, where each player has one of two possible competing behavior strategies. It is necessary to determine the possibility of mutual synchronization of oscillations in the number of players adhering to a particular strategy, build a synchronization region, and also evaluate the dependence of the properties of oscillations on the coupling strength. Methods. In this paper, we study the system of evolutionary games «Battle of the Sexes» interacting through migration. To simulate the evolutionary game dynamics we make use of the stochastic Moran process, as well as the Monte Carlo method to sample game trajectories. Mutual synchronization is defined by the appropriately generalized criteria of frequency and phase locking. Results. It is shown that the system of coupled evolutionary games «Battle of the Sexes» demonstrates mutual synchronization of oscillations under sufficiently strong coupling. In particular, oscillation frequencies of two communities get adjusted to each other and begin to coincide at some interaction parameter, while the oscillations themselves become almost identical. A similar result was also observed for an ensemble of more than two communities. Conclusion. The dependence of the average frequencies of community oscillations on the coupling strength was determined, the adjustment of oscillations with an increase in the coupling strength was demonstrated, thereby showing the possibility of mutual synchronization in the model of coupled evolutionary games «Battle of the Sexes». The region of frequency synchronization was numerically found.
Izvestiya VUZ. Applied Nonlinear Dynamics. 2023;31(5):610-621
610-621
Self-organization dynamics of charge carrier concentration in semiconductors due to the charge injection
Resumo
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.
Izvestiya VUZ. Applied Nonlinear Dynamics. 2023;31(5):622-627
622-627
Mathematical model for epileptic seizures detection on an EEG recording
Resumo
Purpose of this study — analysis of the possibility of using convolutional neural networks as a model for detecting epileptic seizures on real EEG data. Methods. In this paper, wavelet analysis is used for time-frequency analysis. To localize epileptic discharges, the task of detecting them was reduced to the classification task and the ResNet18 architecture of neural network was used. Techniques were used to augment and balance the biomedical data dataset under consideration. Wavelet analysis is used for time-frequency analysis. To localize epileptic discharges, the problem of their detection was reduced to the classification task, and the ResNet18 neural network architecture was used. Techniques were used to augment and balance the considered biomedical dataset. Results. Convolutional neural network can be successfully used to detect epileptic seizures, a method of postprocessing the results of primary detection is proposed to improve the quality of the model. It is shown that the developed model demonstrates high accuracy in comparison with other methods based on classical machine learning algorithms. The value of the F1-score metric reaches 0.44, which is a high value for classification of the real biological data. Conclusion. The presented model based on a convolutional neural network for detecting epileptic seizures on an EEG recording can become the main one in medical decision support systems for epileptologist.
Izvestiya VUZ. Applied Nonlinear Dynamics. 2023;31(5):628-642
628-642
Marking stages of REM and non-REM sleep using recurrent analysis
Resumo
The purpose of this study — to develop a simple technique for labeling sleep stages according to EEG data obtained from half-somnography recordings. To test the work of the method, it will be applied to three groups of subjects: conditionally healthy, patients with Parkinson’s disease, patients with sleep apnea. Methods. In this work, to recognize sleep stages, we use the calculation of a recurrent indicator and its subsequent assessment. It is shown that the stages of REM (Rapid Eye Movement) and non-REM sleep demonstrate different values of the recurrent index. Results. Depending on the range in which the recurrent indicator falls, the stages of REM and non-REM sleep were determined for the subjects, according to their nightly polysomnographic records. For three groups of subjects, the average knowledge of the accuracy of the method was calculated, which for all groups exceeded 72.5%. Conclusion. It is shown that on the basis of recurrent analysis it is possible to create a simple and effective method for recognizing sleep stages. For patients with apnea, the average accuracy of the method is higher than for apparently healthy subjects, for whom, in turn, this value was higher than for patients with Parkinson’s disease. This can be explained by the fact that the variability in the group of statistical characteristics of sleep stages in patients with apnea is lower, and in patients with Parkinson’s disease is higher, compared with apparently healthy subjects.
Izvestiya VUZ. Applied Nonlinear Dynamics. 2023;31(5):643-649
643-649
Oscillatory characteristics in the brain activity of the newborns and their correlation with different gestational ages
Resumo
The purpose of this study is to detect the characteristic features of the oscillatory electrical activity of the brain in early postnatal development, depending on the gestational age of newborns. Methods. The study is based on automatic processing of clinical data from electroencephalography of newborns on the third day after birth. Behavioral characteristics assessed periods of sleep and wakefulness, without a precise division into stages of sleep and various states of wakefulness. The processing of multichannel electroencephalography signals was carried out on the basis of the method of modifying the continuous wavelet transform (CWT), which makes it possible to estimate the average characteristics of the number, duration and energy of oscillatory components (patterns) developing in different frequency ranges. Results. A paradoxical picture has been demonstrated describing the state of sleep and wakefulness in weakly preterm infants. For this group of children, the number and average energy of patterns detected in the frequency ranges from 1 to 20 Hz behave in a reflected way during sleep compared to children born at the usual time. At the same time, the average duration of oscillatory patterns remains unchanged. Conclusion. In the first days of a child’s life, it is possible to detect significant differences in the activity of the brain of newborns with slightly different gestational age during sleep/wake behavioral states. Quantitative estimates of the parameters of oscillatory CWT patterns are promising for use as the basis for systems for automatic processing of neonatal brain activity, additional to amplitude electroencephalography estimates. Such systems may be relevant for the search for early signs of anomalies in the development of the central nervous system.
Izvestiya VUZ. Applied Nonlinear Dynamics. 2023;31(5):650-660
650-660
Classification of brain activity using synolitic networks
Resumo
Because the brain is an extremely complex hypernet of interacting macroscopic subnetworks, full-scale analysis of brain activity is a daunting task. Nevertheless, this task can be greatly simplified by analysing the correspondence between various patterns of macroscopic brain activity, for example, through functional magnetic resonance imaging (fMRI) scans, and the performance of particular cognitive tasks or pathological states. The purpose of this work is to present and validate a methodology of representing fMRI data in the form of graphs that effectively convey valuable insights into the interconnectedness of brain region activity for subsequent classification purposes. Methods. This paper explores the application of synolitic networks in the analysis of brain activity. We propose a method for constructing a graph, the vertices of which reflect fMRI voxels’ values, and the edges and edge weights reflect the relationships between fMRI voxels. Results and Conclusion. Based on the classification of fMRI data by graph properties, the effectiveness of the method in conveying important information for classification in the construction of graphs was shown.
Izvestiya VUZ. Applied Nonlinear Dynamics. 2023;31(5):661-669
661-669
670-671