Pub Date : 2023-01-01DOI: 10.18500/0869-6632-003048
Aleksandr D. Dmitriev, A. Ryzhov, C. Sierra-Teran
The purpose of this paper is to analyse the statistical characteristics of a Direct Chaotic Differentially Coherent communication scheme based on chaotic radio pulses in a communication channel with additive white Gaussian noise, where the chaotic signal is given by different instantaneous distributions. Methods. To achieve this goal, numerical modelling of the noise immunity of Direct Chaotic Differentially Coherent communication is conducted and compared with the results of analytical research. Results. The regularities associated with the use of chaotic signals with various statistical distributions of instantaneous values were studied. The minimum values of energy per bit to white Gaussian noise power spectral density ratio were obtained, providing the required error probabilities. Conclusion. It is shown that the proposed system works efficiently at high values of processing gain, and as the processing gain increases, the dependence of noise immunity on the specific statistical distribution of the chaotic signal is levelled out.
{"title":"Introduction to the statistical theory of differential communication based on chaotic signals","authors":"Aleksandr D. Dmitriev, A. Ryzhov, C. Sierra-Teran","doi":"10.18500/0869-6632-003048","DOIUrl":"https://doi.org/10.18500/0869-6632-003048","url":null,"abstract":"The purpose of this paper is to analyse the statistical characteristics of a Direct Chaotic Differentially Coherent communication scheme based on chaotic radio pulses in a communication channel with additive white Gaussian noise, where the chaotic signal is given by different instantaneous distributions. Methods. To achieve this goal, numerical modelling of the noise immunity of Direct Chaotic Differentially Coherent communication is conducted and compared with the results of analytical research. Results. The regularities associated with the use of chaotic signals with various statistical distributions of instantaneous values were studied. The minimum values of energy per bit to white Gaussian noise power spectral density ratio were obtained, providing the required error probabilities. Conclusion. It is shown that the proposed system works efficiently at high values of processing gain, and as the processing gain increases, the dependence of noise immunity on the specific statistical distribution of the chaotic signal is levelled out.","PeriodicalId":41611,"journal":{"name":"Izvestiya Vysshikh Uchebnykh Zavedeniy-Prikladnaya Nelineynaya Dinamika","volume":"1 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87596781","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-01DOI: 10.18500/0869-6632-003051
V. Aristov, O. Kubryak, I. Stepanyan
The purpose of the study is experimental verification of the proposed EEG analysis method based on the construction of a connectivity graph of the analyzed signal, in which the amplitudes are displayed by vertices, and their relative position relative to each other by arcs. The display of the EEG signal in the graph structure causes the appearance of cyclic structures with the possibility of calculating their numerical characteristics. As a result of the study, criteria for initialization of the initial conditions of the counting algorithm have been developed. The following parameters were calculated: the number of cycles and the Euler number in the EEG recording. Coil representations of graphs are given. The proposed algorithm has a scaling parameter, the choice of which affects the final results. The second free parameter of the proposed algorithm is the degree of artificial signal coarsening. Variants of the algorithm application for multichannel EEG signals with multichannel signal processing by channel-by-channel detection of semantic units and construction of a generalized semantic connectivity graph are considered. An example of an analyzed multichannel EEG signal, which was pre-processed with reduction of all amplitudes to natural numbers in accordance with the calculated characteristics, is given. An example of an EEG of a subject with closed eyes during quiet wakefulness and an EEG of a subject with open eyes is given. In Conclusion, it is shown that the final indicators can vary significantly (from zero to tens of thousands or more) depending on the particular derivation of the EEG channel. Analysis of the cyclic structures of the electroencephalogram seems to be a potential way to assess various human states due to the possibility of distinguishing them using the proposed method. The study has a limited, pilot character.
{"title":"Calculation of the cyclic characteristics of the electroencephalogram for investigation of the electrical activity of the brain","authors":"V. Aristov, O. Kubryak, I. Stepanyan","doi":"10.18500/0869-6632-003051","DOIUrl":"https://doi.org/10.18500/0869-6632-003051","url":null,"abstract":"The purpose of the study is experimental verification of the proposed EEG analysis method based on the construction of a connectivity graph of the analyzed signal, in which the amplitudes are displayed by vertices, and their relative position relative to each other by arcs. The display of the EEG signal in the graph structure causes the appearance of cyclic structures with the possibility of calculating their numerical characteristics. As a result of the study, criteria for initialization of the initial conditions of the counting algorithm have been developed. The following parameters were calculated: the number of cycles and the Euler number in the EEG recording. Coil representations of graphs are given. The proposed algorithm has a scaling parameter, the choice of which affects the final results. The second free parameter of the proposed algorithm is the degree of artificial signal coarsening. Variants of the algorithm application for multichannel EEG signals with multichannel signal processing by channel-by-channel detection of semantic units and construction of a generalized semantic connectivity graph are considered. An example of an analyzed multichannel EEG signal, which was pre-processed with reduction of all amplitudes to natural numbers in accordance with the calculated characteristics, is given. An example of an EEG of a subject with closed eyes during quiet wakefulness and an EEG of a subject with open eyes is given. In Conclusion, it is shown that the final indicators can vary significantly (from zero to tens of thousands or more) depending on the particular derivation of the EEG channel. Analysis of the cyclic structures of the electroencephalogram seems to be a potential way to assess various human states due to the possibility of distinguishing them using the proposed method. The study has a limited, pilot character.","PeriodicalId":41611,"journal":{"name":"Izvestiya Vysshikh Uchebnykh Zavedeniy-Prikladnaya Nelineynaya Dinamika","volume":"143 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76232128","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-01DOI: 10.18500/0869-6632-003040
L. Mogilevich, E. Popova
Purpose of this paper is to study the evolution of longitudinal strain waves in the walls of an annular channel filled with a viscous incompressible fluid. The walls of the channel were represented as coaxial shells with fractional physical nonlinearity. The viscosity of the fluid and its influence on the wave process was taken into account within the study. Metods. The system of two evolutionary equations, which are generalized Schamel equations, was obtained by the two-scale asymptotic expansion method. The fractional nonlinearity of the channel wall material leads to the necessity to use a computational experiment to study the wave dynamics in them. The computational experiment was conducted based on obtaining new difference schemes for the governing equations. These schemes are analogous to the Crank–Nicholson scheme for modeling heat propagation. Results. Numerical simulation showed that over time, the velocity and amplitude of the deformation waves remain unchanged, and the wave propagation direction concurs with the positive direction of the longitudinal axis. The latter specifies that the velocity of the waves is supersonic. For a particular case, the coincidence of the computational experiment with the exact solution is shown. This substantiates the adequacy of the proposed difference scheme for the generalized Schamel equations. In addition, it was shown that solitary deformation waves in the channel walls are solitons.
{"title":"Longitudinal waves in the walls of an annular channel filled with liquid and made of a material with fractional nonlinearity","authors":"L. Mogilevich, E. Popova","doi":"10.18500/0869-6632-003040","DOIUrl":"https://doi.org/10.18500/0869-6632-003040","url":null,"abstract":"Purpose of this paper is to study the evolution of longitudinal strain waves in the walls of an annular channel filled with a viscous incompressible fluid. The walls of the channel were represented as coaxial shells with fractional physical nonlinearity. The viscosity of the fluid and its influence on the wave process was taken into account within the study. Metods. The system of two evolutionary equations, which are generalized Schamel equations, was obtained by the two-scale asymptotic expansion method. The fractional nonlinearity of the channel wall material leads to the necessity to use a computational experiment to study the wave dynamics in them. The computational experiment was conducted based on obtaining new difference schemes for the governing equations. These schemes are analogous to the Crank–Nicholson scheme for modeling heat propagation. Results. Numerical simulation showed that over time, the velocity and amplitude of the deformation waves remain unchanged, and the wave propagation direction concurs with the positive direction of the longitudinal axis. The latter specifies that the velocity of the waves is supersonic. For a particular case, the coincidence of the computational experiment with the exact solution is shown. This substantiates the adequacy of the proposed difference scheme for the generalized Schamel equations. In addition, it was shown that solitary deformation waves in the channel walls are solitons.","PeriodicalId":41611,"journal":{"name":"Izvestiya Vysshikh Uchebnykh Zavedeniy-Prikladnaya Nelineynaya Dinamika","volume":"13 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82380463","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-01DOI: 10.18500/0869-6632-003052
V. Moskvitin, N. Semenova
The purpose of this study is to establish the features of noise propagation and accumulation in a recurrent neural network using a simplified echo network as an example. In this work, we studied the influence of activation function of artificial neurons and the connection matrices between them. Methods. We have considered white Gaussian noise sources. We used additive, multiplicative and mixed noise depending on how the noise is introduced into artificial neurons. The noise impact was estimated using the dispersion (variance) of the output signal. Results. It is shown that the activation function plays a significant role in noise accumulation. Two nonlinear activation functions have been considered: the hyperbolic tangent and the sigmoid function with range form 0 to 1. It is shown that some types of noise are suppressed in the case of the second function. As a result of considering the influence of coupling matrices, it was found that diagonal coupling matrices with a large blurring coefficient lead to less noise accumulation in the echo network reservoir with an increase in the reservoir memory influence. Conclusion. It is shown that activation functions of the form of sigmoid with range from 0 to 1 are suitable for suppressing multiplicative and mixed noise. The accumulation of noise in the reservoir was considered for three types of coupling matrices inside the reservoir: a uniform matrix, a band matrix with a small blurring coefficient, and a band matrix with a large blurring coefficient. It has been found that the band matrix echo networks with a high blurring coefficient accumulates the least noise. This holds for both additive and multiplicative noise.
{"title":"Noise influence on recurrent neural network with nonlinear neurons","authors":"V. Moskvitin, N. Semenova","doi":"10.18500/0869-6632-003052","DOIUrl":"https://doi.org/10.18500/0869-6632-003052","url":null,"abstract":"The purpose of this study is to establish the features of noise propagation and accumulation in a recurrent neural network using a simplified echo network as an example. In this work, we studied the influence of activation function of artificial neurons and the connection matrices between them. Methods. We have considered white Gaussian noise sources. We used additive, multiplicative and mixed noise depending on how the noise is introduced into artificial neurons. The noise impact was estimated using the dispersion (variance) of the output signal. Results. It is shown that the activation function plays a significant role in noise accumulation. Two nonlinear activation functions have been considered: the hyperbolic tangent and the sigmoid function with range form 0 to 1. It is shown that some types of noise are suppressed in the case of the second function. As a result of considering the influence of coupling matrices, it was found that diagonal coupling matrices with a large blurring coefficient lead to less noise accumulation in the echo network reservoir with an increase in the reservoir memory influence. Conclusion. It is shown that activation functions of the form of sigmoid with range from 0 to 1 are suitable for suppressing multiplicative and mixed noise. The accumulation of noise in the reservoir was considered for three types of coupling matrices inside the reservoir: a uniform matrix, a band matrix with a small blurring coefficient, and a band matrix with a large blurring coefficient. It has been found that the band matrix echo networks with a high blurring coefficient accumulates the least noise. This holds for both additive and multiplicative noise.","PeriodicalId":41611,"journal":{"name":"Izvestiya Vysshikh Uchebnykh Zavedeniy-Prikladnaya Nelineynaya Dinamika","volume":"11 3 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83624917","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-01DOI: 10.18500/0869-6632-003026
T. Mokhseni, M. Petrosyan
Methods of differentially coherent information transmission using noise signals are of interest because of the impossibility of implementing the known methods of correlation reception for such signals. With a potentially higher noise immunity compared to the methods of information transmission based on chaotic synchronization, however, they have a feature that does not allow transceivers to be implemented in practice. The transmitter and receiver of the scheme, based on already known methods of differentially coherent transmission, require a time delay comparable to the duration of the transmitted bits. With an analog implementation of the scheme this leads to a physical length of the delay line of tens of meters or more. Previously, the authors proposed and studied a differentially coherent transmission scheme in which there are no long delays. In this scheme, the duration of delays in the transmitter and receiver is determined not by the duration of the bit, but by the decay time of the autocorrelation function of the chaotic signal. Purpose of this work is to experimentally demonstrate the possibility of physical implementation of a direct-chaotic differentially coherent information transmission scheme in a wired communication channel. Methods. For this, a layout of the communication scheme, transmitting a binary data stream in the frequency range from 200 to 500 MHz, was designed and assembled. The layout is an ultrawideband differentially coherent transmitter and receiver connected via a wired channel. Results of the experiment are in full agreement with the previously obtained results of the analytical evaluations, as well as with the data of computer simulation. Conclusion. In the course of the research, a transceiver layout of a differentially coherent ultra-wideband direct chaotic communication scheme was developed, designed and manufactured. For the first time, experiments on the transmission of digital information were carried out on it, and thereby the practical feasibility and operability of the proposed direct chaotic differentially coherent transmission scheme were proved.
{"title":"Experiments on direct chaotic differentially coherent data transmission in a wired communication channel","authors":"T. Mokhseni, M. Petrosyan","doi":"10.18500/0869-6632-003026","DOIUrl":"https://doi.org/10.18500/0869-6632-003026","url":null,"abstract":"Methods of differentially coherent information transmission using noise signals are of interest because of the impossibility of implementing the known methods of correlation reception for such signals. With a potentially higher noise immunity compared to the methods of information transmission based on chaotic synchronization, however, they have a feature that does not allow transceivers to be implemented in practice. The transmitter and receiver of the scheme, based on already known methods of differentially coherent transmission, require a time delay comparable to the duration of the transmitted bits. With an analog implementation of the scheme this leads to a physical length of the delay line of tens of meters or more. Previously, the authors proposed and studied a differentially coherent transmission scheme in which there are no long delays. In this scheme, the duration of delays in the transmitter and receiver is determined not by the duration of the bit, but by the decay time of the autocorrelation function of the chaotic signal. Purpose of this work is to experimentally demonstrate the possibility of physical implementation of a direct-chaotic differentially coherent information transmission scheme in a wired communication channel. Methods. For this, a layout of the communication scheme, transmitting a binary data stream in the frequency range from 200 to 500 MHz, was designed and assembled. The layout is an ultrawideband differentially coherent transmitter and receiver connected via a wired channel. Results of the experiment are in full agreement with the previously obtained results of the analytical evaluations, as well as with the data of computer simulation. Conclusion. In the course of the research, a transceiver layout of a differentially coherent ultra-wideband direct chaotic communication scheme was developed, designed and manufactured. For the first time, experiments on the transmission of digital information were carried out on it, and thereby the practical feasibility and operability of the proposed direct chaotic differentially coherent transmission scheme were proved.","PeriodicalId":41611,"journal":{"name":"Izvestiya Vysshikh Uchebnykh Zavedeniy-Prikladnaya Nelineynaya Dinamika","volume":"20 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82539141","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-01DOI: 10.18500/0869-6632-003036
Ksenia O. Merkulova, D. Postnov
The purpose of this study is to analyze the influence of the shape of the daily illumination profile on the synchronization of rhythms in the sleep–wake state switching model. Normally, the alternation of sleep and wakefulness of a person is synchronized with his circadian rhythm and with the 24-hour rhythm of illumination. There is, however, a lot of experimental evidence of a violation of this synchronism, both in the form of phase failures (for example, during air travel) and in the form of long-term mismatch of rhythms (for example, during shift work in production). Mathematical models of the process of switching between sleep and wakefulness also demonstrate the desynchronization of rhythms and are successfully used to optimize work schedules. At the same time, the influence of a number of factors on this process has not been sufficiently studied, including the nature of changes in illumination during the day. Methods. An analysis of the six-dimensional model under study shows that, in terms of nonlinear dynamics, the problem is reduced to finding and interpreting resonance regions on a three-dimensional torus. For the specific purposes of our work, it turned out to be convenient to estimate the ratio of three periods (24 hours, the circadian period, and the current duration of the sleep–wake cycle) by numerically integrating the model equations on a grid of parameter values using parallel computing technology. The main result of our work is that the presence of round-the-clock low-intensity illumination (that is, the addition of a zero-frequency signal to the daily light cycle) causes the circadian rhythm to desynchronize with respect to the daily one in a significant range of parameters. We have proposed an explanation of this effect based on the structure of the mathematical model. Conclusion. Our results raise at least two serious questions, the first of which is related to the physiological interpretation of one of the main variables of the model, sleep homeostasis, and the second is to refine the assumptions that are used in the model description of the photoreceptor response. In any case, there are interesting prospects for further research.
{"title":"Ambient light at night causes desynchronization of rhythms in the sleep–wake switching model","authors":"Ksenia O. Merkulova, D. Postnov","doi":"10.18500/0869-6632-003036","DOIUrl":"https://doi.org/10.18500/0869-6632-003036","url":null,"abstract":"The purpose of this study is to analyze the influence of the shape of the daily illumination profile on the synchronization of rhythms in the sleep–wake state switching model. Normally, the alternation of sleep and wakefulness of a person is synchronized with his circadian rhythm and with the 24-hour rhythm of illumination. There is, however, a lot of experimental evidence of a violation of this synchronism, both in the form of phase failures (for example, during air travel) and in the form of long-term mismatch of rhythms (for example, during shift work in production). Mathematical models of the process of switching between sleep and wakefulness also demonstrate the desynchronization of rhythms and are successfully used to optimize work schedules. At the same time, the influence of a number of factors on this process has not been sufficiently studied, including the nature of changes in illumination during the day. Methods. An analysis of the six-dimensional model under study shows that, in terms of nonlinear dynamics, the problem is reduced to finding and interpreting resonance regions on a three-dimensional torus. For the specific purposes of our work, it turned out to be convenient to estimate the ratio of three periods (24 hours, the circadian period, and the current duration of the sleep–wake cycle) by numerically integrating the model equations on a grid of parameter values using parallel computing technology. The main result of our work is that the presence of round-the-clock low-intensity illumination (that is, the addition of a zero-frequency signal to the daily light cycle) causes the circadian rhythm to desynchronize with respect to the daily one in a significant range of parameters. We have proposed an explanation of this effect based on the structure of the mathematical model. Conclusion. Our results raise at least two serious questions, the first of which is related to the physiological interpretation of one of the main variables of the model, sleep homeostasis, and the second is to refine the assumptions that are used in the model description of the photoreceptor response. In any case, there are interesting prospects for further research.","PeriodicalId":41611,"journal":{"name":"Izvestiya Vysshikh Uchebnykh Zavedeniy-Prikladnaya Nelineynaya Dinamika","volume":"67 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73603554","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-01DOI: 10.18500/0869-6632-003031
A. Pisarchik, V. Khorev, Artem Badarin, V. Antipov, A. Budarina, A. Hramov
Aim of this study is to compare different experimental paradigms and to determine parameters suitable for conducting a neurophysiological experiment with visual stimuli to assess foreign language proficiency and providing further time series analysis of electrical brain activity to reveal specific biomarkers. Methods. This paper explores the possibilities and limitations of various experimental studies using the metaanalysis paradigm. Statistical approaches are used to determine significance of the results. Results. We review the current state of research in the field of experimental works related to visual stimulus presentation and verbal performance acquisition. Generalizations and analytical estimates of the experimental parameters used in the studies are carried out to provide recommendations for future experimental research. Conclusion. In this area of applied research, we have developed experimental design and algorithms for working with multiple data sources. In addition, experimental encephalographic studies have been carried out, that allowed the optimal temporal structure selection.
{"title":"Methodology of the neurophysiological experiments with visual stimuli to assess foreign language proficiency","authors":"A. Pisarchik, V. Khorev, Artem Badarin, V. Antipov, A. Budarina, A. Hramov","doi":"10.18500/0869-6632-003031","DOIUrl":"https://doi.org/10.18500/0869-6632-003031","url":null,"abstract":"Aim of this study is to compare different experimental paradigms and to determine parameters suitable for conducting a neurophysiological experiment with visual stimuli to assess foreign language proficiency and providing further time series analysis of electrical brain activity to reveal specific biomarkers. Methods. This paper explores the possibilities and limitations of various experimental studies using the metaanalysis paradigm. Statistical approaches are used to determine significance of the results. Results. We review the current state of research in the field of experimental works related to visual stimulus presentation and verbal performance acquisition. Generalizations and analytical estimates of the experimental parameters used in the studies are carried out to provide recommendations for future experimental research. Conclusion. In this area of applied research, we have developed experimental design and algorithms for working with multiple data sources. In addition, experimental encephalographic studies have been carried out, that allowed the optimal temporal structure selection.","PeriodicalId":41611,"journal":{"name":"Izvestiya Vysshikh Uchebnykh Zavedeniy-Prikladnaya Nelineynaya Dinamika","volume":"1 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79026964","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-01DOI: 10.18500/0869-6632-003053
S. Revina, A. Ryabov
The purpose of this work is to find the region of necessary and sufficient conditions for diffusion instability on the parameter plane (τ, d) of the Gierer–Meinhardt system, where τ is the relaxation parameter, d is the dimensionless diffusion coefficient; to derive analytically the dependence of the critical wave number on the characteristic size of the spatial region; to obtain explicit representations of secondary spatially distributed structures, formed as a result of bifurcation of a spatially homogeneous equilibrium position, in the form of series in degrees of supercriticality. Methods. To find the region of Turing instability, methods of linear stability analysis are applied. To find secondary solutions (Turing structures), the Lyapunov– Schmidt method is used in the form developed by V. I. Yudovich. Results. Expressions for the critical diffusion coefficient in terms of the eigenvalues of the Laplace operator for an arbitrary bounded region are obtained. The dependence of the critical diffusion coefficient on the characteristic size of the region is found explicitly in two cases: when the region is an interval and a rectangle. Explicit expressions for the first terms of the expansions of the secondary stationary solutions with respect to the supercriticality parameter are constructed in the one-dimensional case, as well as for a rectangle, when one of the wave numbers is equal to zero. In these cases, sufficient conditions for a soft loss of stability are found, and examples of secondary solutions are given. Conclusion. A general approach is proposed for finding the region of Turing instability and constructing secondary spatially distributed structures. This approach can be applied to a wide class of mathematical models described by a system of two reaction–diffusion equations.
本文的目的是在Gierer-Meinhardt系统的参数平面(τ， d)上找到扩散不稳定的充分必要条件区域，其中τ为松弛参数，d为无量纲扩散系数;解析导出了临界波数与空间区域特征尺寸的关系;获得二次空间分布结构的显式表示，这些结构是由空间均匀平衡位置的分岔形成的，以超临界度级数的形式表示。方法。为了找到图灵不稳定的区域，应用了线性稳定性分析的方法。为了找到次级解(图灵结构)，Lyapunov - Schmidt方法以V. I. Yudovich开发的形式使用。结果。得到了任意有界区域的临界扩散系数的拉普拉斯算子特征值表达式。在两种情况下，临界扩散系数对区域特征尺寸的依赖得到了明确的证明:当区域为区间和矩形时。在一维情况下，以及当其中一个波数等于零时，构造了二次平稳解关于超临界参数展开的第一项的显式表达式。在这些情况下，找到了软稳定性损失的充分条件，并给出了二次解的例子。结论。提出了寻找图灵不稳定区域和构造二次空间分布结构的一般方法。这种方法可以应用于由两个反应扩散方程组成的系统所描述的一类广泛的数学模型。
{"title":"Turing instability in the one-parameter Gierer–Meinhardt system","authors":"S. Revina, A. Ryabov","doi":"10.18500/0869-6632-003053","DOIUrl":"https://doi.org/10.18500/0869-6632-003053","url":null,"abstract":"The purpose of this work is to find the region of necessary and sufficient conditions for diffusion instability on the parameter plane (τ, d) of the Gierer–Meinhardt system, where τ is the relaxation parameter, d is the dimensionless diffusion coefficient; to derive analytically the dependence of the critical wave number on the characteristic size of the spatial region; to obtain explicit representations of secondary spatially distributed structures, formed as a result of bifurcation of a spatially homogeneous equilibrium position, in the form of series in degrees of supercriticality. Methods. To find the region of Turing instability, methods of linear stability analysis are applied. To find secondary solutions (Turing structures), the Lyapunov– Schmidt method is used in the form developed by V. I. Yudovich. Results. Expressions for the critical diffusion coefficient in terms of the eigenvalues of the Laplace operator for an arbitrary bounded region are obtained. The dependence of the critical diffusion coefficient on the characteristic size of the region is found explicitly in two cases: when the region is an interval and a rectangle. Explicit expressions for the first terms of the expansions of the secondary stationary solutions with respect to the supercriticality parameter are constructed in the one-dimensional case, as well as for a rectangle, when one of the wave numbers is equal to zero. In these cases, sufficient conditions for a soft loss of stability are found, and examples of secondary solutions are given. Conclusion. A general approach is proposed for finding the region of Turing instability and constructing secondary spatially distributed structures. This approach can be applied to a wide class of mathematical models described by a system of two reaction–diffusion equations.","PeriodicalId":41611,"journal":{"name":"Izvestiya Vysshikh Uchebnykh Zavedeniy-Prikladnaya Nelineynaya Dinamika","volume":"111 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79207726","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-01DOI: 10.18500/0869-6632-003023
A. Polunovskii
Purpose. New algorithms were consider for functional equations solving using the Feigenbaum equation as an example. This equation is of great interest in the theory of deterministic chaos and is a good illustrative example in the class of functional equations with superposition. Methods. The article proposes three new effective methods for solving functional equations — the method of successive approximations, the method of successive approximations using the fast Fourier transform and the numerical-analytical method using a small parameter. Results. Three new methods for solving functional equations were presented, considered on the example of the Feigenbaum equation. For each of them, the features of their application were investigated, as well as the complexity of the resulting algorithms was estimated. The methods previously used by researchers to solve functional equations are compared with those described in this article. In the description of the latter, the numerical-analytical method, several coefficients of expansions of the universal Feigenbaum constants were written out. Conclusion. The obtained algorithms, based on simple iteration methods, allow solving functional equations with superposition without the need to reverse the Jacobi matrix. This feature greatly simplifies the use of computer memory and gives a gain in the operating time of the algorithms in question, compared with previously used ones. Also, the latter, numerically-analytical method made it possible to obtain sequentially the coefficients of expansions of the universal Feigenbaum constants, which in fact can be an analytical representation of these constants.
{"title":"Effective algorithms for solving functional equations with superposition on the example of the Feigenbaum equation","authors":"A. Polunovskii","doi":"10.18500/0869-6632-003023","DOIUrl":"https://doi.org/10.18500/0869-6632-003023","url":null,"abstract":"Purpose. New algorithms were consider for functional equations solving using the Feigenbaum equation as an example. This equation is of great interest in the theory of deterministic chaos and is a good illustrative example in the class of functional equations with superposition. Methods. The article proposes three new effective methods for solving functional equations — the method of successive approximations, the method of successive approximations using the fast Fourier transform and the numerical-analytical method using a small parameter. Results. Three new methods for solving functional equations were presented, considered on the example of the Feigenbaum equation. For each of them, the features of their application were investigated, as well as the complexity of the resulting algorithms was estimated. The methods previously used by researchers to solve functional equations are compared with those described in this article. In the description of the latter, the numerical-analytical method, several coefficients of expansions of the universal Feigenbaum constants were written out. Conclusion. The obtained algorithms, based on simple iteration methods, allow solving functional equations with superposition without the need to reverse the Jacobi matrix. This feature greatly simplifies the use of computer memory and gives a gain in the operating time of the algorithms in question, compared with previously used ones. Also, the latter, numerically-analytical method made it possible to obtain sequentially the coefficients of expansions of the universal Feigenbaum constants, which in fact can be an analytical representation of these constants.","PeriodicalId":41611,"journal":{"name":"Izvestiya Vysshikh Uchebnykh Zavedeniy-Prikladnaya Nelineynaya Dinamika","volume":"3 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75062834","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-01DOI: 10.18500/0869-6632-003056
Olga Vershinina, Mihail Ivanchenko
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.
{"title":"Mutual synchronization of oscillations in a system of coupled evolutionary games","authors":"Olga Vershinina, Mihail Ivanchenko","doi":"10.18500/0869-6632-003056","DOIUrl":"https://doi.org/10.18500/0869-6632-003056","url":null,"abstract":"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.","PeriodicalId":41611,"journal":{"name":"Izvestiya Vysshikh Uchebnykh Zavedeniy-Prikladnaya Nelineynaya Dinamika","volume":"53 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85394789","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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