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Block diagram of the experimental system of coupled electronic oscillators with delayed feedback for the 1∕III type of coupling. DL-1 and DL-2 are the delay lines, ND-1 and ND-2 are the nonlinear devices, ADC-1 and ADC-2 are the analog-to-digital converters, and DAC-1 and DAC-2 are the digital-to-analog converters of the first and the second oscillator, respectively. ADC is a two-channel analog-to-digital converter and PC is a computer.
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We propose a method for estimation of coupling between the systems governed by scalar time-delay differential equations of the Mackey-Glass type from the observed time series data. The method allows one to detect the presence of certain types of linear coupling between two time-delay systems, to define the type, strength, and direction of coupling,...
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... next example is the method application to experimen- tal time series gained from two coupled electronic oscillators with delayed feedback. A block diagram of the experimental setup is shown in Fig. 5. The delay of the signal V 1 t for time 1 and the delay of the signal V 2 t for time 2 are provided by the delay lines DL-1 and DL-2, respectively, constructed using digital elements or computer. The delay line DL-1 was constructed in the following way: the analog- to-digital converter ADC-1 and the digital-to-analog con- verter DAC-1 ...
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... record the signals V 1 t and V 2 t using a two-channel analog-to-digital converter ADC Fig. 5 with the sampling frequency f s = 10 kHz at 1 =23 ms, 2 = 31.7 ms, R 1 C 1 = 0.48 ms, R 2 C 2 = 1.01ms, k 1 = −0.1 and k 2 = 0.1 . To measure the resistance and capacitance of the filter elements we used the universal digital multimeter E7-8 Russia. It has the accuracy of 0.3% for the measurement of the capacitance and the accuracy of ...
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A method for the estimation of the Lyapunov exponent corresponding to enslaved phase dynamics from time series has been proposed. It is valid for both nonautonomous systems demonstrating periodic dynamics in the presence of noise and coupled chaotic oscillators and allows us to estimate precisely enough the value of this Lyapunov exponent in the su...
Citations
... One of them has a constant delay time and is in the turbulent chaos regime, while the other has a variable delay time and is in the laminar chaos regime. Time-delay systems may be coupled in various ways that differ both in the type of coupling (linear, diffusion, delayed) and in the position of the point within a ring time-delay system at which the signal from the other system is fed into the ring [18]. For example, the coupling signal may be fed into the system characterized by Eq. (1) between an inertial element (filter) and a delay line, between a delay line and a nonlinear element, or between a nonlinear element and a filter. ...
... For example, the coupling signal may be fed into the system characterized by Eq. (1) between an inertial element (filter) and a delay line, between a delay line and a nonlinear element, or between a nonlinear element and a filter. Each of these cases is characterized by its own equation [18]. Figure 1, a presents the block diagram of two unidirectionally coupled time-delay systems (1) and (3) for the case when the diffusion coupling signal is fed into the slave system between a nonlinear element and a filter. ...
The possibility of the existence of laminar chaos in coupled time-delayed feedback systems is investigated. The cases of unidirectional and mutual coupling of time-delay systems are considered. It is shown for the first time that laminar chaos can exist not only in a system with a variable delay time, but also in a system with a constant delay time, if it is coupled with a system in the regime of laminar chaos. Keywords: Time-delay systems, Laminar chaos, Coupled oscillators, Synchronization
... Рассмотрим теперь две системы с запаздыванием, из которых одна имеет постоянное время задержки и находится в режиме турбулентного хаоса, а другая имеет переменное время задержки и находится в режиме ламинарного хаоса, и свяжем эти системы между собой. Системы с запаздыванием могут быть связаны между собой различными способами, которые отличаются не только типом связи (линейная, диффузионная, запаздывающая), но и тем, в какую точку кольцевой системы с запаздыванием подается сигнал от другой системы [18]. Так, в систему, описываемую уравнением (1), сигнал связи можно подать между инерционным элементом (фильтром) и линией задержки, между линией задержки и нелинейным элементом, между нелинейным элементом и фильтром. ...
... Так, в систему, описываемую уравнением (1), сигнал связи можно подать между инерционным элементом (фильтром) и линией задержки, между линией задержки и нелинейным элементом, между нелинейным элементом и фильтром. Каждый из этих случаев будет описываться своим уравнением [18]. ...
The possibility of the existence of laminar chaos in coupled time-delayed feedback systems is investigated. The cases of unidirectional and mutual coupling of time-delay systems are considered. It is shown for the first time that laminar chaos can exist not only in a system with a variable delay time, but also in a system with a constant delay time, if it is coupled with a system in the regime of laminar chaos.
... The target function for optimization in such a case is usually constructed using the difference between the state vector components and their approximated values. Another approach was first proposed in [37] and with some changes used in [17,18]. It exploits the target function that is constructed taking into account the continuity of nonlinear function. ...
An approach to solve the inverse problem of the reconstruction of the network of time-delay oscillators from their time series is proposed and studied in the case of the nonstationary connectivity matrix. Adaptive couplings have not been considered yet for this particular reconstruction problem. The problem of coupling identification is reduced to linear optimization of a specially constructed target function. This function is introduced taking into account the continuity of the nonlinear functions of oscillators and does not exploit the mean squared difference between the model and observed time series. The proposed approach allows us to minimize the number of estimated parameters and gives asymptotically unbiased estimates for a large class of nonlinear functions. The approach efficiency is demonstrated for the network composed of time-delayed feedback oscillators with a random architecture of constant and adaptive couplings in the absence of a priori knowledge about the connectivity structure and its evolution. The proposed technique extends the application area of the considered class of methods.
... The general algorithms like proposed in [22] occurred to be inefficient [24], model dimension was too large, and amount of experimental data seemed to be not enough to estimate model parameters. Some progress was available when the idea to use a priori information [25] was converted into a number of special approaches to particular classes of systems [26], [27], [28], [29], [30], [31], [32]. Here, we mainly follow the approach for reconstruction of nonlinear oscillators from scalar experimental series proposed in [33] for van der Pol oscillators with many additional corrections and changes to fit the specifics of the considered system. ...
Phase-locked loops (PLLs) are now widely used in communication systems and have been a classic system for more than 60 years. Well-known mathematical models of such systems are constructed in a number of approximations, so questions about how they describe the experimental dynamics qualitatively and quantitatively, and how the accuracy of the model description depends on the behavior mode, remain open. One of the most direct approaches to the verification of any model is its reconstruction from the time series obtained in experiment. If it is possible to fit the model to experimental data and the resulting parameter values are close to the expected values (calculated from the first principles), the quantitative correspondence between the model and the physical object is nearly proved. In this paper, for the first time, the equations of the PLL model with a bandpass filter are reconstructed from the experimental signals of the generator in various modes. The reconstruction showed that the model known in the literature generally describes the experimental dynamics in regular and chaotic regimes. The relative error of parameter estimation is between 2% and 50% for different regimes and parameters. The reconstructed nonlinear function of phase is not harmonic and highly asymmetric in contrary to the model one.
... • an approach was developed for some general purpose (e. g., see [6]) or for some particular class of systems like nonautonomous ordinary differential equations (ODEs, [7]), time delayed systems [8], or stochastic phase oscillators [9]. ...
... However, both techniques occurred to be hardly to apply to experimental data. This was recognized, and specific techniques for some narrow classes of systems in special cases were developed [13], including nonautonomous systems [7,14] and time delayed systems [8]. Since the approaches based on general functional series for approximation of nonlinear functions are usually very inefficient, some efforts were made for removal of superfluous terms [15]. ...
... In this study the new, simple and numerically effective approach for reconstruction of networks of generalized Van der Pol oscillators [37] with unknown nonlinear functions is proposed. It is mostly based on the idea of nonparametric reconstruction of a nonlinear function for individual nodes, proposed for two coupled time-delay systems in [8], and generalized for larger ensembles in [38]. This idea aims to reduce the parametrization in comparison to the direct approaches such as [39], for which all nonlinear functions of individual nodes have to be parametrized in order to be reconstructed. ...
The method for reconstruction of nonlinear potential, dissipation and coupling functions for all elements of ensembles of coupled generalized Van der Pol oscillators (including Rayleigh and Bonhoeffer–Van der Pol oscillators) from multivariate time series is proposed. It is based on the idea of using the smoothness of reconstructed potential function as a criterion for quality of reconstruction. Therefore, no parametrization for potential is required and an arbitrary potential function unique for each oscillator in the ensemble can be considered. Different approaches to reconstruction of dissipation are studied. Linear and nonlinear coupling between oscillators is considered, including coupling by derivatives. Effects of measurement noise, contaminating the series, are taken into account.
... We couple the oscillators (1) via the mean field G(t), which acts on each element in the network and provides global coupling. The signal G(t) can be fed into the ring -feedback oscillator at various points [50] indicated by numerals 1, 2, and 3 in Fig. 1. Depending on the point at which the mean field acts on oscillators, their dynamics is described by one of the following equations: ...
We study the collective dynamics of oscillators in a network of identical bistable time-delayed feedback systems globally coupled via the mean field. The influence of delay and inertial properties of the mean field on the collective behavior of globally coupled oscillators is investigated. A variety of oscillation regimes in the network results from the presence of bistable states with substantially different frequencies in coupled oscillators. In the physical experiment and numerical simulation we demonstrate the existence of chimeralike states, in which some of the oscillators in the network exhibit synchronous oscillations, while all other oscillators remain asynchronous.
... где τ – время запаздывания, общее для обоих осцилляторов; λ x и λ y – параметры нелинейности; k – параметр связи. Сходная система уже рассматривалась в работе[12]. Параметр инерционности в нашем случае был взят равным 1, поэтому он отсутствует в уравнениях (1). ...
... В данной работе время запаздывания считается известным. К настоящему времени предложено уже довольно много подходов к его реконструкции для систем как в хаотическом[16], так и периодическом[14,17]режимах, а также для связанных систем[12,18]. Однако, даже если используемый метод даёт не совсем точное значение времени запаздывания, можно повторять процедуру реконструкции, используя полученную этими методам оценку запаздывания как пробное значение. ...
Time-delayed systems, including coupled ones, became popular models of different physical and biological objects. Often One or few variables of such models cannot be directly measured, these variables are called hidden variables. However, reconstruction of models from experimental signals in presence of hidden variables can be very suitable for model verification and indirect measurement. Current study considers reconstruction of parameters of both systems and hidden variable of the driving system from time series of the driven system in ensemble of two unidirectionally coupled first order time-delayed systems. Initial condition approach was used; this method considers initial conditions for hidden variables as additional unknown parameters. The method was adapted for time-delayed systems: vector of initial conditions was used instead of a single initial condition. Time series of the driven system, parameters of nonlinear function of both systems and the coupling coefficient were shown to be reconstructable from a time series of driven system in periodical regime, if starting guesses for the hidden variable were set using a priori informationabout the model. The space of starting guesses for parameters was studied; the probability of successful reconstruction was shown to be approximately 1/4 for starting guesses for parameters distant from their true values in 50 % of their absolute values. The fundamental possibility of reconstruction of system with one time delay in presence of hidden variables from scalar time series was shown. © 2017 Izvestiya Vysshikh Uchebnykh Zavedeniy. Prikladnaya Nelineynaya Dinamika.
... The reconstruction problem becomes even more difficult in the presence of interactions between the time-delay systems and requires development of alternative methods [26]. The architecture and strengths of connections between the network elements define their possible synchronous behavior [27,28]. ...
... The action of other elements of the ensemble on the time-delay system under consideration disturbs it and results in the disappearance of some extrema in the system time series and appearance of new ones. This effect is especially pronounced in the case of linear coupling of time-delay systems studied in Ref. [26]. In the case of diffusive coupling between the timedelay systems, the method based on the statistical analysis of extrema in the chaotic time series can also be applied to the delay time recovery. ...
We propose a method for reconstructing model differential equations with time delay for ensembles of coupled time-delay systems from their time series. The method has made it possible to recover the parameters of elements of the ensemble as well as the architecture and strength of couplings in ensembles of nonidentical systems with delay with an arbitrary number of unidirectional and bidirectional couplings between them. The effectiveness of the method is demonstrated for chaotic and periodic time series of model equations for ensembles of diffusively coupled systems with time delay in the presence of noise, as well as experimental time series for resistively coupled radiotechnical oscillators with delayed feedback.
... The reconstruction problem becomes even more difficult in the presence of interactions between the time-delay systems and requires development of alternative methods [26]. The architecture and strengths of connections between the network elements define their possible synchronous behavior [27,28]. ...
... The action of other elements of the ensemble on the time-delay system under consideration disturbs it and results in the disappearance of some extrema in the system time series and appearance of new ones. This effect is especially pronounced in the case of linear coupling of time-delay systems studied in Ref. [26]. In the case of diffusive coupling between the timedelay systems, the method based on the statistical analysis of extrema in the chaotic time series can also be applied to the delay time recovery. ...
We propose a method to recover from time series the parameters of coupled time-delay systems and the architecture of couplings between them. The method is based on a reconstruction of model delay-differential equations and estimation of statistical significance of couplings. It can be applied to networks composed of nonidentical nodes with an arbitrary number of unidirectional and bidirectional couplings. We test our method on chaotic and periodic time series produced by model equations of ensembles of diffusively coupled time-delay systems in the presence of noise, and apply it to experimental time series obtained from electronic oscillators with delayed feedback coupled by resistors.
... The effects of its variability on complexity and information exchange are shown inFigure 5. The figure shows net transfer entropy (Figure 5A), differences in sample entropy at fine (Figure 5B) and coarse time scales (Figure 5D) as functions of the time delay T. Note that, when we deal with real data, such relations cannot be observed as typically the true values of T are not known (see, however , Prokhorov and Ponomarenko, 2005; Silchenko et al., 2010; Vicente et al., 2011 for the attempts in recovering time delays in coupling). What we can observe is the correlations between the net transfer entropy and the differences in sample entropy shown in Figures 5C,E. ...
Variability in source dynamics across the sources in an activated network may be indicative of how the information is processed within a network. Information-theoretic tools allow one not only to characterize local brain dynamics but also to describe interactions between distributed brain activity. This study follows such a framework and explores the relations between signal variability and asymmetry in mutual interdependencies in a data-driven pipeline of non-linear analysis of neuromagnetic sources reconstructed from human magnetoencephalographic (MEG) data collected as a reaction to a face recognition task. Asymmetry in non-linear interdependencies in the network was analyzed using transfer entropy, which quantifies predictive information transfer between the sources. Variability of the source activity was estimated using multi-scale entropy, quantifying the rate of which information is generated. The empirical results are supported by an analysis of synthetic data based on the dynamics of coupled systems with time delay in coupling. We found that the amount of information transferred from one source to another was correlated with the difference in variability between the dynamics of these two sources, with the directionality of net information transfer depending on the time scale at which the sample entropy was computed. The results based on synthetic data suggest that both time delay and strength of coupling can contribute to the relations between variability of brain signals and information transfer between them. Our findings support the previous attempts to characterize functional organization of the activated brain, based on a combination of non-linear dynamics and temporal features of brain connectivity, such as time delay.
