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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.

Background and Objectives: In this paper, we consider the general case of the establishment/destruction of a completely synchronous state of phase oscillator networks with topologies of links such as “ring” and “small world”. The natural frequencies of the node oscillators are supposed to be distributed along the network according to an arbitrary law. Materials and Methods: The network of Kuramoto oscillators, consisting of 1000 node elements, was considered as the system under study theoretically and numerically. Results: The influence of quantitative characteristics of the dependence of natural frequencies on the number (spatial coordinate) of the oscillator on the onset of a completely synchronous state of the network of phase oscillators as well as the mechanism of the transition to a completely synchronous regime has been studied. An analytical expression forthe critical value ofthe coupling parameter correspondingtothe establishment of a fully synchronous regime within the network under consideration has been deduced. The theoretical results obtained have been compared with the results of the direct numerical simulation of the oscillator network behavior, with the excellent agreement being observed. Conclusion: It has been found that the dependence of the natural frequencies of oscillators on the spatial coordinate (or, on the number of the oscillator in the network) in the case of networks with the topology of links such as “ring” and “small world” determines completely the properties of such networks from the point of view of establishing the phase synchronization. Having based on the “ring” and “small world” network properties, it is possible to solve not only the problem of finding the critical value of the coupling parameter for the known frequency dependences on the coordinate, but also the problem of synthesizing such networks with the predetermined properties.

Purpose of this study is to investigate the problem of how typical (or, conversely, unique) is the phenomenon of explosive synchronization in networks of nonlinear oscillators with topologies of links such as “ring” and “small world”, and, in turn, how the partial frequencies of the interacting oscillators must correlate with each other for the phenomenon of explosive synchronization in these networks can be possible. Methods. In this paper, we use an analytical description of the synchronous behavior of networks of nonlinear elements with “ring” and “small world” link topologies. To confirm the obtained results the numerical simulation is used. Results. It is shown that in networks of nonlinear oscillators with topologies of links such as “ring” and “small world”, the phenomenon of explosive synchronization can be observed for the different distributions of partial frequencies of network oscillators. Conclusion. The paper considers an analytical description of the behavior of network oscillators with “ring” and “small world” topologies of links and shows that the phenomenon of explosive synchronization in such networks is atypical, but not unique.

Aim of this work is to study the possibility of existence of multistability near the boundary of generalized synchronization in systems with complex attractor topology. Unidirectionally coupled Lorentz systems have been chosen as an object of study, and a modified auxiliary system method has been used to detect the presence of the synchronous regime. Result of the work is a proof of the presence of multistability near the boundary of generalized synchronization in unidirectionally coupled systems with a complex topology of attractor. For this purpose, the basins of attraction of the synchronous and asynchronous states of interacting Lorenz systems have been obtained for the value of the coupling parameter corresponding to the realization of the intermittent generalized synchronization regime in the system under study, and the dependence of the multistability measure on the value of the coupling parameter has also been calculated. It is shown that in the regime of intermittent generalized synchronization the measure of multistability turns out to be positive, which is an additional confirmation of the presence of multistability in this case.

Congratulations to Aleksei Aleksandrovich Koronovskii on his anniversary.

Background and Objectives: The goal of the paper is to study the influence of noise on generalized synchronization in unidirectionally coupled systems with a complex topology of attractor. As the systems under study, two models of chaotic systems with two-sheeted topologies of attractors such as Lorenz and Chen systems are considered. Materials and Methods: For the synchronous regime detection the method of calculating the spectrum of Lyapunov exponents for coupled systems and the auxiliary system approach are used. Results: It has been shown that the dependences of the threshold for the onset of the generalized synchronization regime on the noise intensity do not practically change both for Lorenz and Chen systems. Conclusion: Using the examples of Lorenz oscillators and Chen systems we have found that the generalized synchronization regime in systems with a complex topology of attractor is stable to external noise. This behavior of systems is determined by the weak influence of noise on the structure of attractors of interacting systems, which is confirmed by constructing their phase portraits for different values of the noise intensity

The purpose of this work is to study the possibility of the existence of explosive chaotic synchronization phenomenon in small world networks of nonlinear oscillators in the same way as it has been observed for complex networks of nonlinear elements with a random or scale-free topologies of links between network nodes. Methods. In this work, along with numerical modeling, an analytical approach has been used to describe the behavior of network nonlinear elements below the threshold of the occurrence of a completely synchronous state of all network oscillators. Results. It has been shown that in networks of nonlinear oscillators with topologies of links between nodes being the “ring” or “small world” type, with an increase in the coupling strength, an abrupt transition to the completely synchronous dynamics of all network oscillators is possible, just as it happens within the random and scale-free networks where the explosive synchronization phenomenon has been discovered and studied. With the help of theoretical consideration, a mechanism leading to the emergence of this abrupt transition to a synchronous state in ring and small-world networks has been revealed. This mechanism is associated with the formation of two independent synchronous clusters coexisting within network. Conclusion. The paper considers the mechanisms leading to a sharp “explosive” transition to completely synchronous dynamics of all elements in ring and small world networks of nonlinear oscillators, analytical relations are obtained that describe this transition and make it possible to explain the revealed phenomenon.

The aim of the paper consists in the development of universal method for the detection of characteristic phases of the behavior in systems with complex topology of attractor being in the regime of intermittent generalized synchronization. The method is based on an analysis of the location of representation points on the attractors of interacting systems coupled unidirectionally or mutually. The result of this work is the verification of the performance of the proposed method on systems with unidirectional coupling (two unidirectionally coupled Lorenz oscillators being in chaotic regime) that allow the analysis of intermittency using the auxiliary system method. It was found that the jump of the representation points to different sheets of attractors of interacting systems precedes the appearance of the turbulent phase of the behavior detected using the auxiliary system method. Using both methods, the statistical characteristics of intermittency, i.e. the distributions of the laminar phase lengths for several fixed values of the coupling parameter, were calculated and they were compared with each other. It was found that in all considered cases the results of both methods almost exactly coincide with each other, while the distributions of the laminar phase lengths obey the exponential laws, which is not typical for systems with a simple enough topology of attractor. It was assumed that in systems with a complex topology of attractor a new type of intermittency called by jump intermittency is observed.