Synchronization in time-varying networks: a matrix measure approach

Desynchronizing oscillators coupled in multi-cluster networks through adaptively controlling partial networks

This article introduces an adaptive control scheme with a feedback delay, specifically designed for controlling partial networks, to achieve desynchronization in a coupled network with two or multiple clusters. The proposed scheme's effectiveness is validated through several representative examples of coupled neuronal networks with two interconnected clusters. The efficacy of this scheme is attributed to the rigorous and numerical analyses on the corresponding transcendental characteristic equation, which includes time delay and other network parameters. In addition to investigating the impact of time delay and inter-connectivity on the stability of an incoherent state, we also rigorously find that controlling only one cluster cannot realize the desynchronization in the coupled oscillators within three or more clusters. All these, we believe, can deepen the understanding of the deep brain stimulation techniques presently used in the clinical treatment of neurodegenerative diseases and suggest future avenues for enhancing these clinical techniques through adaptive feedback settings.

The Study for Synchronization between Two Coupled FitzHugh-Nagumo Neurons Based on the Laplace Transform and the Adomian Decomposition Method

The synchronization between two coupled FitzHugh-Nagumo (FHN) neurons with or without external current is studied by using the Laplace transform and the Adomian decomposition method. Different from other researches, the synchronization error system is expressed as sets of Volterra integral equations based on the convolution theorem in the Laplace transform. Then, it is easy to analytically obtain the conditions that synchronization errors disappear based on the successive approximation method in integral equation theorem, the correctness of which is verified by numerical simulations. Furthermore, the synchronous dynamics of the two coupled FHN neurons also can be written in the form of Volterra integral equations, which is more convenient to analytically solve by using the Adomian decomposition method. It is found that the occurrence of synchronization between the two FHN neurons only depends on the coupling strength and is irrelevant to the external current. Only synchronous rest state in the two FHN neurons without external current can be achieved, while synchronous spikes appear if the external current is not zero.

A Synchronization Criterion for Two Hindmarsh-Rose Neurons with Linear and Nonlinear Coupling Functions Based on the Laplace Transform Method

In this paper, an analytical criterion is proposed to investigate the synchronization between two Hindmarsh-Rose neurons with linear and nonlinear coupling functions based on the Laplace transform method. Different from previous works, the synchronization error system is expressed in its integral form, which is more convenient to analyze. The synchronization problem of two HR coupled neurons is ultimately converted into the stability problem of roots to a nonlinear algebraic equation. Then, an analytical criterion for synchronization between the two HR neurons can be given by using the Routh-Hurwitz criterion. Numerical simulations show that the synchronization criterion derived in this paper is valid, regardless of the periodic spikes or burst-spike chaotic behavior of the two HR neurons. Furthermore, the analytical results have almost the same accuracy as the conditional Lyapunov method. In addition, the calculation quantities always are small no matter the linear and nonlinear coupling functions, which show that the approach presented in this paper is easy to be developed to study synchronization between a large number of HR neurons.

Eliminating synchronization of coupled neurons adaptively by using feedback coupling with heterogeneous delays

In this paper, we present an adaptive scheme involving heterogeneous delay interactions to suppress synchronization in a large population of oscillators. We analytically investigate the incoherent state stability regions for several specific kinds of distributions for delays. Interestingly, we find that, among the distributions that we discuss, the exponential distribution may offer great convenience to the performance of our adaptive scheme because this distribution renders an unbounded stability region. Moreover, we demonstrate our scheme in the realization of synchronization elimination in some representative, realistic neuronal networks, which makes it possible to deepen the understanding and even refine the existing techniques of deep brain stimulation in the treatment of some synchronization-induced mental disorders.

Random temporal connections promote network synchronization

We report a phenomenon of collective dynamics on discrete-time complex networks: a random temporal interaction matrix even of zero or/and small average is able to significantly enhance synchronization with probability one. According to current knowledge, there is no verifiably sufficient criterion for the phenomenon. We use the standard method of synchronization analytics and the theory of stochastic processes to establish a criterion, by which we rigorously and accurately depict how synchronization occurring with probability one is affected by the statistical characteristics of the random temporal connections such as the strength and topology of the connections as well as their probability distributions. We also illustrate the enhancement phenomenon using physical and biological complex dynamical networks.

Influence of Time Delay in Signal Transmission on Synchronization between Two Coupled FitzHugh-Nagumo Neurons

In this paper, the energy method is employed to analytically investigate the influence of time delay in signal transmission on synchronization between two coupled FitzHugh-Nagumo (FHN) neurons. Unlike pre-existing methods that deal with synchronization problems, our major idea is to consider the change rate of the energy of the synchronization error system, since the original system’s synchronization is equivalent to the disappearance of the energy of the error system. In rewriting the original coupled system in the corresponding energy coordinates based on the energy method, we find that the change rate of energy of the error system can be divided into two parts (periodic and non-periodic). The synchronization criterion for the original system can then be obtained by letting the non-periodic part of the change rate of the energy be less than zero. The correctness of the analysis is illustrated with numerical simulations. Our analytical results show that time delay in signal transmission has very significant effects on the synchronization between two FHN neurons. If the time delay in signal transmission is not taken into account in the two coupled FHN neurons, synchronous spikes cannot be achieved in the system for any given coupling strength. By adjusting the value of the time delay in signal transmission, the neural system can freely switch between neural rest and synchronous spikes. This means that time delay in signal transmission is crucial for the occurrence of synchronous spikes in the FHN neural system, which contributes to our understanding of the interaction between neurons. We analytically show the influence of the time delay on the synchronization between two FHN neurons, which was seldom considered by other researchers.

Network synchronization with periodic coupling

The synchronization behavior of networked chaotic oscillators with periodic coupling is investigated. It is observed in simulations that the network synchronizability could be significantly influenced by tuning the coupling frequency, even making the network alternating between the synchronous and nonsynchronous states. Using the master stability function method, we conduct a detailed analysis of the influence of coupling frequency on network synchronizability and find that the network synchronizability is maximized at some characteristic frequencies comparable to the intrinsic frequency of the local dynamics. Moreover, it is found that as the amplitude of the coupling increases, the characteristic frequencies are gradually decreased. Using the finite-time Lyapunov exponent technique, we investigate further the mechanism for the maximized synchronizability and find that at the characteristic frequencies the power spectrum of the finite-time Lyapunov exponent is abruptly changed from the localized to broad distributions. When this feature is absent or not prominent, the network synchronizability is less influenced by the periodic coupling. Our study shows the efficiency of finite-time Lyapunov exponent in exploring the synchronization behavior of temporally coupled oscillators and sheds lights on the interplay between the system dynamics and structure in general temporal networks.

Synchronization of moving oscillators in three dimensional space

We investigate the macroscopic behavior of a dynamical network consisting of a time-evolving wiring of interactions among a group of random walkers. We assume that each walker (agent) has an oscillator and show that depending upon the nature of interaction, synchronization arises where each of the individual oscillators are allowed to move in such a random walk manner in a finite region of three dimensional space. Here, the vision range of each oscillator decides the number of oscillators with which it interacts. The live interaction between the oscillators is of intermediate type (i.e., not local as well as not global) and may or may not be bidirectional. We analytically derive the density dependent threshold of coupling strength for synchronization using linear stability analysis and numerically verify the obtained analytical results. Additionally, we explore the concept of basin stability, a nonlinear measure based on volumes of basin of attractions, to investigate how stable the synchronous state is under large perturbations. The synchronization phenomenon is analyzed taking limit cycle and chaotic oscillators for wide ranges of parameters like interaction strength k between the walkers, speed of movement v, and vision range r.

Synchronization in time-varying networks

We study the stability of the synchronized state in time-varying complex networks using the concept of basin stability, which is a nonlocal and nonlinear measure of stability that can be easily applied to high-dimensional systems [P. J. Menck, J. Heitzig, N. Marwan, and J. Kurths, Nature Phys. 9, 89 (2013)]. The time-varying character is included by stochastically rewiring each link with the average frequency f. We find that the time taken to reach synchronization is lowered and the stability range of the synchronized state increases considerably in dynamic networks. Further we uncover that small-world networks are much more sensitive to link changes than random ones, with the time-varying character of the network having a significant effect at much lower rewiring frequencies. At very high rewiring frequencies, random networks perform better than small-world networks and the synchronized state is stable over a much wider window of coupling strengths. Lastly we show that the stability range of the synchronized state may be quite different for small and large perturbations, and so the linear stability analysis and the basin stability criterion provide complementary indicators of stability.

Experimental evidence of synchronization of time-varying dynamical network

We investigate synchronization of time varying networks and stability conditions. We derive interesting relations between the critical coupling constants for synchronization and switching times for time-varying and time average networks. The relations are based on the additive property of Lyapunov exponents and are verified experimentally in electronic circuit.

Induced synchronization of a mobile agent network by phase locking

We investigate synchronization issues of a set of mobile agents in plane, where each agent carries an identical chaotic oscillator and interacts with its neighbors through a blinking coupling mechanism. We discuss the effect of blinking pattern on synchronization of the related network. In particular, we show that phase locking of the blinking behavior can dramatically enhance synchronizability of the mobile agent network, while the random blinking pattern works little. Also, we show that by assessing the convergence time, the mobile agent networks with different blinking periods and duty ratios share a common idle duration which is independent of both the blinking period and the corresponding duty ratio.

Coordinate transformation and matrix measure approach for synchronization of complex networks

Global synchronization in complex networks has attracted considerable interest in various fields. There are mainly two analytical approaches for studying such time-varying networks. The first approach is Lyapunov function-based methods. For such an approach, the connected-graph-stability (CGS) method arguably gives the best results. Nevertheless, CGS is limited to the networks with cooperative couplings. The matrix measure approach (MMA) proposed by Chen, although having a wider range of applications in the network topologies than that of CGS, works for smaller numbers of nodes in most network topologies. The approach also has a limitation with networks having partial-state coupling. Other than giving yet another MMA, we introduce a new and, in some cases, optimal coordinate transformation to study such networks. Our approach fixes all the drawbacks of CGS and MMA. In addition, by merely checking the structure of the vector field of the individual oscillator, we shall be able to determine if the system is globally synchronized. In summary, our results can be applied to rather general time-varying networks with a large number of nodes.

Matrix-measure criterion for synchronization in coupled-map networks

We present conditions for the local and global synchronization in coupled-map networks using the matrix measure approach. In contrast to many existing synchronization conditions, the proposed synchronization criteria do not depend on the solution of the synchronous state and give less limitation on the network connections. Numerical simulations of the coupled quadratic maps demonstrate the potentials of our main results.

Synchronization with on-off coupling: Role of time scales in network dynamics

We consider the problem of synchronizing a general complex network by means of the on-off coupling strategy; in this case, the on-off time scale is varied from a very small to a very large value. In particular, we find that when the time scale is comparable to that of node dynamics, synchronization can also be achieved and greatly optimized for the upper bound of the stability region which nearly disappears, and the synchronization speed is accelerated a lot, independent of network topologies. Our study indicates that the time scale for network variation is of crucial importance for network dynamics and synchronization under the comparable time scale which is much more advantageous over other time scales. Both analysis and experiments confirm the conclusions.

Synchronization in small-world networks

In this paper we consider complete synchronization in small-world networks of identical Rössler oscillators. By applying a simple but effective dynamical optimization coupling scheme, we realize complete synchronization in networks with undelayed or delayed couplings, as well as ensuring that all oscillators have uniform intensities during the transition to synchronization. Further, we obtain the coupling matrix with much better synchronizability in a certain range of the probability p for adding long-range connections. Direct numerical simulations fully verify the efficiency of our mechanism.