Periodic two-cluster synchronization modes in fully coupled networks of nonlinear oscillators

Theoretical and Mathematical Physics - We consider special systems of ordinary differential equations, the so-called fully coupled networks of nonlinear oscillators. For a given class of systems,...     

One- and two-cluster synchronized dynamics of non-diffusively coupled Tchebycheff map networks

We use the master stability formalism to discuss one- and two-cluster synchronization of coupled Tchebycheff map networks. For diffusively coupled map systems, the one-cluster synchronized dynamics is given by the behaviour of the individual maps, and the coupling only determines the stability of the coherent state. For the case of non-diffusive coupling and for two-cluster synchronization, the synchronized dynamics on networks is different from the behaviour of the single individual map. Depending on the coupling, we study numerically the characteristics of various forms of the resulting synchronized dynamics. The stability properties of the respective one-cluster synchronized states are discussed for arbitrary network structures. For the case of two-cluster synchronization on bipartite networks we also present analytical expressions for fixed points and zig-zag patterns, and explicitly determine the linear stability of these orbits for the special case of ring-networks.     

Bursting synchronization in scale-free networks

Neuronal networks in some areas of the brain cortex present the scale-free property, i.e., the neuron connectivity is distributed according to a power-law, such that neurons are more likely to couple with other already well-connected ones. Neuron activity presents two timescales, a fast one related to action-potential spiking, and a slow timescale in which bursting takes place. Some pathological conditions are related with the synchronization of the bursting activity in a weak sense, meaning the adjustment of the bursting phase due to coupling. Hence it has been proposed that an externally applied time-periodic signal be applied in order to control undesirable synchronized bursting rhythms. We investigated this kind of intervention using a two-dimensional map to describe neurons with spiking–bursting activity in a scale-free network.     

Two-sided networks for completely simple semigroups

Let S be a completely simple semigroup represented as a Rees matrix semigroup M(I,G,P) with normalized sandwich matrix P. On the congruence lattice C(S) of S we consider the relations T _i, K and T _r which identify congruences with the same left trace, kernel and right trace, respectively. These are equivalences whose classes are intervals. The upper and lower ends of these intervals induce the following operators on C(S) T_l, K, T_r, t_l, k and t_r .We construct here the semigroup generated by these operators as a homomorphic image of a semigroup given by generators and relations and demonstrate the minimality of the latter.     

Adaptive synchronization in tree-like dynamical networks

Synchronization in an array of coupled identical nonlinear dynamical systems have attracted increasing attention from various fields of science and engineering. In this paper, we investigate the synchronization phenomenon in tree-like dynamical networks. Based on the LaSalle invariant principle, a simple and systematic adaptive control scheme with variable coupling strength is proposed for the synchronization of tree-like dynamical networks without any knowledge of the concrete structure of isolate system. This result indicates that synchronization can be achieved for strong enough coupling if there exists a system (located at the root of the tree) which directly or indirectly influences all other systems. Furthermore, the main result is applied to several Lorenz chaotic systems coupled by a tree. And numerical simulations are also given to show the effectiveness of the proposed synchronization method.     

Clock synchronization in symmetric stochastic networks

We consider a stochastic model of clock synchronization in a wireless network of $N$ sensors interacting with one dedicated accurate time server. For large $N$ we find an estimate of the final time sychronization error for global and relative synchronization. The main results concern the behavior of the network on different timescales $t_{N}\rightarrow \infty $ , $N\rightarrow \infty $ . We discuss the existence of phase transitions and find the exact timescales for which an effective clock synchronization of the system takes place.     

Finite-time stochastic synchronization of complex networks

In this paper, we study the finite-time stochastic synchronization problem for complex networks with stochastic noise perturbations. By using finite-time stability theorem, inequality techniques, the properties of Weiner process and adding suitable controllers, sufficient conditions are obtained to ensure finite-time stochastic synchronization for the complex networks. The effects of control parameters on synchronization speed and time are also analyzed. The results of this paper are applicable to both directed and undirected weighted networks while do not need to know any information about eigenvalues of coupling matrix. Since finite-time synchronization means the optimality in convergence time and has better robustness and disturbance rejection properties, the results of this paper are important. A numerical example shows the effectiveness of our new results.     

Global synchronization for directed complex networks

Synchronization for generic directed networks is investigated in this paper. Based on Lyapunov stability theory, a sufficient condition for synchronization of directed networks is derived. Compared with other approaches, where weighted path-lengths are required, or the complex eigenvalues of an asymmetric matrix, Jacobian matrix or Kronecker product need to be calculated, the sufficient condition in this paper is simple and convenient to use. Two typical examples of directed networks with chaotic nodes are finally demonstrated to verify the theoretical results and the effectiveness of the proposed synchronization scheme.     

分数阶复杂网络的自适应同步

运用自适应控制方法,研究了分数阶复杂网络的同步.通过构造一种简单的Lyapuno函数,得到同步准则.最后通过数值例子表明所提出方法的有效性.     

Periodic perturbation of genetic oscillations

Kinetics of gene expression may be oscillatory due to the feedbacks between the mRNA, non-coding RNA and protein synthesis. In complex genetic networks, the kinetic oscillations generated in one subnetwork may influence oscillations in another subnetwork. To clarify what may happen in such situations, we have performed a mean-field analysis and Monte Carlo simulations of periodic perturbation of the oscillatory kinetics of the simplest genetic network including a gene with negative regulation of the mRNA production by protein, obtained via mRNA translation and two steps of conversion. Our analysis shows universal and specific features of the kinetics under consideration. Our simulations indicate that due to fluctuations only some of these features can really be observed. Specifically, the main frequencies obtained by the Fourier expansion of the mean-field and Monte Carlo kinetics are found to be often similar except that the Monte Carlo distribution of frequencies near the main frequencies is somewhat wider.     

Linear generalized synchronization between two complex networks

In this paper, the linear generalized synchronization between two complex networks is investigated. Based on the Lyapunov stability theory, a simple criterion for linear generalized synchronization between two networks with the same connection topologies is attained by using the nonlinear control method, which can widen the application range of the generalized synchronization methods. The feasibility of the proposed scheme is proved in theory and numerical simulations further demonstrate the effectiveness of it.     

Generalized synchronization between two different complex networks

In this paper, generalized synchronization (GS) between two coupled complex networks is theoretically and numerically studied, where the node vectors in different networks are not the same, and the numbers of nodes of both networks are not necessarily equal. First, a sufficient criterion for GS, one kind of outer synchronizations, of two coupled networks is established based on the auxiliary system method and the Lyapunov stability theory. Numerical examples are also included which coincide with the theoretical analysis.     

Complete synchronization on multi-layer center dynamical networks

In this paper, complete synchronization of three-layer center networks is studied. By using linear stability analysis approach, several different coupling schemes of three-layer center networks with the Logistic map local dynamics are discussed, and the stability conditions for synchronization are illustrated via some examples.     

Impulsive synchronization seeking in general complex delayed dynamical networks

The present paper investigates the issues of impulsive synchronization seeking in general complex delayed dynamical networks with nonsymmetrical coupling. By establishing the extended Halanay differential inequality on impulsive delayed dynamical systems, some simple yet generic sufficient conditions for global exponential synchronization of the impulsive controlled delayed dynamical networks are derived analytically. Compared with some existing works, the distinctive features of these sufficient conditions indicate two aspects: on the one hand, these sufficient conditions can provide an effective impulsive control scheme to synchronize an arbitrary given delayed dynamical network to a desired synchronization state even if the original given network may be asynchronous itself. On the other hand, the controlled synchronization state can be selected as a weighted average of all the states in the network for the purpose of practical control strategy, which reveals the contributions and influences of various nodes in synchronization seeking processes of the dynamical networks. It is shown that impulses play an important role in making the delayed dynamical networks globally exponentially synchronized. Furthermore, the results are applied to a typical nearest-neighbor unidirectional time-delay coupled networks composed of chaotic FHN neuron oscillators, and numerical simulations are given to demonstrate the effectiveness of the proposed control methodology.     

Periodic points of solvable group actions on 1-arcwise connected continua

It is shown that every continuous action of a solvable group on a 1-arcwise connected continuum must have a fixed point or have a 2-periodic point.     

Position-specific information in social networks: Are you connected?

Individuals in social networks often imperfectly monitor others’ network relationships and have incomplete information about the value of forming new relationships. This paper introduces the Generalized Conjectural Equilibrium (GCE) concept for such settings and completely characterizes the set of GCE networks when players observe only local parts of the network. Incomplete information and imperfect monitoring generate different types of inefficiency. These inefficiencies increase in number and scope as network observation becomes more localized. These results suggest that actual social networks will be structured inefficiently in general.     

Periodic solutions for nonautonomous discrete-time neural networks

In this paper, we theoretically prove the existence of periodic solutions for a nonautonomous discrete-time neural networks by using the topological degree theory. Sufficient conditions are also obtained for the existence of an asymptotically stable periodic solution. As a special case, we obtain the existence of a fixed point to the corresponding autonomous discrete-time neural networks which corrects the error in [W.R. Zhao, W. Lin, R.S. Liu, J. Ruan, Asymptotical stability in discrete-time neural networks, IEEE Trans. Circuits Syst. I 49 (2002) 1516–1520]. Numerical simulations are given at the end of the paper.     

Observer-based lag synchronization between two different complex networks

In this paper, some new criteria for lag synchronization between two or more complex networks are proposed based on the theory of state observer. Some adaptive controllers are designed to make the drive and response systems achieve lag synchronization, no matter whether the nodes in the two systems are with the same dynamical character or the coupling configuration matrices are nonidentical. In addition, based on the output coupling, the amount of coupling variables between two connected nodes is flexible, which can save a lot of channel resources, simplify the network topology and has more significant meanings in engineering applications. At last, the effects of the lag synchronization criteria are verified through some simulation experiments.