Generalized synchronization in linearly coupled time periodic systems

We consider the synchronization of a network of linearly coupled and not necessarily identical oscillators. We present an approach to the existence of the synchronization manifold which is based on some results developed by R. Smith for the study of periodic solutions of ODEs. Our framework allows the study of a large class of systems and does not assume that they are small perturbations of linear systems. Moreover, it provides a practical way to compute estimations on the parameters of the system for which generalized synchronization occurs. Additionally, we give a new proof of the main result of R. Smith on invariant manifolds using Wazewski's principle. Several examples of application are presented.     

Group synchronization in complex dynamical networks with different types of oscillators and adaptive coupling schemes

This paper focus on schemes and corresponding criteria for group synchronization in complex dynamical networks consisted of different group of chaotic oscillators. The global asymptotically stable criteria for a linearly or adaptively coupled network are derived to ensure each group of oscillators synchronize to the same behavior. Theoretical analysis and numerical simulation results show that the group synchronization can be guaranteed by enhancing the external coupling strength whenever there are connections or not within the groups under the “same input” condition. All of the results are proved rigorously. Finally, a network with three groups, a scale-free sub-network, a small-world sub-network and a ring sub-network, is illustrated, and the corresponding numerical simulations verify the theoretical analysis.     

ADAPTIVE PINNING SYNCHRONIZATION OF COUPLED NEURAL NETWORKS WITH MIXED DELAYS AND VECTOR-FORM STOCHASTIC PERTURBATIONS＊

In this article,we consider the global chaotic synchronization of general coupled neural networks,in which subsystems have both discrete and distributed delays.Stochastic perturbations between subsyste...     

Broad-scale small-world network topology induces optimal synchronization of flexible oscillators

The discovery of small-world and scale-free properties of many man-made and natural complex networks has attracted increasing attention. Of particular interest is how the structural properties of a network facilitate and constrain its dynamical behavior. In this paper we study the synchronization of weakly coupled limit-cycle oscillators in dependence on the network topology as well as the dynamical features of individual oscillators. We show that flexible oscillators, characterized by near zero values of divergence, express maximal correlation in broad-scale small-world networks, whereas the non-flexible (rigid) oscillators are best correlated in more heterogeneous scale-free networks. We found that the synchronization behavior is governed by the interplay between the networks global efficiency and the mutual frequency adaptation. The latter differs for flexible and rigid oscillators. The results are discussed in terms of evolutionary advantages of broad-scale small-world networks in biological systems.     

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.     

Exponential synchronization of stochastic coupled oscillators networks with delays

This paper investigates the exponential synchronization problem of coupled oscillators networks with disturbances and time-varying delays. On basis of graph theory and stochastic analysis theory, a feedback control law is designed to achieve exponential synchronization. By constructing a global Lyapunov function for error network, both pth moment exponential synchronization and almost sure exponential synchronization of drive-response networks are obtained. Finally, a numerical example is given to show the effectiveness of the proposed criteria.     

Asymptotically stable invariant manifold for coupled nonlinear parabolic-hyperbolic partial differential equations

This article considers a coupled system of nonlinear parabolic and hyperbolic partial differential equations which arises in the study of wave phenomena which are heat generating or temperature related. Under appropriate conditions, for example high thermal diffusivity, it is proved that there exists an invariant manifold for the full system of equations. The asymptotic stability of the invariant manifold is also considered. Moreover, it is shown that an equilibrium which is asymptotically stable for flows on the invariant manifold will be asymptotically stable for the full system.     

Heteroclinic Ratchets in Networks of Coupled Oscillators

We analyze an example system of four coupled phase oscillators and discover a novel phenomenon that we call a “heteroclinic ratchet”; a particular type of robust heteroclinic network on a torus where connections wind in only one direction. The coupling structure has only one symmetry, but there are a number of invariant subspaces and degenerate bifurcations forced by the coupling structure, and we investigate these. We show that the system can have a robust attracting heteroclinic network that responds to a specific detuning Δ between certain pairs of oscillators by a breaking of phase locking for arbitrary Δ>0 but not for Δ≤0. Similarly, arbitrary small noise results in asymmetric desynchronization of certain pairs of oscillators, where particular oscillators have always larger frequency after the loss of synchronization. We call this heteroclinic network a heteroclinic ratchet because of its resemblance to a mechanical ratchet in terms of its dynamical consequences. We show that the existence of heteroclinic ratchets does not depend on symmetry or number of oscillators but depends on the specific connection structure of the coupled system.     

Numerical Analysis of the Synchronization of Coupled Chemical Oscillators

A reaction–diffusion model describing a system of coupled oscillators is constructed and investigated. The oscillators in this study are chemical oscillators that represent an oscillatory heterogeneous catalytic reaction in a granular catalyst layer. The oscillators are arranged serially in the reagent stream and are coupled through the gaseous phase. The dynamic behavior of the system is investigated as a function of the main external parameter — the partial pressure of one of the reagents in the gaseous phase. Existence regions of regular and chaotic oscillations are identified. Synchronization conditions are established for the oscillations in such a chain of coupled chemical oscillators.     

Bursting synchronization in networks with long-range coupling mediated by a diffusing chemical substance

Many networks of physical and biological interest are characterized by a long-range coupling mediated by a chemical which diffuses through a medium in which oscillators are embedded. We considered a one-dimensional model for this effect for which the diffusion is fast enough so as to be implemented through a coupling whose intensity decays exponentially with the lattice distance. In particular, we analyzed the bursting synchronization of neurons described by two timescales (spiking and bursting activity), and coupled through such a long-range interaction network. One of the advantages of the model is that one can pass from a local (Laplacian) type of coupling to a global (all-to-all) one by varying a single parameter in the interaction term. We characterized bursting synchronization using an order parameter which undergoes a transition as the coupling parameters are changed through a critical value. We also investigated the role of an external time-periodic signal on the bursting synchronization properties of the network. We show potential applications in the control of pathological rhythms in biological neural networks.     

Multistable remote synchronization in a star-like network of non-identical oscillators

In the context of networks of coupled oscillators, remote synchronization happens when phase difference between non-adjacent units become constant, even though there is no global phase-locking. We study such regime considering a star-like network of Stuart-Landau oscillators. As previous works, our setup comprises peripheral nodes with different but close natural frequencies and the central node frequency detuned from them. The main contribution here is to numerically report multistability under intermediate coupling values: some initial condition yield remote synchronization, with quasi-periodic motion; while others do not converge to synchronized states. By using a Gaussian distribution to select the initial phases of the oscillators, we found that relatively small value of the standard deviation and absolute value of the mean of this distribution far from a specific range of values seem to favor remote synchronization in the multistability region. This phenomenon is extensively analyzed for both cases, considering a fixed coupling value.     

Analysis of generalized projective synchronization for a chaotic gyroscope with a periodic gyroscope

In this paper, a simple nonlinear controller is applied to investigate the generalized projective synchronization for a controlled chaotic gyroscope with a periodic gyroscope dynamical system. The necessary and sufficient conditions for generalized projective synchronization are developed through the theory of discontinuous dynamical systems. The synchronization invariant domain from the synchronization conditions is presented. The parameter maps are explored for a better understanding of the synchronicity of two gyroscopes with different motions. Finally, the partial and full generalized projective synchronizations of two nonlinear coupled gyroscope systems are carried out to verify the effectiveness of the scheme.     

n维耦合振子格点系统的同步性

本文通过在相空间中引入新范数的方法研究了Dirichlet,Neumann,周期三个不同边界条件下带有周期外力的n维二阶耦合振子格点系统的同步性.在Dirichlet边界条件下,如果系统非线性项的一阶偏导数有界,则当耦合系数足够大时,系统是有界耗散的并且任意两个解之间是同步的.在Neumann与周期边界条件下,如果不同子系统的外力之间的差和不同子系统的非线性项之间的差都比较小,并且系统是有界耗散的,则当耦合系数足够大时,系统任意一个解的任意两个分量之间是渐近同步的.这两种情况下,当耦合系数c_1→+∞,c_2→+∞时,系统任意一个解的任意两个分量之间是同步的.     

Winding Numbers and Average Frequencies in Phase Oscillator Networks

We study networks of coupled phase oscillators and show that network architecture can force relations between average frequencies of the oscillators. The main tool of our analysis is the coupled cell theory developed by Stewart, Golubitsky, Pivato, and Torok, which provides precise relations between network architecture and the corresponding class of ODEs in R^M and gives conditions for the flow-invariance of certain polydiagonal subspaces for all coupled systems with a given network architecture. The theory generalizes the notion of fixed-point subspaces for subgroups of network symmetries and directly extends to networks of coupled phase oscillators. For systems of coupled phase oscillators (but not generally for ODEs in R^M, where M ≥ 2), invariant polydiagonal subsets of codimension one arise naturally and strongly restrict the network dynamics. We say that two oscillators i and j coevolve if the polydiagonal θ_i = θ_j is flow-invariant, and show that the average frequencies of these oscillators must be equal. Given a network architecture, it is shown that coupled cell theory provides a direct way of testing how coevolving oscillators form collections with closely related dynamics. We give a generalization of these results to synchronous clusters of phase oscillators using quotient networks, and discuss implications for networks of spiking cells and those connected through buffers that implement coupling dynamics.     

Synchronization of stochastic coupled systems via feedback control based on discrete-time state observations

In this paper, we formulate and investigate the synchronization of stochastic coupled systems via feedback control based on discrete-time state observations (SCSFD). The discrete-time state feedback control is used in the drift parts of response system. Combining Lyapunov method with graph theory, the upper bound of duration between two consecutive state observations is provided. And a global Lyapunov function of SCSFD is presented, which derives some sufficient criteria to guarantee the synchronization of drive–response systems in the sense of mean-square asymptotical synchronization. In addition, the theoretical results are applied to stochastic coupled oscillators and second-order Kuramoto oscillators. Finally, two numerical examples are given to verify the effectiveness of the theoretical results.     

A New Approach to Synchronization Analysis of Linearly Coupled Map Lattices

In this paper, a new approach to analyze synchronization of linearly coupled map lattices (LCMLs) is presented. A reference vector x(t) is introduced as the projection of the trajectory of the coupled system on the synchronization manifold. The stability analysis of the synchronization manifold can be regarded as investigating the difference between the trajectory and the projection. By this method, some criteria are given for both local and global synchronization. These criteria indicate that the left and right eigenvectors corresponding to the eigenvalue "0" of the coupling matrix play key roles in the stability of synchronization manifold for the coupled system. Moreover, it is revealed that the stability of synchronization manifold for the coupled system is different from the stability for dynamical system in usual sense. That is, the solution of the coupled system does not converge to a certain knowable s(t) satisfying s(t 1) = f(s(t)) but to the reference vector on the synchronization manifold, which in fact is a certain weighted average of each xi(t) for i = 1, ... ,m, but not a solution s(t) satisfying s(t 1) = f(s(t)).     

广义同步化流形的Holder连续性

证明了两个不同的混沌系统线性耦合时能实现广义同步化,在一定条件下广义同步化流形是Holder连续的.采用的思想是Temam的无穷维动力系统的惯性流形理论的改进.在线性耦合下两个混沌系统具有吸收集和吸引子的基础上,通过定义在一个函数类上的映射满足Schauder不动点定理,从而得到广义同步化流形,该广义同步化流形具有不变性.又证明了存在分数维的指数吸引子,指数吸引子与广义同步流形的交集具有指数吸引性.数值仿真证实了理论的正确性.在驱动系统和响应系统外引入辅助系统,辅助系统与响应系统的完全同步化表明了驱动系统和响应系统的广义同步化.     

Synchronous dynamics of two coupled oscillators with inhibitory-to-inhibitory connection

We investigate the behaviour of a neural network model consisting of two coupled oscillators with delays and inhibitory-to-inhibitory connections. We consider the absolute synchronization and show that the connection topology of the network plays a fundamental role in classifying the rich dynamics and bifurcation phenomena. Regarding eigenvalues of the connection matrix as bifurcation parameters, we obtain codimension one bifurcations (including fold bifurcation and Hopf bifurcation) and codimension two bifurcation (including fold-Hopf bifurcations and Hopf–Hopf bifurcations). Based on the normal form theory and center manifold reduction, we obtain detailed information about the bifurcation direction and stability of various bifurcated equilibria as well as periodic solutions with some kinds of spatio-temporal patterns. Numerical simulation is also given to support the obtained results.     

Emergence of a common generalized synchronization manifold in network motifs of structurally different time-delay systems

We point out the existence of a transition from partial to global generalized synchronization (GS) in symmetrically coupled structurally different time-delay systems of different orders using the auxiliary system approach and the mutual false nearest neighbor method. The present authors have recently reported that there exists a common GS manifold even in an ensemble of structurally nonidentical scalar time-delay systems with different fractal dimensions and shown that GS occurs simultaneously with phase synchronization (PS). In this paper we confirm that the above result is not confined just to scalar one-dimensional time-delay systems alone but there exists a similar type of transition even in the case of time-delay systems with different orders. We calculate the maximal transverse Lyapunov exponent to evaluate the asymptotic stability of the complete synchronization manifold of each of the main and the corresponding auxiliary systems, which in turn ensures the stability of the GS manifold between the main systems. Further we estimate the correlation coefficient and the correlation of probability of recurrence to establish the relation between GS and PS. We also calculate the mutual false nearest neighbor parameter which doubly confirms the occurrence of the global GS manifold.     

Various synchronization phenomena in bidirectionally coupled double scroll circuits

In this paper, the various cases of synchronization phenomena investigated in a system of two bidirectionally coupled double scroll circuits, were studied. Complete synchronization, inverse lag synchronization, and inverse π-lag synchronization are the observed synchronization phenomena, as the coupling factor is varied. The inverse lag synchronization phenomenon in mutually coupled identical oscillators is presented for the first time. As the coupling factor is increased, the system undergoes a transition from chaotic desynchronization to chaotic complete synchronization, while inverse lag synchronization and inverse π-lag synchronization are observed for greater values of the coupling factor, depending on the initial conditions of the state variables of the system. Inverse π-lag synchronization in coupled nonlinear oscillators is a special case of lag synchronization, which is also presented for the first time.