Generalized synchronization in complex dynamical networks via adaptive couplings

This paper investigates generalized synchronization of three typical classes of complex dynamical networks: scale-free networks, small-world networks, and interpolating networks. The proposed synchronization strategy is to adjust adaptively a node’s coupling strength based on the node’s local generalized synchronization information. By taking the auxiliary-system approach and using the Lyapunov function method, we prove that for any given initial coupling strengths, the generalized synchronization can take place in complex networks consisting of nonidentical dynamical systems. It is demonstrated that the coupling strengths are affected by topologies of the networks. Furthermore, it is found that there are hierarchical features in the processes of generalized synchronization in scale-free networks because of their highly heterogeneous distributions of connection degree. Finally, we discuss in detail how a network’s degree of heterogeneity affects its generalization synchronization behavior.     

Measure synchronization in hybrid quantum-classical systems

Measure synchronization in hybrid quantum-classical systems is investigated in this paper. The dynamics of the classical subsystem is described by the Hamiltonian equations, while the dynamics of the quantum subsystem is governed by the Schrödinger equation. By increasing the coupling strength in between the quantum and classical subsystems, we reveal the existence of measure synchronization in coupled quantum-classical dynamics under energy conservation for the hybrid systems.     

模态分解法在非恒同耦合系统同步研究中的推广

通过改变耦合函数将模态分解法进行了推广,应用于非恒同耦合系统同步的研究.详细研究了周期吸引子、概周期吸引子等非恒同耦合系统的同步,得到了同步的局部渐近稳定性条件.并进行了数值模拟,发现同步时动力学现象丰富.概周期吸引子耦合系统会出现稳定的周期、概周期同步解,由于耦合周期吸引子耦合系统会出现多个稳定的周期同步解,且其吸引域差别较大,均出现了同步的多值性.同时也验证了该方法的正确性.     

Thermodynamics based on the principle of least abbreviated action: Entropy production in a network of coupled oscillators

We present some novel thermodynamic ideas based on the Maupertuis principle. By considering Hamiltonians written in terms of appropriate action-angle variables we show that thermal states can be characterized by the action variables and by their evolution in time when the system is nonintegrable. We propose dynamical definitions for the equilibrium temperature and entropy as well as an expression for the nonequilibrium entropy valid for isolated systems with many degrees of freedom. This entropy is shown to increase in the relaxation to equilibrium of macroscopic systems with short-range interactions, which constitutes a dynamical justification of the Second Law of Thermodynamics. Several examples are worked out to show that this formalism yields the right microcanonical (equilibrium) quantities. The relevance of this approach to nonequilibrium situations is illustrated with an application to a network of coupled oscillators (Kuramoto model). We provide an expression for the entropy production in this system finding that its positive value is directly related to dissipation at the steady state in attaining order through synchronization.     

2D quantum gravity from quantum entanglement

In quantum systems with many degrees of freedom the replica method is a useful tool to study the entanglement of arbitrary spatial regions. We apply it in a way that allows them to backreact. As a consequence, they become dynamical subsystems whose position, form, and extension are determined by their interaction with the whole system. We analyze, in particular, quantum spin chains described at criticality by a conformal field theory. Its coupling to the Gibbs' ensemble of all possible subsystems is relevant and drives the system into a new fixed point which is argued to be that of the 2D quantum gravity coupled to this system. Numerical experiments on the critical Ising model show that the new critical exponents agree with those predicted by the formula of Knizhnik, Polyakov, and Zamolodchikov.     

Experimental investigation of high-quality synchronization of coupled oscillators

We describe two experiments in which we investigate the synchronization of coupled periodic oscillators. Each experimental system consists of two identical coupled electronic periodic oscillators that display bursts of desynchronization events similar to those observed previously in coupled chaotic systems. We measure the degree of synchronization as a function of coupling strength. In the first experiment, high-quality synchronization is achieved for all coupling strengths above a critical value. In the second experiment, no high-quality synchronization is observed. We compare our results to the predictions of the several proposed criteria for synchronization. We find that none of the criteria accurately predict the range of coupling strengths over which high-quality synchronization is observed. (c) 2000 American Institute of Physics.     

Resonance Analyses for a Noisy Coupled Brusselator Model

We discuss the dynamical behavior of a chemical network arising from the coupling of two Brusselators established by the relationship between products and substrates. Our interest is to investigate the coherence resonance(CR)phenomena caused by noise for a coupled Brusselator model in the vicinity of the Hopf bifurcation, which can be determined by the signal-to-noise ratio(SNR). The CR in two coupled Brusselators will be considered in the presence of the Gaussian colored noise and two uncorrelated Gaussian white noises. Simulation results show that,for the case of single noise, the SNR characterizing the degree of temporal regularity of coupled model reaches a maximum value at some optimal noise levels, and the noise intensity can enhance the CR phenomena of both subsystems with a similar trend but in different resonance degrees. Meanwhile, effects of noise intensities on CR of the second subsystem are opposite for the systems under two uncorrelated Gaussian white noises. Moreover,we find that CR might be a general phenomenon in coupled systems.     

Towards a dynamical temperature of finite Hamiltonian systems

With the aid of numerical simulations of the β Fermi-Pasta-Ulam (FPU) system, we compare the different definitions of dynamical temperature for Hamiltonian systems. We have shown that each definition gives different values of temperature for a system with a small number of degrees of freedom (DOF). Only for systems with a sufficiently large number of DOF, do all the definitions of dynamical temperature approach the same value.     

Generalized <Emphasis Type="Italic">N</Emphasis>-coupled maps with invariant measure in Bose-Mesner algebra perspective

By choosing a dynamical system with d different couplings, one can rearrange a system based on the graph with a given vertex dependent on the dynamical system elements. The relation between the dynamical elements (coupling) is replaced by a relation between the vertexes. Based on the E ₀ transverse projection operator, we addressed synchronization problem of an array of the linearly coupled map lattices of identical discrete time systems. The synchronization rate is determined by the second largest eigenvalue of the transition probability matrix. Algebraic properties of the Bose-Mesner algebra with an associated scheme with definite spectrum has been used in order to study the stability of the coupled map lattice. Associated schemes play a key role and may lead to analytical methods in studying the stability of the dynamical systems. The relation between the coupling parameters and the chaotic region is presented. It is shown that the feasible region is analytically determined by the number of couplings (i.e. by increasing the number of coupled maps, the feasible region is restricted). It is very easy to apply our criteria to the system being studied and they encompass a wide range of coupling schemes including most of the popularly used ones in the literature.      

INVERSE SYNCHRONIZATION OF CHAOTIC SYSTEMS IN AN ERBIUM-DOPED FIBRE DUAL-RING LASER USING THE MUTUAL COUPLING METHOD

Inverse synchronization of chaos is a type of synchronization in which the dynamical variables of two chaotic systems are inversely equal. In this paper, we present a scheme for inverse synchronization of two chaotic systems in an erbium-doped fibre dual-ring laser using the mutual coupling method. For realistic values of the systems, we demonstrate two kinds of results, as follows. (1) Two independent identical chaotic systems can go into inversely synchronized chaotic oscillation for coupling greater than 0.03. (2) When some parameter of one system varies, the state of the coupled systems could go into some periodic states directly or by inverse bifurcation. Simultaneously, they will lose the synchronization as the parameter changes.     

Synchronization processes in complex networks

During the last decades the emergence of collective dynamics in large networks of coupled units has been investigated in fields such as optics, chemistry, biology and ecology. Recently, complex networks have provided a challenging framework for the study of synchronization of dynamical units, based on the interplay between complexity in the overall topology and local dynamical properties of the coupled units. In this work, we review the constructive role played by such complex wirings for the synchronization of networks of coupled dynamical systems. We review the main techniques that have been proposed for assessing the propensity for synchronization (synchronizability) of a given networked system. We will also describe the main applications, especially in the view of selecting the optimal topology in the coupling configuration that provides enhancement of the synchronization features.     

Crowd synchrony and quorum sensing in delay-coupled lasers

Crowd synchrony and quorum sensing arise when a large number of dynamical elements communicate with each other via a common information pool. Previous evidence has shown that this type of coupling leads to synchronization, when coupling is instantaneous and the number of coupled elements is large enough. Here we consider a situation in which the transmission of information between the system components and the coupling pool is not instantaneous. To that end, we model a system of semiconductor lasers optically coupled to a central laser with a delay. Our results show that, even though the lasers are nonidentical due to their distinct optical frequencies, zero-lag synchronization arises. By changing a system parameter, we can switch between two different types of synchronization transition. The dependence of the transition with respect to the delay-coupling parameters is studied.     

Diffusion in Energy Conserving Coupled Maps

We consider a dynamical system consisting of subsystems indexed by a lattice. Each subsystem has one conserved degree of freedom (“energy”) the rest being uniformly hyperbolic. The subsystems are weakly coupled together so that the sum of the subsystem energies remains conserved. We prove that the subsystem energies satisfy the diffusion equation in a suitable scaling limit.     

Synchronization experiments with an atmospheric global circulation model

Synchronization in a chaotic system with many degrees of freedom is investigated by coupling two identical global atmospheric circulation models. Starting from different initial conditions, the two submodels show complete synchronization as well as noncomplete synchronization depending on the coupling strength. The relatively low value of the coupling strength threshold for complete synchronization indicates the potential importance of synchronization mechanisms involved in climate variability. In addition, the results suggest synchronization experiments as a valuable additional method to analyze complex dynamical models, e.g., to estimate the largest Lyapunov exponent. (c) 2001 American Institute of Physics.     

Generalized projective synchronization of two coupled complex networks of different sizes

We investigate a new generalized projective synchronization between two complex dynamical networks of different sizes. To the best of our knowledge, most of the current studies on projective synchronization have dealt with coupled networks of the same size. By generalized projective synchronization, we mean that the states of the nodes in each network can realize complete synchronization, and the states of a pair of nodes from both networks can achieve projective synchronization. Using the stability theory of the dynamical system, several sufficient conditions for guaranteeing the existence of the generalized projective synchronization under feedback control and adaptive control are obtained. As an example, we use Chua's circuits to demonstrate the effectiveness of our proposed approach.     

Synchronous behavior of coupled systems with discrete time

The dynamics of one-way coupled systems with discrete time is considered. The behavior of the coupled logistic maps is compared to the dynamics of maps obtained using the Poincaré sectioning procedure applied to the coupled continuous-time systems in the phase synchronization regime. The behavior (previously considered as asynchronous) of the coupled maps that appears when the complete synchronization regime is broken as the coupling parameter decreases, corresponds to the phase synchronization of flow systems, and should be considered as a synchronous regime. A quantitative measure of the degree of synchronism for the interacting systems with discrete time is proposed.     

Dynamical hysteresis and spatial synchronization in coupled non-identical chaotic oscillators

We identify a novel phenomenon in distinct (namely non-identical) coupled chaotic systems, which we term dynamical hysteresis. This behavior, which appears to be universal, is defined in terms of the system dynamics (quantified for example through the Lyapunov exponents), and arises from the presence of at least two coexisting stable attractors over a finite range of coupling, with a change of stability outside this range. Further characterization via mutual synchronization indices reveals that one attractor corresponds to spatially synchronized oscillators, while the other corresponds to desynchronized oscillators. Dynamical hysteresis may thus help to understand critical aspects of the dynamical behavior of complex biological systems, e.g. seizures in the epileptic brain can be viewed as transitions between different dynamical phases caused by time dependence in the brain’s internal coupling.     

Synchronization of networks

We study the synchronization of coupled dynamical systems on networks. The dynamics is governed by a local nonlinear oscillator for each node of the network and interactions connecting different nodes via the links of the network. We consider existence and stability conditions for both single- and multi-cluster synchronization. For networks with time-varying topology we compare the synchronization properties of these networks with the corresponding time-average network. We find that if the different coupling matrices corresponding to the time-varying networks commute with each other then the stability of the synchronized state for both the time-varying and the time-average topologies are approximately the same. On the other hand, for non-commuting coupling matrices the stability of the synchronized state for the time-varying topology is in general better than the time-average topology.      

化学自催化混沌反应模型中的耦合作用与混沌同步

选用混沌自催化反应作为子系统 ,构造了耦合自催化反应系统 ,研究了耦合变量、耦合系数对混沌动力学行为的影响 ,给出了不同耦合系数下系统的动力学特征 ,探讨了耦合作用机制 .结果表明 ,耦合作用能明显地改变子系统的动力学行为 ,强化系统间的相关性 .耦合后的混沌运动受到调整与抑制 ,耦合强度加大时 ,呈现出混沌运动轨线的周期化 ,耦合系数大于临界值 ,两子系统实现了完全的同步 .不同变量的耦合时 ,影响最大的是第二种变量 .对于三种物质均有耦合时 ,更容易出现混沌的抑制、运动状态的锁相与周期化和混沌的完全同步 .     

Nonlocal Measure Synchronization in Coupled Bosonic Josephson Junctions

We investigate the measure synchronization （MS） in two coupled bosonic Josephson junctions. By tuning up the coupling between the two dynamical systems, in addition to the normal MS, a nonlocal MS （NLMS） state is observed. Furthermore, with the dynamic stability analysis, we present the exact analytical solution of the transition point to NLMS.