Synchronization of complex dynamical networks with nonidentical nodes

This Letter investigates the synchronization problem of a complex network with nonidentical nodes, and proposes two effective control schemes to synchronize the network onto any smooth goal dynamics. By applying open-loop control to all nodes and placing adaptive feedback injections on a small fraction of network nodes, a low-dimensional sufficient condition is derived to guarantee the global synchronization of the complex network with nonidentical nodes. By introducing impulsive effects to the open-loop controlled network, another synchronization scheme is developed for the network composed of nonidentical nodes, and an upper bound of impulsive intervals is estimated to ensure the global stability of the synchronization process. Numerical simulations are given to verify the theoretical results.     

Synchronization of complex community networks with nonidentical nodes and adaptive coupling strength

In this Letter, the complex dynamical networks with community structure and nonidentical nodes are considered. The globally asymptotical synchronization of the time-delayed complex community networks onto any uniformly smooth state is studied. Some simple and useful criteria are derived by constructing an effective control scheme and adjusting automatically the adaptive coupling strength. Finally, the developed techniques are applied to two complex community networks which are respectively synchronized to a chaotic trajectory and a periodic orbit, and numerical simulations are provided to show the feasibility of the developed methods.     

Inertial ranges for turbulent solutions of complex Ginzburg-Landau equations

We consider the spatially periodic, complex Ginzburg-Landau (CGL) equation in regimes close to that of a critical or supercritical focusing non-linear Schrödinger (NLS) equation, which is known to have solutions that exhibit self-similar blow-up. We use the NLS blow-up solutions as a template to develop a theory of how nearly self-similar intermittent burst events can create a power-law inertial range in the time-averaged wave-number spectrum of CGL solutions. Numerical experiments in one dimension with a quintic (critical) and septant (supercritical) non-linearity show a that power-law inertial range emerges which differs from that predicted by the theory. However, as one approaches the NLS limit in the supercritical case, a second power-law inertial range is seen to emerge that agrees with the theory.     

Synchronization in coupled nonidentical incommensurate fractional-order systems

Synchronization of fractional-order nonlinear systems has received considerable attention for many research activities in recent years. In this Letter, we consider the synchronization between two nonidentical fractional-order systems. Based on the open-plus-closed-loop control method, a general coupling applied to the response system is proposed for synchronizing two nonidentical incommensurate fractional-order systems. We also derive a local stability criterion for such synchronization behavior by utilizing the stability theory of linear incommensurate fractional-order differential equations. Feasibility of the proposed coupling scheme is illustrated through numerical simulations of a limit cycle system, a chaotic system and a hyperchaotic system.     

Ginzburg-Landau equations for anisotropic superconductors

We use the framework of a general quasiclassical theory of superconductivity which allows for arbitrary gap and Fermi surface anisotropy and for impurity scattering in Born approximation. We derive general Ginzburg-Landau integro-differential equations, which comprise all previous limiting cases considered in the literature. From these equations more specialized Ginzburg-Landau equations may easily be derived.     

Rotating superconductors: Ginzburg-Landau equations

Superconductors put into rotation develope a spontaneous internal magnetic field (the “London field”). In this paper Ginzburg Landau equations for order parameter, field, and current distributions for superconductors in rotation are derived. Two simple examples are discussed: the massive cylinder and the “Little and Parks geometry”: a thin film of superconducting material deposited on a cylinder of normal material. A dependence of T _c on rotational frequency is predicted. The magnitude of the effect is estimated and should be observable. Received 28 May 2001     

Generalized chaotic synchronization in coupled Ginzburg-Landau equations

Generalized synchronization is analyzed in unidirectionally coupled oscillatory systems exhibiting spatiotemporal chaotic behavior described by Ginzburg-Landau equations. Several types of coupling between the systems are analyzed. The largest spatial Lyapunov exponent is proposed as a new characteristic of the state of a distributed system, and its calculation is described for a distributed oscillatory system. Partial generalized synchronization is introduced as a new type of chaotic synchronization in spatially nonuniform distributed systems. The physical mechanisms responsible for the onset of generalized chaotic synchronization in spatially distributed oscillatory systems are elucidated. It is shown that the onset of generalized chaotic synchronization is described by a modified Ginzburg-Landau equation with additional dissipation irrespective of the type of coupling. The effect of noise on the onset of a generalized synchronization regime in coupled distributed systems is analyzed.     

Solitary pulses in linearly coupled Ginzburg-Landau equations

This article presents a brief review of dynamical models based on systems of linearly coupled complex Ginzburg-Landau (CGL) equations. In the simplest case, the system features linear gain, cubic nonlinearity (possibly combined with cubic loss), and group-velocity dispersion (GVD) in one equation, while the other equation is linear, featuring only intrinsic linear loss. The system models a dual-core fiber laser, with a parallel-coupled active core and an additional stabilizing passive (lossy) one. The model gives rise to exact analytical solutions for stationary solitary pulses (SPs). The article presents basic results concerning stability of the SPs; interactions between pulses are also considered, as are dark solitons (holes). In the case of the anomalous GVD, an unstable stationary SP may transform itself, via the Hopf bifurcation, into a stable localized breather. Various generalizations of the basic system are briefly reviewed too, including a model with quadratic (second-harmonic-generating) nonlinearity and a recently introduced model of a different but related type, based on linearly coupled CGL equations with cubic-quintic nonlinearity. The latter system features spontaneous symmetry breaking of stationary SPs, and also the formation of stable breathers.     

Synchronization of Spatiotemporal Chaos in Coupled Complex Ginzburg-Landau Oscillators

Synchronization of spatiotemporal distributed system is investigated by considering the model of 1D dif-fusively coupled complex Ginzburg-Landau oscillators. An itinerant approach is suggested to randomly move turbulentsignal injections in the space of spatiotemporal chaos. Our numerical simulations show that perfect turbulence synchro-nization can be achieved with properly selected itinerant time and coupling intensity.     

Derivation of Cahn-Hilliard equations from Ginzburg-Landau models

The generalized Cahn-Hilliard equation is obtained as the hydrodynamic limit from a stochastic Ginzburg-Landau model. The associated large-deviation principle is also proved. In the one-dimensional case, we prove a related result about the scaling limit of conservative Langevin dynamics of an SOS surface.     

耦合复金兹堡-朗道(Ginzburg-Landau)方程中的模螺旋波

以双层耦合复金兹堡-朗道(Ginzburg-Landau)方程系统为时空模型, 研究了其中的模螺旋波, 讨论了这种特殊波动现象的稳定条件和相关影响因素. 模螺旋波与该类时空系统中常见的相螺旋波相比, 其中心不存在缺陷点, 同时仅在其变量的振幅部分(而非相位部分) 表现为螺旋结构. 本文通过数值方法研究了耦合复金兹堡-朗道方程中产生模螺旋波所需要的初始和参数条件.研究表明, 当双层耦合系统的初始斑图之间的差距较大时, 才能够产生模螺旋波; 同时观察到系统在参数不匹配的条件下会发生相螺旋波向模螺旋波的转变.通过对同步函数的计算, 发现该转变过程具有非连续性.     

ArbitraryN-vortex solutions to the first order Ginzburg-Landau equations

We prove that a set ofN not necessarily distinct points in the plane determine a unique, real analytic solution to the first order Ginzburg-Landau equations with vortex numberN. This solution has the property that the Higgs field vanishes only at the points in the set and the order of vanishing at a given point is determined by the multiplicity of that point in the set. We prove further that these are the onlyC solutions to the first order Ginzburg-Landau equations.This work is supported in part through funds provided under Contract PHY 77-18762     

Bistability in the complex Ginzburg-Landau equation with drift

Properties of the complex Ginzburg-Landau equation with drift are studied focusing on the Benjamin-Feir stable regime. On a finite interval with Neumann boundary conditions the equation exhibits bistability between a spatially uniform time-periodic state and a variety of nonuniform states with complex time dependence. The origin of this behavior is identified and contrasted with the bistable behavior present with periodic boundary conditions and no drift.     

Synchronization in complex networks

Synchronization processes in populations of locally interacting elements are the focus of intense research in physical, biological, chemical, technological and social systems. The many efforts devoted to understanding synchronization phenomena in natural systems now take advantage of the recent theory of complex networks. In this review, we report the advances in the comprehension of synchronization phenomena when oscillating elements are constrained to interact in a complex network topology. We also take an overview of the new emergent features coming out from the interplay between the structure and the function of the underlying patterns of connections. Extensive numerical work as well as analytical approaches to the problem are presented. Finally, we review several applications of synchronization in complex networks to different disciplines: biological systems and neuroscience, engineering and computer science, and economy and social sciences.