The relation between generalized synchronization and phase synchronization is investigated. It was claimed that generalized synchronization always leads to phase synchronization, and phase synchronization is a weaker form than generalized synchronization. We propose examples that generalized synchronization can be weaker than phase synchronization, depending on parameter misfits. Moreover, generalized synchronization does not always lead to phase synchronization.

The model of nonlocally coupled identical phase oscillators on complex networks is investigated. We find the existence of chimera states in which identical oscillators evolve into distinct coherent and incoherent groups. We find that the coherent group of chimera states always contains the same oscillators no matter what the initial conditions are. The properties of chimera states and their dependence on parameters are investigated on both scale-free networks and Erdös-Rényi networks.

The behaviors of coupled oscillators, each of which has periodic motion with random natural frequency in the absence of coupling, are investigated. Some novel collective phenomena are revealed. At the onset of instability of the phase-locking state, simultaneous phase slips of all oscillators and quantized phase shifts in these phase slips are observed. By increasing the coupling, a bifurcation tree from high-dimensional quasiperiodicity to chaos to quasiperiodicity and periodicity is found. Different orders of phase synchronizations of chaotic oscillators and chaotic clusters play the key role for constructing this tree structure. [S0031-9007(98)07916-2]

Generalized synchronization in an array of mutually (bidirectionally) coupled nonidentical chaotic oscillators is studied. Coupled Lorenz oscillators and coupled Lorenz-Rossler oscillators are adopted as our working models. With increasing the coupling strengths, the system experiences a cascade of transitions from the partial to the global generalized synchronizations, i.e., different oscillators are gradually entrained through a clustering process. This scenario of transitions reveals an intrinsic self-organized order in groups of interacting units, which generalizes the idea of generalized synchronizations in drive-response systems. Recently synchronization behaviors in coupled or driven chaotic elements have been extensively exploited both theoretically and experimentally in the context of many specific problems such as laser dynamics ͓1͔, electronic circuits ͓2͔, chemical and biological systems ͓3͔, and secure communications ͓4͔, due to its theoretical importance and application perspectives. The entrainment of coupled or driven limit cycles ͑periodic oscillators͒ has long been a widely studied topic, while the synchronization of chaotic oscillators was an open area due to the presence of the intrinsic nonlinearity ͓5͔. People had thought synchronization of coupled chaotic oscillators cannot be attained because chaotic systems exhibit the exponential instability of nearby orbits ͑the socalled butterfly effect͒. However, this has been changed since it was shown by Pecora and Carroll and others ͓6͔ that two interacting identical chaotic oscillators can achieve synchronization ͓the complete or identical synchronization ͑CS͔͒ i.e., they evolve on the synchronized manifold, X 1 (t)ϭX 2 (t) ϭX(t), even though they individually possess the exponential instability of neighboring orbits. Due to the complicated feature of chaotic motion, there should be different levels of synchronized order. This study arouse extensive interest in synchronized entrainment of chaotic oscillators, and different degrees of synchronizations were found. CS appears only when interacting systems are identical. For two different chaotic oscillators it was found in 1995 for the drive-response systems that although CS can never be attained, the so-called generalized synchronization ͑GS͒ could be achieved, i.e., an emergence of some functional relation between the states of response and drive, i.e., X 2 (t)ϭG͓X 1 (t)͔, can be observed ͓7͔. In 1996 Rosenblum and co-workers observed the entrainment of phases for two coupled chaotic oscillators with small parameter mismatches, if an appropriate phase variable can be defined. This locking of phases admits the chaoticity and uncorrelation of oscillation amplitudes for the two chaotic oscillators. They call this locking behavior the phase synchronization ͑PS͒ ͓8͔. The emergence of PS indicates an order in some degrees of freedom in coupled oscillator systems. Later on, the lag synchronization ͑LS͒ was found ͓9͔ for stronger coupling strengths, i.e., the two states are identified by a temporal shift,...

The remarkable phenomenon of chimera state in systems of non-locally coupled, identical oscillators has attracted a great deal of recent theoretical and experimental interests. In such a state, different groups of oscillators can exhibit characteristically distinct types of dynamical behaviors, in spite of identity of the oscillators. But how robust are chimera states against random perturbations to the structure of the underlying network? We address this fundamental issue by studying the effects of random removal of links on the probability for chimera states. Using direct numerical calculations and two independent theoretical approaches, we find that the likelihood of chimera state decreases with the probability of random-link removal. A striking finding is that, even when a large number of links are removed so that chimera states are deemed not possible, in the state space there are generally both coherent and incoherent regions. The regime of chimera state is a particular case in which the oscillators in the coherent region happen to be synchronized or phase-locked.

Chaos should occur often in gene regulatory networks (GRNs) which have been widely described by nonlinear coupled ordinary differential equations, if their dimensions are no less than 3. It is therefore puzzling that chaos has never been reported in GRNs in nature and is also extremely rare in models of GRNs. On the other hand, the topic of motifs has attracted great attention in studying biological networks, and network motifs are suggested to be elementary building blocks that carry out some key functions in the network. In this paper, chaotic motifs (subnetworks with chaos) in GRNs are systematically investigated. The conclusion is that: (i) chaos can only appear through competitions between different oscillatory modes with rivaling intensities. Conditions required for chaotic GRNs are found to be very strict, which make chaotic GRNs extremely rare. (ii) Chaotic motifs are explored as the simplest few-node structures capable of producing chaos, and serve as the intrinsic source of chaos of random few-node GRNs. Several optimal motifs causing chaos with atypically high probability are figured out. (iii) Moreover, we discovered that a number of special oscillators can never produce chaos. These structures bring some advantages on rhythmic functions and may help us understand the robustness of diverse biological rhythms. (iv) The methods of dominant phase-advanced driving (DPAD) and DPAD time fraction are proposed to quantitatively identify chaotic motifs and to explain the origin of chaotic behaviors in GRNs.

Nowadays massive amount of data are available for analysis in natural and social systems. Inferring system structures from the data, i.e., the inverse problem, has become one of the central issues in many disciplines and interdisciplinary studies. In this Letter, we study the inverse problem of stochastic dynamic complex networks. We derive analytically a simple and universal inference formula called double correlation matrix (DCM) method. Numerical simulations confirm that the DCM method can accurately depict both network structures and noise correlations by using available kinetic data only. This inference performance was never regarded possible by theoretical derivation, numerical computation and experimental design.PACS numbers: 89.75. Hc, 05.10.Gg, 05.45.Tp Introduction. In recent decades, large scale of data sets have been accumulated in various and wide fields, in particular in social and biological systems [1][2][3][4]. There are massive amount of data available for utilization, however, the system structures yielding these data are often not clear [5,6]. Therefore, deducing the connectivity of systems from these data, i.e., the inverse problem, turns to be today one of the central issues in interdisciplinary fields [7][8][9][10][11][12][13]. A typical example of inference efforts is a recent project of the Dialogue on Reverse Engineering Assessment and Methods (DREAM) which has attracted extensive attention for reconstructing gene regulatory networks from high-throughput microarray data [14,15]. Similar goals have been also pursued in other fields, such as neural networks [16], ecosystems [17], chemical reactions [18,19] and so on. Most of biological and social systems contain many units which evolve collectively with very complicated interaction structures represented by complex networks [20][21][22]. Mathematically, the dynamics of these complex systems are extensively described by sets of coupled ordinary differential equations (ODEs) [23][24][25][26]. The inverse problems of these systems can thus be interpreted as to retrieve the interaction Jacobian matrices from the measurable data of dynamical variables of networks. So far, a wide range of network inference methods have been proposed to address this issue in diverse fields. Available methods can be classified into several broad categories [14,[27][28][29]: Bayesian networks and probabilistic graphical models, which maximize a scoring function over alternative network models [30,31]; regression techniques, which fit the data to a priori models [32]; integrative bioinformatics approaches, which combine data from a number of independent ex- * Electronic address: ganghu@bnu.edu.cn perimental clues [33,34]; statistical methods, which rely on a variety of measures of pairwise correlations or mutual information and other methods [27,35,36].The complexity of networks can hinder the attempt to solve the inverse problems [33]. Moreover, the network dynamics are inevitably perturbed by many uncontrollable impacts, called noise, and these random and unkn...

Kink dynamics of the damped Frenkel-Kontorova (discrete sine-Gordon) chain driven by a constant external force are investigated. Resonant steplike transitions of the average velocity occur due to the competitions between the moving kinks and their radiated phasonlike modes. A mean-field consideration is introduced to give a precise prediction of the resonant steps. Slip-stick motion and spatiotemporal dynamics on those resonant steps are discussed. Our results can be applied to studies of the fluxon dynamics of 1D Josephson-junction arrays and ladders, dislocations, tribology and other fields.