General stability analysis of synchronized dynamics in coupled systems

Effects of time delay on the synchronized states of globally coupled network

The effects of the time delay on the stability of different synchronized states of a globally coupled network are investigated. Conditions for the stability of the synchronized fixed points, synchronized periodic orbits, or synchronized chaos in a network of globally coupled chaotic smooth maps over a ring lattice with a homogeneous delay are derived analytically. Our analysis reveals that the stability properties of the synchronized dynamics are significantly different for odd and even time delays. The conditions for the stability of a synchronized fixed point and synchronized period-2 orbits for both odd and even delays are determined analytically. The range of parameter values for the stability of synchronized chaos has been calculated for a unit delay. All theoretical results are illustrated with the help of numerical examples.

Mixed outer synchronization of coupled complex networks with time-varying coupling delay

In this paper, the problem of outer synchronization between two complex networks with the same topological structure and time-varying coupling delay is investigated. In particular, we introduce a new type of outer synchronization behavior, i.e., mixed outer synchronization (MOS), in which different state variables of the corresponding nodes can evolve into complete synchronization, antisynchronization, and even amplitude death simultaneously for an appropriate choice of the scaling matrix. A novel nonfragile linear state feedback controller is designed to realize the MOS between two networks and proved analytically by using Lyapunov-Krasovskii stability theory. Finally, numerical simulations are provided to demonstrate the feasibility and efficacy of our proposed control approach.

Scaling and synchronization in a ring of diffusively coupled nonlinear oscillators

Chaos synchronization in a ring of diffusively coupled nonlinear oscillators driven by an external identical oscillator is studied. Based on numerical simulations we show that by introducing additional couplings at (mN(c)+1)-th oscillators in the ring, where m is an integer and N(c) is the maximum number of synchronized oscillators in the ring with a single coupling, the maximum number of oscillators that can be synchronized can be increased considerably beyond the limit restricted by size instability. We also demonstrate that there exists an exponential relation between the number of oscillators that can support stable synchronization in the ring with the external drive and the critical coupling strength ε(c) with a scaling exponent γ. The critical coupling strength is calculated by numerically estimating the synchronization error and is also confirmed from the conditional Lyapunov exponents of the coupled systems. We find that the same scaling relation exists for m couplings between the drive and the ring. Further, we have examined the robustness of the synchronous states against Gaussian white noise and found that the synchronization error exhibits a power-law decay as a function of the noise intensity indicating the existence of both noise-enhanced and noise-induced synchronizations depending on the value of the coupling strength ε. In addition, we have found that ε(c) shows an exponential decay as a function of the number of additional couplings. These results are demonstrated using the paradigmatic models of Rössler and Lorenz oscillators.

Cluster synchronization in networks of coupled nonidentical dynamical systems

In this paper, we study cluster synchronization in networks of coupled nonidentical dynamical systems. The vertices in the same cluster have the same dynamics of uncoupled node system but the uncoupled node systems in different clusters are different. We present conditions guaranteeing cluster synchronization and investigate the relation between cluster synchronization and the unweighted graph topology. We indicate that two conditions play key roles for cluster synchronization: the common intercluster coupling condition and the intracluster communication. From the latter one, we interpret the two cluster synchronization schemes by whether the edges of communication paths lie in inter- or intracluster. By this way, we classify clusters according to whether the communications between pairs of vertices in the same cluster still hold if the set of edges inter- or intracluster edges is removed. Also, we propose adaptive feedback algorithms to adapting the weights of the underlying graph, which can synchronize any bi-directed networks satisfying the conditions of common intercluster coupling and intracluster communication. We also give several numerical examples to illustrate the theoretical results.

Coordinate transformation and matrix measure approach for synchronization of complex networks

Global synchronization in complex networks has attracted considerable interest in various fields. There are mainly two analytical approaches for studying such time-varying networks. The first approach is Lyapunov function-based methods. For such an approach, the connected-graph-stability (CGS) method arguably gives the best results. Nevertheless, CGS is limited to the networks with cooperative couplings. The matrix measure approach (MMA) proposed by Chen, although having a wider range of applications in the network topologies than that of CGS, works for smaller numbers of nodes in most network topologies. The approach also has a limitation with networks having partial-state coupling. Other than giving yet another MMA, we introduce a new and, in some cases, optimal coordinate transformation to study such networks. Our approach fixes all the drawbacks of CGS and MMA. In addition, by merely checking the structure of the vector field of the individual oscillator, we shall be able to determine if the system is globally synchronized. In summary, our results can be applied to rather general time-varying networks with a large number of nodes.

Generalized outer synchronization between complex dynamical networks

In this paper, the problem of generalized outer synchronization between two completely different complex dynamical networks is investigated. With a nonlinear control scheme, a sufficient criterion for this generalized outer synchronization is derived based on Barbalat's lemma. Two corollaries are also obtained, which contains the situations studied in two lately published papers as special cases. Numerical simulations further demonstrate the feasibility and effectiveness of the theoretical results.

Critical lattice size limit for synchronized chaotic state in one- and two-dimensional diffusively coupled map lattices

We consider diffusively coupled map lattices with P neighbors (where P is arbitrary) and study the stability of the synchronized state. We show that there exists a critical lattice size beyond which the synchronized state is unstable. This generalizes earlier results for nearest neighbor coupling. We confirm the analytical results by performing numerical simulations on coupled map lattices with logistic map at each node. The above analysis is also extended to two-dimensional P -neighbor diffusively coupled map lattices.

Synchronization in time-varying networks: a matrix measure approach

Synchronization in complex networks has attracted lots of interest in various fields. We consider synchronization in time-varying networks, in which the weights of links are time varying. We propose a useful approach--i.e., the matrix measure approach--to derive some analytically sufficient conditions for synchronization in time-varying networks. These conditions are less conservative than many existing synchronization conditions. Theoretical analysis and numerical simulations of different networks verify our main results.

Synchronization in networks with random interactions: theory and applications

Synchronization is an emergent property in networks of interacting dynamical elements. Here we review some recent results on synchronization in randomly coupled networks. Asymptotical behavior of random matrices is summarized and its impact on the synchronization of network dynamics is presented. Robert May's results on the stability of equilibrium points in linear dynamics are first extended to systems with time delayed coupling and then nonlinear systems where the synchronized dynamics can be periodic or chaotic. Finally, applications of our results to neuroscience, in particular, networks of Hodgkin-Huxley neurons, are included.

Synchronization in asymmetrically coupled networks with node balance

We study global stability of synchronization in asymmetrically connected networks of limit-cycle or chaotic oscillators. We extend the connection graph stability method to directed graphs with node balance, the property that all nodes in the network have equal input and output weight sums. We obtain the same upper bound for synchronization in asymmetrically connected networks as in the network with a symmetrized matrix, provided that the condition of node balance is satisfied. In terms of graphs, the symmetrization operation amounts to replacing each directed edge by an undirected edge of half the coupling strength. It should be stressed that without node balance this property in general does not hold.

Bifurcation and chaos in coupled ratchets exhibiting synchronized dynamics

The bifurcation and chaotic behavior of unidirectionally coupled deterministic ratchets is studied as a function of the driving force amplitude and frequency . A classification of the various types of bifurcations likely to be encountered in this system was done by examining the stability of the steady state in linear response as well as constructing a two-parameter phase diagram in the plane. Numerical explorations revealed varieties of bifurcation sequences including quasiperiodic route to chaos. Besides, the familiar period-doubling and crises route to chaos exhibited by the one-dimensional ratchet were also found. In addition, the coupled ratchets display symmetry-breaking, saddle-nodes and bubbles of bifurcations. Chaotic behavior is characterized by using the Lyapunov exponent spectrum; while a perusal of the phase space projected in the Poincaré cross section confirms some of the striking features.

Desynchronized wave patterns in synchronized chaotic regions of coupled map lattices

We analyze the size limits of coupled map lattices with diffusive coupling at the crossover of low-dimensional to high-dimensional chaos. We investigate the existence of standing-wave-type periodic patterns, within the low-dimensional limit, in addition to the stable synchronous chaotic states depending upon the initial conditions. Further, we bring out a controlling mechanism to explain the emergence of standing-wave patterns in the coupled map lattices. Finally, we give an analytic expression in terms of the unstable periodic orbits of the isolated map to represent the standing-wave patterns.

Synchronized clusters in coupled map networks. II. Stability analysis

We study self-organized and driven synchronization in some simple coupled map networks, namely globally coupled networks and complete bipartite networks, using both linear stability analysis and Lyapunov function approach and determine stability conditions for synchronization. The phase diagrams for the networks studied here have features very similar to the different kinds of structurally similar networks studied in Part I. Lyapunov function approach shows that when any two nodes are in driven synchronization, all the coupling terms in the difference between the variables of these two nodes cancel out, whereas when they are in self-organized synchronization, the direct coupling term between the two nodes adds an extra term while the other couplings cancel out. We also discuss the conditions for the occurrence of a floating node and suggest that the fluctuations of the conditional Lyapunov exponent about zero can be a criterion for its occurrence.

Does synchronization of networks of chaotic maps lead to control?

We consider networks of chaotic maps with different network topologies. In each case, they are coupled in such a way as to generate synchronized chaotic solutions. By using the methods of control of chaos we are controlling a single map into a predetermined trajectory. We analyze the reaction of the network to such a control. Specifically we show that a line of one-dimensional logistic maps that are unidirectionally coupled can be controlled from the first oscillator whereas a ring of diffusively coupled maps cannot be controlled for more than 5 maps. We show that rings with more elements can be controlled if every third map is controlled. The dependence of unidirectionally coupled maps on noise is studied. The noise level leads to a finite synchronization lengths for which maps can be controlled by a single location. A two-dimensional lattice is also studied.

Synchronization and symmetry-breaking bifurcations in constructive networks of coupled chaotic oscillators

The spatiotemporal dynamics of networks based on a ring of coupled oscillators with regular shortcuts beyond the nearest-neighbor couplings is studied by using master stability equations and numerical simulations. The generic criterion for dynamic synchronization has been extended to arbitrary network topologies with zero row-sum. The symmetry-breaking oscillation patterns that resulted from the Hopf bifurcation from synchronous states are analyzed by the symmetry group theory.