Generalized connection graph method for synchronization in asymmetrical networks

doi:10.1016/j.physd.2006.09.014

Physica D: Nonlinear Phenomena

Volume 224, Issues 1–2, December 2006, Pages 42-51

https://doi.org/10.1016/j.physd.2006.09.014 Get rights and content

Abstract

We present a general framework for studying global complete synchronization in networks of dynamical systems with asymmetrical connections. We extend the connection graph stability method, originally developed for symmetrically coupled networks, to the general asymmetrical case. The principal new component of the method is the transformation of the directed connection graph into an undirected graph. In our method for symmetrically coupled networks we have to choose a path between each pair of nodes. The extension of the method to asymmetrical coupling consists in symmetrizing the graph and associating a weight to each path. This weight involves the “node unbalance” of the two nodes. This quantity is defined to be the difference between the sum of connection coefficients of the outgoing edges and the sum of the connection coefficients of the incoming edges to the node. The synchronization condition for this symmetrized-and-weighted network then also guarantees synchronization in the original asymmetrical network.

Introduction

The increasing interest in synchronization in limit-cycle and chaotic dynamical systems [1], [2], [3] has led many researchers to consider the phenomenon of synchronization in large complex networks of coupled oscillators (see, e.g. [4], [5] for a sampling of this large field).

Much of this research has been inspired by technological and biological examples, including coupled synchronized lasers [6], [7], networks of computer clocks [8], and synchronized neuronal firing [9], [10]. Networks of identical or slightly non-identical oscillators often synchronize, or else form synchronous patterns that depend on the symmetry of the underlying network [11], [12].

The strongest form of synchrony in oscillator networks is complete synchronization when all oscillators do the same thing at the same time. An important problem in the study of complete synchrony is how the stability of a synchronized behavior, where the behavior could be a fixed point, a limit cycle or a chaotic attractor, is influenced by the network topology and kind of interaction. This problem was intensively studied for networks of biological oscillators [13], [14], [15], [16], and more generally of limit-cycle and chaotic oscillators[17], [18], [19], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], [43], [44].

Most methods for determining stability of synchronization in linearly coupled networks of chaotic systems are based on the calculation of the eigenvalues of the connection matrix and a term depending mainly on the dynamics of the individual oscillators (see, e.g., [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27]). Pecora and Carroll [22] developed a general approach to the local stability of complete synchronization for any linear coupling network architecture. This approach, called the Master Stability function, is based on the calculation of the maximum Lyapunov exponent for the least stable transversal mode of the synchronous manifold and the eigenvalues of the connection matrix. This powerful method is widely used in local stability studies of synchronization in complex oscillator networks [28], [29], [30], [31], [32], [33]. Global stability results based on the calculation of the connection matrix eigenvalues were also derived for oscillator networks coupled via undirected [34], [35] and directed graphs [36]. These studies show that both local and global stabilities of complete synchronization depend on the eigenvalues of the Laplacian connection matrix.

We have previously developed an alternate way to establish synchrony which does not depend on explicit knowledge of the spectrum of the connection matrix [40]. This connection graph method combines the Lyapunov function approach with graph theoretical reasoning. It guarantees complete synchronization from arbitrary initial conditions and not just local stability of the synchronization manifold. It is also applicable to time-dependent networks. This approach was originally developed for undirected graphs and applied to global synchronization in complex networks [41], [42]. More recently, we showed that the method can be directly applied to asymmetrically coupled networks with node balance [43], [44]. Node balance means that the sum of the coupling coefficients of all edges directed to a node equals the sum of the coupling coefficients of all the edges directed outward from the node. We proved that for node balanced networks it is sufficient to symmetrize all connections by replacing a unidirectional coupling with a bidirectional coupling of half the coupling strength. The bound for global synchronization in this undirected network then holds also for the original directed network.

In this paper we extend our approach to networks with arbitrary asymmetrical connections. The connection graph of such a network is directed and the coupling coefficient from node $i$ to node $j$ is in general different from the coupling coefficient for the reverse direction. The new ingredient of the method is the transformation of the directed connection graph into an undirected weighted graph. This is done by symmetrizing the graph and associating a weight to each edge of the undirected graph and to each path between any two nodes. This weight involves the “node unbalance” of the two nodes. This quantity is defined to be the difference between the sum of connection coefficients of the outgoing edges and the sum of the connection coefficients of the incoming edges to the node. As in the case of node-balanced networks, the synchronization criterion derived for this symmetrical network then guarantees synchronization in the asymmetrical directed network.

The layout of this paper is as follows. First, in Section 2, we state the problem under consideration. Then, in Section 3, we derive a graph-based criterion for global synchronization in asymmetrically coupled networks and formulate the main theorem of our method. In Section 4, we show how to apply the generalized connection graph method to several examples of concrete networks. We start with the simplest network of two unidirectionally coupled oscillators, then we continue with a star-configuration and a directed network with an irregular topology. In Section 5, a brief discussion of the obtained results is given.

Section snippets

Systems under study

We consider a network of $n$ interacting nonlinear $l$ -dimensional dynamical systems (oscillators). We assume that the individual oscillators are all identical, even though our results can be generalized to slightly non-identical systems. The composed dynamical system is described by the $n \times l$ ordinary differential equations ${\dot{x}}_{i} = F (x_{i}) + \sum_{k = 1}^{n} d_{i k} (t) P x_{k}, i = 1, \dots, n,$ where $x_{i} = (x_{i}^{1}, \dots, x_{i}^{l})$ is the $l$ -vector containing the coordinates of the $i$ th oscillator, the function $F : R^{l} \to R^{l}$ is nonlinear and capable of

Stability system for the difference variables

Since we are interested in complete synchronization, we introduce the difference variables $X_{i j} = x_{j} - x_{i}$ for any $i$ and $j$ . Similarly to our previous works [40], [43], we can write the stability system for the difference variables ${\dot{X}}_{i j} = F (x_{j}) - F (x_{i}) + \sum_{k = 1}^{n} {d_{j k} P X_{j k} - d_{i k} P X_{i k}}, i, j = 1, \dots, n .$ The function difference $F (x_{j}) - F (x_{i})$ can be rewritten in a compact vector form $F (x_{j}) - F (x_{i}) = [\int_{0}^{1} D F (β x_{j} + (1 - β) x_{i}) d β] X_{i j},$ where $D F$ is an $l \times l$ Jacobi matrix of $F$ . Hence, we obtain ${\dot{X}}_{i j} = [\int_{0}^{1} D F (β x_{j} + (1 - β) x_{i}) d β] X_{i j} + \sum_{k = 1}^{n} {d_{j k} P X_{j k} - d_{i k} P X_{i}$

Examples: Application of the method

To find an upper bound for the synchronization threshold in concrete networks, we should follow the steps of the above study.

Conclusions

We have given a sufficient condition for global complete synchronization in an arbitrary network of diffusively coupled identical dynamical systems. The condition is composed of a set of inequalities which have to be satisfied, one inequality for each edge of the connection graph. Each inequality involves a term that depends only on the individual dynamical systems, namely the coupling strength that guarantees global synchronizing of two systems. The other terms of the inequality depend only on

Acknowledgments

I.B. acknowledges the financial support of the Georgia State University Research Initiation Program (Grant FY07) and a Cariplo Foundation fellowship. V.B. acknowledges the support from the RFBR (grant No. 05-01-00509), NWO-RFBR (grant No. 047-017-018), and RFBR-MF (grant No. 05-02-19815). M.H. acknowledges the support from the SNSF (grant No. 200021-112081) and EU Commission (FP6-NEST project N 517133).

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Generalized connection graph method for synchronization in asymmetrical networks

Abstract

Introduction

Section snippets

Systems under study

Stability system for the difference variables

Examples: Application of the method

Conclusions

Acknowledgments

Math. Biosci.

Physica D

Physica D

Physica D

Physica D

Prog. Theor. Phys.

Radiophys. Quantum Electron.

Phys. Rev. Lett.

Phys. Rev. A

Nature

IEEE Trans. Communications

Proc. Natl. Acad. Sci. USA

Nonlinearity

SIAM J. Appl. Dyn. Syst.

Bull. Amer. Math. Soc.

Physica D

Phys. Rev. Lett.

IEEE Trans. Circ. Syst. -I: Fundam. Theory Appl.

Phys. Rev. Lett.

Chaos