- Research Article
- Open Access

# Global Phase Synchronization for a Class of Dynamical Complex Networks with Time-Varying Coupling Delays

- XinBin Li
^{1}and - Haiyan Jing
^{1}Email author

**2010**:295805

https://doi.org/10.1155/2010/295805

© XinBin Li and Haiyan Jing. 2010

**Received:**25 September 2010**Accepted:**22 November 2010**Published:**2 December 2010

## Abstract

Global phase synchronization for a class of dynamical complex networks composed of multiinput multioutput pendulum-like systems with time-varying coupling delays is investigated. The problem of the global phase synchronization for the complex networks is equivalent to the problem of the asymptotical stability for the corresponding error dynamical networks. For reducing the conservation, no linearization technique is involved, but by Kronecker product, the problem of the asymptotical stability of the high dimensional error dynamical networks is reduced to the same problem of a class of low dimensional error systems. The delay-dependent criteria guaranteeing global asymptotical stability for the error dynamical complex networks in terms of Liner Matrix Inequalities (LMIs) are derived based on free-weighting matrices technique and Lyapunov function. According to the convex characterization, a simple criterion is proposed. A numerical example is provided to demonstrate the effectiveness of the proposed results.

## Keywords

- Phase Synchronization
- Global Asymptotical Stability
- Dynamical Complex Network
- Global Synchronization
- Dimensional Error

## 1. Introduction

Over the recent decades, dynamical complex networks are increasingly used to model a variety of phenomena of nature in power system, biological system, traffic system, and so on [1]. Many of these networks exhibit complexity in the overall topological properties and dynamical properties of the network nodes and the coupled units. The complex nature of the networks has resulted in a series of important research problems. In particular, one significant and interesting phenomenon is the synchronization of all its dynamics.

The pendulum-like system is a special kind of nonlinear system with periodic nonlinearity and multiple equilibria [2]. In practical engineering, there are many kinds of nonlinear pendulum-like system, such as phase-locked loops and various synchronous machines. With the development of the modern industry and control technique, all kinds of rotating electrical machines play more and more important roles in industry. Therefore, the pendulum-like system is worth being researched not only of academic significant, but also of practical value.

Recently, the coupled pendulum-like systems attract more and more researchers' attentions. Anticipating synchronization in a class of nonlinear dynamical systems is investigated in [3]. In [4], the global asymptotical stability and generalized synchronization of phase synchronous dynamical networks composed of multiinput multioutput pendulum-like systems via linear interconnections are investigated. Of particular note is that the global synchronization of the dynamical complex network composed of the pendulum-like systems is different from that of the general complex networks. The global synchronization of the dynamical complex networks composed of the pendulum-like systems is defined as phase synchronization introduced in [4]. But all of literatures above are not involving the coupling delays. However, time delay is unavoidably encountered, and it is the main cause of instability and poor performance of a system. Besides, the time-varying delays should be considered because they are more general than the constant cones. Thus, it is important and necessary to study the global synchronization of the dynamical complex networks composed of pendulum-like systems with time-varying coupling delays. In fact, the synchronization of the dynamical complex networks can be transformed into the global asymptotical stability of the corresponding error dynamical systems. In this paper, through studying the asymptotical stability of the corresponding error dynamical networks, several criteria guaranteeing the global phase synchronization of the dynamical network composed of multiinput multioutput pendulum-like systems with time-varying coupling delays are given. The effectiveness of the proved results is illustrated by a concrete example.

The rest of this paper is organized as follows. In Section 2, some preliminary results necessary for successive development are introduced. Section 3 contains our main results. In this section, we give some criteria guaranteeing the phase synchronization of the dynamical complex networks composed of the multiinput multioutput pendulum-like systems with time-varying coupling delays. The effectiveness of the proposed results is illustrated with a numerical example given in Section 4, and a brief conclusion is given in Section 5.

If not explicitly stated, matrices are assumed to have compatible dimensions.

## 2. Preliminaries

where variables and denote the state variables. , , , and are constant matrices. The continuously differentiable vector function and is -periodic with finite number of zeros on the interval ( ). The system equation (2.1) with -periodic is called a pendulum-like system.

Proposition 2.1 (see [2]).

If the solution of the pendulum-like system (2.1) is bounded, then the functions ( ), where belongs to a solution of (2.1), are uniformly continuous on .

The validity of this assertion follows from the facts that is locally Lipschitz continuous and is bounded on .

Lemma 2.2 (see [2]).

Lemma 2.3 (see [2]).

## 3. Main Results

Lemma 3.1 (Wu [5]).

The eigenvalues of an irreducible matrix with ( ) satisfy the following properties.

(i)0 is an eigenvalue of associated the eigenvector .

(ii)If for , , and , then the real parts of all eigenvalues of are less than or equal to 0, and all possible eigenvalues with zero part are the real eigenvalue 0. In fact, 0 is an eigenvalue of with multiplicity 1.

where is the maximum eigenvalue of multiply 1, and is the eigenvalue of multiply ( ) satisfying and .

Definition 3.2 (see [4]).

Under such circumstances, system (3.7) could be regarded as a pendulum-like system with state delay. Thus, the synchronization problem of the dynamical network (3.1) can be transformed into the global asymptotical stability problem of the corresponding error dynamical system.

Theorem 3.3.

where , and is a selected orthogonal matrix satisfying , where is defined as (3.3). Then the delayed pendulum-like system (3.7) with time-varying coupling delay satisfying (3.2) is global asymptotic, stable and the corresponding dynamical network (3.1) achieves phase synchronization.

Proof.

Since , , then the last three parts are all less than 0. So if , then .

The conditions (3.40) and (3.43) show that every solution of the pendulum-like system (3.15) converges to a certain equilibrium with ( ). Namely, the pendulum-like system (3.7) is global asymptotic stable.

Remark 3.4.

It is shown from the formula (3.3) that the coupling matrix has different eigenvalues. Therefore, it is just needed to examine LMIs groups in (3.9) and (3.10). In addition, according to the convex properties of LMI [7], groups of LMIs corresponding to can be written as a linear combination of the tow groups of LMIs with the second-maximum and the minimum eigenvalue . Therefore, above criterion only needs to examine three groups of LMIs corresponding to the largest, second largest, and the smallest distinct eigenvalues of , respectively. Furthermore, note that the system (3.1) with just corresponds to the synchronous manifold, which is not required to be verified. Hence, if (3.9) and (3.10) hold for and , the nonlinear pendulum-like dynamical network will achieve phase synchronization.

Corollary 3.5.

where , , , , , , , , and defined as the Theorem 3.3. Then, the dynamical network (3.1) with time-varying coupling delay satisfying (3.2) achieves phase synchronization.

## 4. Numerical Example

The example given in this section is based on concrete systems studied in the theory of interconnected phase-locked loops (PLLs), which are frequently observed in electrical and engineering aspects. PLL could be treated as a representative for pendulum-like system, where model is described by (2.1) after certain simplifications [8].

## 5. Conclusion

In this paper, the effects of interconnections between two independent second-order pendulum-like systems have been investigated. Linear interconnection and a class of input and output interconnections have been involved. Some frequency domain and LMI conditions of dichotomy for interconnected pendulum-like systems have been established. Examples show that input and output interchange presented here can result in great changes in some practical systems. For example, chaotic phenomenon in partial variables may appear by adding interconnections between two independent second-order pendulum-like systems which are dichotomous. Since the solution is unbounded, there is no chaotic phenomenon in plane phase space. However, chaotic phenomenon appears on the cylindrical surface of cylindrical phase space, here we call it the chaos on cylindrical surface, which was never studied by now. It shows the complexity of physical property in concrete systems even they are dichotomous. This also indicates that it is possible for the existence of chaotic attractors in pendulum-like systems.

## Declarations

### Acknowledgments

This work is supported by National Natural Science Foundation of China under Grant no. 60874026 and Natural Science Foundation of Heibei province, China under Grant no. 07M007.

## Authors’ Affiliations

## References

- Wang XF, Xiang L, Chen GR:
*The Theory and Application of Complex Network*. Tsinghua University, Beijing, China; 2006.Google Scholar - Leonov GA, Ponomarenko DV, Smirnova VB:
*Frequency-Domain Methods for Nonlinear Analysis. Theory and Applications, World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises*.*Volume 9*. World Scientific, River Edge, NJ, USA; 1996:xii+498.MATHGoogle Scholar - Xu S, Yang Y: Predicting dynamic behavior via anticipating synchronization in coupled pendulum-like systems.
*Journal of Physics A*2009, 42(33):-15.View ArticleMathSciNetMATHGoogle Scholar - Xu S, Yang Y: Global asymptotical stability and generalized synchronization of phase synchronous dynamical networks.
*Nonlinear Dynamics*2010, 59(3):485–496. 10.1007/s11071-009-9555-3MathSciNetView ArticleMATHGoogle Scholar - Wu CW:
*Synchronization in Coupled Chaotic Circuits and Systems, World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises*.*Volume 41*. World Scientific, River Edge, NJ, USA; 2002:xii+174.Google Scholar - Wu CW, Chua LO: Application of Kronecker products to the analysis of systems with uniform linear coupling.
*IEEE Transactions on Circuits and Systems I*1995, 42(10):775–778. 10.1109/81.473586MathSciNetView ArticleGoogle Scholar - Boyd S, ELGhaoui L, Feron E, Balakrishnam V:
*Linear Matrix Inequalities in Systems and Control*. SIAM, Philadelphia, Pa, USA; 1994.View ArticleGoogle Scholar - Yang Y, Fu R, Huang L: Robust analysis and synthesis for a class of uncertain nonlinear systems with multiple equilibria.
*Systems & Control Letters*2004, 53(2):89–105. 10.1016/j.sysconle.2004.02.024MathSciNetView ArticleMATHGoogle Scholar - Duan Z, Wang J-Z, Huang L: Special decentralized control problems and effectiveness of parameter-dependent Lyapunov function method.
*Proceedings of the American Control Conference (ACC '05), July 2005, Portland, Ore, USA*3: 1697–1702.Google Scholar - Yang Y, Duan Z, Huang L: Nonexistence of periodic solutions in a class of dynamical systems with cylindrical phase space.
*International Journal of Bifurcation and Chaos in Applied Sciences and Engineering*2005, 15(4):1423–1431. 10.1142/S0218127405012648MathSciNetView ArticleMATHGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.