- About this Journal ·
- Abstracting and Indexing ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Recently Accepted Articles ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents

Mathematical Problems in Engineering

Volume 2012 (2012), Article ID 106830, 9 pages

http://dx.doi.org/10.1155/2012/106830

## Lag Synchronization of Coupled Multidelay Systems

Information Security Center, Beijing University of Posts and Telecommunications, P.O. Box 145, Beijing 100876, China

Received 23 February 2012; Revised 15 April 2012; Accepted 20 May 2012

Academic Editor: Mohamed A. Zohdy

Copyright © 2012 Luo Qun et al. 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.

#### Abstract

Chaos synchronization is an active topic, and its possible applications have been studied extensively. In this paper we present an improved method for lag synchronization of chaotic systems with coupled multidelay. The Lyapunov theory is used to consider the sufficient condition for synchronization. The specific examples will demonstrate and verify the effectiveness of the proposed approach.

#### 1. Introduction

Since synchronization of chaotic systems was first realized by Fujisaka and Yamada [1] and Pecora and Carroll [2], chaos synchronization has received increasing interest and has become an active research topic. Currently its possible applications in various fields are in great interest, for example, applications to control theory [3], telecommunications [4–7], biology [8, 9], lasers [10, 11], secure communications [12], and so on.

Roughly speaking, chaotic communication schemes rely on the synchronization technique: the information signal is mixed at the master, and a driving signal is then generated and is sent to the slave; as a result, their chaotic trajectories remain in step with each other during temporal evolution. Besides identical synchronization [13], several new types of chaos synchronization of coupled oscillators have occurred, that is, generalized synchronization [12], phase synchronization [14], lag synchronization [15], anticipation synchronization [16], and projective synchronization [17].

Lag synchronization can be realized when the strength of the coupling between phase-synchronized oscillators is increased. There, the driving signal is constituted by the sum of multiple nonlinear transformations of delayed state variable [18]. Master and slave's formulas are in the form of single delay [19, 20] and multidelay [21–23]. From the application point of view, this new multidelay synchronization, different from conventional synchronization without lag, offers a significant advantage in terms of security of communication. Since the constructed state variable of the master system with lag becomes more complex than that of the conventional system, multilag systems achieve high security. Intruders cannot reconstruct the attractors of driving signal by using conventional reconstruction methods [24, 25] so as not to decipher the transferred message.

In the present paper, we proposed a systematic and rigorous scheme for lag synchronization of coupled multidelay systems based on the Lyapunov stability theory. Furthermore, the zero solution of lag synchronization differential equation is globally asymptotically stable. The effectiveness of the proposed scheme is confirmed by the numerical simulation of specific example.

#### 2. The Schemes of Lag Synchronization

##### 2.1. The Proposed Lag Synchronization Model

Lag synchronization was first investigated by Rosenblum [15], and it can be considered that the state variable of the slave is delayed by the positive time lag in comparison with that of the master while their amplitudes follow each other, that is, .

We consider the following model of lag synchronization.

Master:

Driving signal:

Slave: where coefficients , , and are positive integers. State variables , and are three continuous nonlinear functions. The driving signal in (2.2) is constituted by the sum of multiple nonlinear transformations of delayed state variable; is added with . The polynomial is added to the right side of , forming the slave equation which is shown as (2.3).

##### 2.2. Proof for the Lag Synchronization Model

The desired synchronization manifold is expressed by the following relation as , where is a lag time.

We choose suitable to satisfy as .

*Assumption 2.1. *, where is integer. When , ,

*Assumption 2.2. *When , ,
When , ,

*Assumption 2.3. *Nonlinearity satisfies Lipshitz condition; that is, there exists a positive constant for all time variables and , such that .

Here we give the sufficient condition for system synchronization.

Theorem 2.4. *If the system (2.1), (2.2), and (2.3) satisfies Assumptions 2.1, 2.2, and 2.3 and if
**
then .*

*Proof. *The dynamics of synchronization error is

By applying Assumption 2.1, (2.8) can be rewritten as

By applying Assumption 2.2, if and , we have
where as well as synchronization established, in fact, reduces during establishing the synchronization regime.

From (2.10) and (2.9) we get

Define a Lyapunov function [26] as

Then, we obtain

Here, we have

According to Assumption 2.1, we have

In our model, can be rewritten as . By Assumption 2.3, we have

According to , where , we get

Finally, we obtain

The proof is completed.

*Note 1. *The advantages of our lag synchronization model are as follows.(1)The nonlinear function satisfies , so the zero solution of lag synchronization error system is globally asymptotically stable. The condition for synchronization is easy to be realized.(2)We can choose nonlinear function in many ways, and vary as changes. Moreover, the format of function can be different even if is the same value.(3)In order to enhance the complexity of the system, can be different positive integers, and the number of multiple time delays can be chosen as many values.

#### 3. Numerical Simulations

The following example will demonstrate synchronization between systems with multidelay. Functions of systems are chosen from the set of . Let us consider synchronization model with the master's and slave's equations defined as.

Master:

Slave: Therefore the equation for driving signal is chosen as.

Driving signal: where , , , satisfy .

According to (2.4)–(2.6), the relation of the delays and parameters is expressed as , , , , , , , , , , , , , .

The value of parameters for simulation is adopted as , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , .

In Figure 1, the portrait of versus illustrates that the lag synchronization of coupled partly nonidentical systems is established. However their trajectories do not remain in step with each other during a short part of evolution, because they are not in synchronization as .

It is clear to observe from Figure 2 that synchronization error leaps at a sudden as and vanishes eventually in a short time. Then stays at zero.

As shown in Figure 3, the slave's state variable is retarded with the time length of in comparison with master's. The desired lag synchronization is realized.

#### 4. Conclusions

In this paper, we have presented a lag synchronization model as well as researched on it. Based on Lyapunov theory, the sufficient conditions of the synchronization model are given. Simulation results of the lag synchronization model are provided to illustrate the effectiveness and feasibility of the proposed method.

#### Acknowledgments

This work is supported by the National Natural Science Foundation of China (Grants nos. 61170269, 61121061), the Specialized Research Fund for the Doctoral Program of Higher Education (Grant no. 20100005110002) and the Fundamental Research Funds for the Central Universities (Grant no. BUPT2011RC0211).

#### References

- H. Fujisaka and T. Yamada, “Stability theory of synchronized motion in coupled-oscillator systems,”
*Progress of Theoretical Physics*, vol. 69, no. 1, pp. 32–47, 1983. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - L. M. Pecora and T. L. Carroll, “Synchronization in chaotic systems,”
*Physical Review Letters*, vol. 64, no. 8, pp. 821–824, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - K. Pyragas, “Continuous control of chaos by self-controlling feedback,”
*Physics Letters A*, vol. 170, no. 6, pp. 421–428, 1992. View at Scopus - K. M. Cuomo and A. V. Oppenheim, “Circuit implementation of synchronized chaos with applications to communications,”
*Physical Review Letters*, vol. 71, no. 1, pp. 65–68, 1993. View at Publisher · View at Google Scholar · View at Scopus - C. W. Wu and L. O. Chua, “A unified framework for synchronization and control of dynamical systems,”
*International Journal of Bifurcation and Chaos in Applied Sciences and Engineering*, vol. 4, no. 4, pp. 979–998, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - R. Brown, N. F. Rulkov, and E. R. Tracy, “Modeling and synchronizing chaotic systems from time-series data,”
*Physical Review E*, vol. 49, no. 5, pp. 3784–3800, 1994. View at Publisher · View at Google Scholar · View at Scopus - L. Kocarev and U. Parlitz, “General approach for chaotic synchronization with applications to communication,”
*Physical Review Letters*, vol. 74, no. 25, pp. 5028–5031, 1995. View at Publisher · View at Google Scholar · View at Scopus - S. K. Han, C. Kurrer, and Y. Kuramoto, “Dephasing and bursting in coupled neural oscillators,”
*Physical Review Letters*, vol. 75, no. 17, pp. 3190–3193, 1995. View at Publisher · View at Google Scholar · View at Scopus - R. C. Elson, A. I. Selverston, R. Huerta, N. F. Rulkov, M. I. Rabinovich, and H. D. I. Abarbanel, “Synchronous behavior of two coupled biological neurons,”
*Physical Review Letters*, vol. 81, no. 25, pp. 5692–5695, 1998. View at Scopus - L. Fabiny, P. Colet, R. Roy, and D. Lenstra, “Coherence and phase dynamics of spatially coupled solid-state lasers,”
*Physical Review A*, vol. 47, no. 5, pp. 4287–4296, 1993. View at Publisher · View at Google Scholar · View at Scopus - W. Yu, J. Cao, K.-W. Wong, and J. Lü, “New communication schemes based on adaptive synchronization,”
*Chaos*, vol. 17, no. 3, p. 033114, 13, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - N. F. Rulkov, M. M. Sushchik, L. S. Tsimring, and H. D. I. Abarbanel, “Generalized synchronization of chaos in directionally coupled chaotic systems,”
*Physical Review E*, vol. 51, no. 2, pp. 980–994, 1995. View at Publisher · View at Google Scholar · View at Scopus - D. Huang and R. Guo, “Identifying parameter by identical synchronization between different systems,”
*Chaos*, vol. 14, no. 1, pp. 152–159, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. G. Rosenblum, A. S. Pikovsky, and J. Kurths, “Phase synchronization of chaotic oscillators,”
*Physical Review Letters*, vol. 76, no. 11, pp. 1804–1807, 1996. View at Scopus - M. G. Rosenblum, A. S. Pikovsky, and J. Kurths, “From phase to lag synchronization in coupled chaotic oscillators,”
*Physical Review Letters*, vol. 78, no. 22, pp. 4193–4196, 1997. View at Scopus - H. U. Voss, “Anticipating chaotic synchronization,”
*Physical Review E*, vol. 61, no. 5 A, pp. 5115–5119, 2000. View at Scopus - R. Mainieri and J. Rehacek, “Projective synchronization in three-dimensional chaotic systems,”
*Physical Review Letters*, vol. 82, no. 15, pp. 3042–3045, 1999. View at Scopus - K. Pyragas, “Syncronization of coupled timedelaysystems: analytical estimations,”
*Physical Review E*, vol. 58, pp. 3067–3071, 1998. - E. M. Shahverdiev, S. Sivaprakasam, and K. A. Shore, “Lag synchronization in time-delayed systems,”
*Physics Letters, Section A*, vol. 292, no. 6, pp. 320–324, 2002. View at Publisher · View at Google Scholar · View at Scopus - E. M. Shahverdiev, S. Sivaprakasam, and K. A. Shore, “Lag times and parameter mismatches in synchronization of unidirectionally coupled chaotic external cavity semiconductor lasers,”
*Physical Review E*, vol. 66, no. 3, Article ID 037202, pp. 037202/1–037202/4, 2002. View at Publisher · View at Google Scholar · View at Scopus - E. M. Shahverdiev, R. A. Nuriev, R. H. Hashimov, and K. A. Shore, “Chaos synchronization between the Mackey-Glass systems with multiple time delays,”
*Chaos, Solitons and Fractals*, vol. 29, no. 4, pp. 854–861, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - C. W. Wu and L. O. Chua, “A simple way to synchronize chaotic systems with applications to secure communication systems,”
*International Journal of Bifurcation and Chaos*, vol. 3, no. 6, pp. 1619–1627, 1993. - T. M. Hoang and M. Nakagawa, “Enhancing security for chaos-based communication system with change in synchronization manifolds' delay and in encoder's parameters,”
*Journal of the Physical Society of Japan*, vol. 75, no. 6, Article ID 064801, 2006. View at Publisher · View at Google Scholar · View at Scopus - T. M. Hoang, D. T. Minh, and M. Nakagawa, “Chaos synchronization of multi-delay feedback systems with multi-delay driving signal,”
*Journal of the Physical Society of Japan*, vol. 74, no. 8, pp. 2374–2378, 2005. View at Publisher · View at Google Scholar · View at Scopus - T. M. Hoang and M. Nakagawa, “Synchronization of coupled nonidentical multidelay feedback systems,”
*Physics Letters, Section A*, vol. 363, no. 3, pp. 218–224, 2007. View at Publisher · View at Google Scholar · View at Scopus - H. K. Khalil,
*Nonlinear Systems*, Prentice Hall, 3rd edition, 2002.