Lag Synchronization of Coupled Multidelay Systems

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 𝜏_𝑑 in comparison with that of the master while their amplitudes follow each other, that is, lim_𝑡→∞|𝑦(𝑡)−𝑥(𝑡−𝜏_𝑑)|=0.

We consider the following model of lag synchronization.

Master: 𝑑𝑥(𝑡)(𝑡)+𝑑𝑡=−𝛼𝑥^𝑃_𝑖=1𝑚_𝑖𝑓_𝑖𝑥𝑡−𝛼_𝑖. (2.1)

Driving signal: DS(𝑡)=^𝑄_𝑖=1𝑘_𝑖𝑔_𝑖𝑥𝑡−𝛽_𝑖+𝑊𝑥𝑡−𝜏_𝑑. (2.2)

Slave: 𝑑𝑦(𝑡)(𝑡)+𝑑𝑡=−𝛼𝑦^𝑅_𝑖=1𝑛_𝑖ℎ_𝑖𝑦𝑡−𝛾_𝑖+DS(𝑡)−𝑊𝑦(𝑡), (2.3) where coefficients 𝛼,𝑚_𝑖,𝑘_𝑖,𝑛_𝑖,𝛼_𝑖,𝛽_𝑖,𝛾_𝑖, 𝑊∈ℜ, and 𝑃,𝑄,𝑅 are positive integers. State variables 𝑥,𝑦∈ℜ, and 𝑓_𝑖(⋅),𝑔_𝑖(⋅),ℎ_𝑖(⋅)∈ℜ→ℜ are three continuous nonlinear functions. The driving signal DS(𝑡) in (2.2) is constituted by the sum of multiple nonlinear transformations of delayed state variable; ∑^𝑄_𝑖=1𝑘_𝑖𝑔_𝑖[𝑥(𝑡−𝛽_𝑖)] is added with 𝑊𝑥(𝑡−𝜏_𝑑). The polynomial −𝑊𝑦(𝑡) is added to the right side of 𝑑𝑦(𝑡)/𝑑𝑡=−𝛼𝑦(𝑡)+∑^𝑅_𝑖=1𝑛_𝑖ℎ_𝑖[𝑦(𝑡−𝛾_𝑖)]+DS(𝑡), 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 𝑦(𝑡)→𝑥(𝑡−𝜏_𝑑) as 𝑡→∞, where 𝜏_𝑑 is a lag time.

We choose suitable DS(𝑡) to satisfy 𝑒(𝑡)=𝑦(𝑡)−𝑥(𝑡−𝜏_𝑑)→0 as 𝑡→∞.

Assumption 2.1. 𝑄=𝑃+𝑅−𝐼, 𝐼<min{𝑃,𝑄,𝑅} where 𝐼 is integer. When 𝑖=1,…,𝐼, 𝑓_𝑖≡𝑔_𝑖≡ℎ_𝑖, 𝛾_𝑖=𝛼_𝑖,𝛽_𝑖=𝛼_𝑖+𝜏_𝑑,𝑘_𝑖=𝑚_𝑖−𝑛_𝑖. (2.4)

Assumption 2.2. When 𝑗=1,…,(𝑃−𝐼), 𝑔_𝐼+𝑗≡𝑓_𝐼+𝑗, 𝑘_𝐼+𝑗=𝑚_𝐼+𝑗,𝛽_𝐼+𝑗=𝛼_𝐼+𝑗+𝜏_𝑑. (2.5) When 𝑗=1,…,(𝑅−𝐼), 𝑔_𝑃+𝑗≡ℎ_𝐼+𝑗, 𝑘_𝑃+𝑗=−𝑛_𝐼+𝑗,𝛽_𝑃+𝑗=𝛾_𝐼+𝑗+𝜏_𝑑. (2.6)

Assumption 2.3. Nonlinearity 𝑓_𝑖(𝑖=1,…,𝐼) satisfies Lipshitz condition; that is, there exists a positive constant 𝐿 for all time variables 𝑎 and 𝑏, such that |𝑓_𝑖(𝑎+𝑏)−𝑓_𝑖(𝑎)|≤𝐿|𝑏|(𝑖=1,…,𝐼).
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 1−𝛼−𝑊+21𝐼+2^𝐼_𝑖=1𝑛²_𝑖𝐿²<0, (2.7) then lim_𝑡→∞[𝑦(𝑡)−𝑥(𝑡−𝜏_𝑑)]=0.

Proof. The dynamics of synchronization error is 𝑑𝑒(𝑡)=(𝑡)𝑑𝑡𝑑𝑦−𝑑𝑡𝑑𝑥𝑡−𝜏_𝑑(𝑡)+𝑑𝑡=−𝛼𝑒^𝑅_𝑖=1𝑛_𝑖ℎ_𝑖𝑦𝑡−𝛾_𝑖+^𝑄_𝑖=1𝑘_𝑖𝑔_𝑖𝑥𝑡−𝛽_𝑖+𝑊𝑥𝑡−𝜏_𝑑(𝑡)−−𝑊𝑦_𝑃𝑖=1𝑚_𝑖𝑓_𝑖𝑥𝑡−𝛼_𝑖−𝜏_𝑑. (2.8)
By applying Assumption 2.1, (2.8) can be rewritten as 𝑑𝑒(𝑡)(𝑡)(𝑡)+𝑑𝑡=−𝛼𝑒−𝑊𝑒^𝐼_𝑖=1𝑛_𝑖𝑓_𝑖𝑦𝑡−𝛾_𝑖−^𝐼_𝑖=1𝑚_𝑖−𝑘_𝑖𝑓_𝑖𝑥𝑡−𝛼_𝑖−𝜏_𝑑+^𝑅_𝑖=𝐼+1𝑛_𝑖ℎ_𝑖𝑦𝑡−𝛾_𝑖+^𝑄_𝑖=𝐼+1𝑘_𝑖𝑔_𝑖𝑥𝑡−𝛽_𝑖−^𝑃_𝑖=𝐼+1𝑚_𝑖𝑓_𝑖𝑥𝑡−𝛼_𝑖−𝜏_𝑑. (2.9)
By applying Assumption 2.2, if 𝑄−𝐼=(𝑃−𝐼)+(𝑅−𝐼) and 𝑦(𝑡−𝛾_𝑖)=𝑥(𝑡−𝛾_𝑖−𝜏_𝑑)+𝑒(𝑡−𝛾_𝑖), we have ^𝑅_𝑖=𝐼+1𝑛_𝑖ℎ_𝑖𝑦𝑡−𝛾_𝑖+^𝑄_𝑖=𝐼+1𝑘_𝑖𝑔_𝑖𝑥𝑡−𝛽_𝑖−^𝑃_𝑖=𝐼+1𝑚_𝑖𝑓_𝑖𝑥𝑡−𝛼_𝑖−𝜏_𝑑=^𝑅−𝐼_𝑗=1𝑛_𝐼+𝑗ℎ_𝐼+𝑗𝑥𝑡−𝛾_𝐼+𝑗−𝜏_𝑑+𝑒𝑡−𝛾_𝐼+𝑗+^𝑃−𝐼_𝑗=1𝑘_𝐼+𝑗𝑔_𝐼+𝑗𝑥𝑡−𝛽_𝐼+𝑗+^𝑅−𝐼_𝑗=1𝑘_𝑃+𝑗𝑔_𝑃+𝑗𝑥𝑡−𝛽_𝑃+𝑗−^𝑃−𝐼_𝑗=1𝑚_𝐼+𝑗𝑓_𝐼+𝑗𝑥𝑡−𝛼_𝐼+𝑗−𝜏_𝑑=0, (2.10) where 𝑒(𝑡−𝛾_𝐼+𝑗)=0 as well as synchronization established, in fact, 𝑒(𝑡−𝛾_𝐼+𝑗) reduces during establishing the synchronization regime.
From (2.10) and (2.9) we get 𝑑𝑒(𝑡)=()𝑒(𝑡)+𝑑𝑡−𝛼−𝑊^𝐼_𝑖=1𝑛_𝑖𝑓_𝑖𝑥𝑡−𝛼_𝑖−𝜏_𝑑+𝑒𝑡−𝛼_𝑖−𝑓_𝑖𝑥𝑡−𝛼_𝑖−𝜏_𝑑. (2.11)
Define a Lyapunov function [26] as 1𝑉=2𝑒²(𝑡)+12^𝐼_𝑖=1^𝑡_{𝑡−𝛼𝑖}𝑒²(𝑠)𝑑𝑠. (2.12)
Then, we obtain 𝑑𝑉(𝑡)(𝑡)𝑑𝑡=𝑒𝑑𝑒+1𝑑𝑡2^𝐼_𝑖=1𝑒²(𝑡)−12^𝐼_𝑖=1𝑒²𝑡−𝛼_𝑖. (2.13)
Here, we have 𝑑𝑉=()𝑒𝑑𝑡−𝛼−𝑊²(𝑡)+^𝐼_𝑖=1𝑛_𝑖𝑒(𝑡)𝑓_𝑖𝑥𝑡−𝛼_𝑖−𝜏_𝑑+𝑒𝑡−𝛼_𝑖−^𝐼_𝑖=1𝑚_𝑖−𝑘_𝑖𝑒(𝑡)𝑓_𝑖𝑥𝑡−𝛼_𝑖−𝜏_𝑑+12^𝐼_𝑖=1𝑒²(𝑡)−12^𝐼_𝑖=1𝑒²𝑡−𝛼_𝑖. (2.14)
According to Assumption 2.1, we have 𝑑𝑉=()𝑒𝑑𝑡−𝛼−𝑊²(𝑡)+^𝐼_𝑖=1𝑛_𝑖𝑒(𝑡)𝑓_𝑖𝑥𝑡−𝛼_𝑖−𝜏_𝑑+𝑒𝑡−𝛼_𝑖−𝑓_𝑖𝑥𝑡−𝛼_𝑖−𝜏_𝑑+12^𝐼_𝑖=1𝑒²(𝑡)−12^𝐼_𝑖=1𝑒²𝑡−𝛼_𝑖. (2.15)
In our model, (1/2)∑^𝐼_𝑖=1𝑒²(𝑡) can be rewritten as (1/2)𝐼𝑒²(𝑡). By Assumption 2.3, we have 𝑑𝑉≤1𝑑𝑡−𝛼−𝑊+2𝐼𝑒²(𝑡)+^𝐼_𝑖=1𝑛_𝑖𝑒(𝑡)𝐿𝑒𝑡−𝛼_𝑖−12^𝐼_𝑖=1𝑒²𝑡−𝛼_𝑖. (2.16)
According to 2𝑥𝑦≤𝑥²+𝑦², where 𝑥,𝑦∈ℜ, we get 𝑑𝑉≤1𝑑𝑡−𝛼−𝑊+2𝐼𝑒²(𝑡)+12^𝐼_𝑖=1𝑛²_𝑖𝐿²𝑒²(𝑡)+12^𝐼_𝑖=1𝑒²𝑡−𝛼_𝑖−12^𝐼_𝑖=1𝑒²𝑡−𝛼_𝑖. (2.17)
Finally, we obtain 𝑑𝑉≤1𝑑𝑡−𝛼−𝑊+21𝐼+2^𝐼_𝑖=1𝑛²_𝑖𝐿²𝑒²(𝑡). (2.18)
The proof is completed.

Note 1. The advantages of our lag synchronization model are as follows.(1)The nonlinear function 𝑓(⋅) satisfies |𝑓(𝑎+𝑏)−𝑓(𝑎)|≤𝐿|𝑏|, 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 𝑓_𝑖,𝑔_𝑖,ℎ_𝑖 vary as 𝑖 changes. Moreover, the format of function can be different even if 𝑖 is the same value.(3)In order to enhance the complexity of the system, 𝑃,𝑄,𝑅 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 {sin𝑢,𝑢/(1+𝑢⁸),𝑢/(1+𝑢¹⁰)}. Let us consider synchronization model with the master's and slave's equations defined as.

Master: 𝑑𝑥(𝑡)(𝑡)𝑑𝑡=−𝛼𝑥+𝑚₁sin𝑥𝑡−𝛼₁+𝑚₂sin𝑥𝑡−𝛼₂+𝑚₃sin𝑥−𝑡−𝛼₃+𝑚₄𝑥𝑡−𝛼₄1+𝑥⁸𝑡−𝛼₄+𝑚₅𝑥𝑡−𝛼₅𝛼+𝑥¹⁰𝑡−𝛼₅. (3.1)

Slave: 𝑑𝑦(𝑡)(𝑡)𝑑𝑡=−𝛼𝑦+𝑛₁sin𝑦𝑡−𝛾₁+𝑛₂sin𝑦𝑡−𝛾₂+𝑛₃sin𝑦𝑡−𝛾₃+𝑛₄sin𝑦𝑡−𝛾₄+DS(𝑡)−𝑊𝑦(𝑡). (3.2) Therefore the equation for driving signal is chosen as.

Driving signal: DS(𝑡)=𝑘₁sin𝑥𝑡−𝛽₁+𝑘₂sin𝑥𝑡−𝛽₂+𝑘₃sin𝑥𝑡−𝛽₃+𝑘₄𝑥𝑡−𝛽₄1+𝑥⁸𝑡−𝛽₄+𝑘₅𝑥𝑡−𝛽₅1+𝑥¹⁰𝑡−𝛽₅+𝑘₆sin𝑥𝑡−𝛽₆+𝑘₇sin𝑥𝑡−𝛽₇+𝑊𝑥𝑡−𝜏_𝑑, (3.3) where 𝑃=5, 𝑄=7, 𝑅=4, 𝐼=2 satisfy 𝑄=𝑃+𝑅−𝐼.

According to (2.4)–(2.6), the relation of the delays and parameters is expressed as 𝑚₁−𝑘₁=𝑛₁, 𝑚₂−𝑘₂=𝑛₂, 𝑘₃=𝑚₃, 𝑘₄=𝑚₄, 𝑘₅=𝑚₅, 𝑘₆=−𝑛₃, 𝑘₇=−𝑛₄, 𝛽₁=𝛼₁+𝜏_𝑑=𝛾₁+𝜏_𝑑, 𝛽₂=𝛼₂+𝜏_𝑑=𝛾₂+𝜏_𝑑, 𝛽₃=𝛼₃+𝜏_𝑑, 𝛽₄=𝛼₄+𝜏_𝑑, 𝛽₅=𝛼₅+𝜏_𝑑, 𝛽₆=𝛾₃+𝜏_𝑑, 𝛽₇=𝛾₄+𝜏_𝑑.

The value of parameters for simulation is adopted as 𝛼=2.0, 𝑊=100, 𝑚₁=−15.4, 𝑚₂=−16.0, 𝑚₃=−0.35, 𝑚₄=−20.0, 𝑚₅=−18.5, 𝑛₁=−0.2, 𝑛₂=−0.1, 𝑛₃=−0.25, 𝑛₄=−0.4, 𝑘₁=−15.2, 𝑘₂=−15.9, 𝑘₃=−0.35, 𝑘₄=−20.0, 𝑘₅=−18.5, 𝑘₆=0.25, 𝑘₇=0.4, 𝜏_𝑑=3.0, 𝛼₁=3.4, 𝛼₂=4.5, 𝛼₃=6.5, 𝛼₄=5.3, 𝛼₅=2.9, 𝛾₁=3.4, 𝛾₂=4.5, 𝛾₃=2.0, 𝛾₄=7.3, 𝛽₁=6.4, 𝛽₂=7.5, 𝛽₃=9.5, 𝛽₄=8.3, 𝛽₅=5.9, 𝛽₆=5.0, 𝛽₇=10.3.

In Figure 1, the portrait of 𝑥(𝑡−𝜏_𝑑) versus 𝑦(𝑡) 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 𝑡<𝜏_𝑑.

Figure 1 

Portrait of 𝑥(𝑡−3.0) versus 𝑦(𝑡).

It is clear to observe from Figure 2 that synchronization error 𝑒(𝑡) leaps at a sudden as 𝜏_𝑑=3.0 and vanishes eventually in a short time. Then 𝑒(𝑡) stays at zero.

Figure 2 

Synchronization error 𝑒(𝑡)=𝑦(𝑡)−𝑥(𝑡−3.0).

As shown in Figure 3, the slave's state variable is retarded with the time length of 𝜏_𝑑=3.0 in comparison with master's. The desired lag synchronization is realized.

Figure 3 

Time series of state variables 𝑥(𝑡) and 𝑦(𝑡).

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).