# Lag quasisynchronization of coupled delayed systems with parameter mismatch by periodically intermittent control

## Authors

- First Online:

- Received:
- Accepted:

DOI: 10.1007/s11071-012-0673-y

- Cite this article as:
- Huang, J., Li, C., Huang, T. et al. Nonlinear Dyn (2013) 71: 469. doi:10.1007/s11071-012-0673-y

- 15 Citations
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## Abstract

This paper further investigates the lag synchronization of coupled delayed systems with parameter mismatch. Different from the most existing results, we formulate the intermittent control system that governs the dynamics of the synchronization error. As a result of parameter mismatch, complete lag synchronization cannot be achieved. In this paper, a lag quasisynchronization scheme is proposed to ensure that coupled systems are in a state of lag synchronization with an error level. We estimate the error bound of lag synchronization by rigorously theoretical analysis. Numerical simulations are presented to verify the theoretical results.

### Keywords

Chaotic systemsIntermittent controlLag quasisynchronizationParameter mismatchTime delay## 1 Introduction

Since the pioneering works of Pecora and Carroll [1], the idea of synchronization of chaotic systems has received a great deal of interest. Over the past decades, many types of chaos synchronization have been presented, e.g., complete synchronization [1, 2], generalized synchronization [3], projective synchronization [4], phase synchronization [5], lag synchronization [6], and anticipating synchronization [7]. For complete synchronization, the master’s state *x*(*t*) and the slave’s state *y*(*t*) are identical, i.e., *y*(*t*)→*x*(*t*). Generalized synchronization is usually described as the presence of some function relation between the slave’s states and the master’s, i.e., there is a function *g* such that *y*(*t*)→*g*(*x*(*t*)). For projective synchronization, there is a scale factor in the amplitude of the master’s state variable and that of the slave’s, i.e., *y*(*t*)→*αx*(*t*). Phase synchronization, which indicates the difference between the phase of the master’s state and that of the slave’s, is constant during interaction, but their amplitudes remain chaotic and uncorrelated, i.e., *nϕ*_{x}−*mϕ*_{y}=*const* (*n* and *m* are integers). For lag synchronization, the state of the slave system is retarded with time delay *τ* in compared to the state of the master system, i.e., *y*(*t*)≈*x*_{τ}(*t*)≡*x*(*t*−*τ*) with positive *τ*. Furthermore, for anticipating synchronization, the synchronization manifold is *y*(*t*)→*x*(*t*+*τ*), where *τ* is the time delay of the driven system.

On the other side, for the finite speed of signals, it has been found that the complete synchronization of chaotic systems is nearly impossible. It is just synchronization lag that makes lag synchronization practically available. For example, in a call, the receiver hears the voice at time *t*, which is a past state on the transmitter side at time *t*−*τ* [8]. Therefore, in many cases, it is more rational to require the slave system to synchronize with the master system within a time delay *τ*. Hence, it is of great importance to study lag synchronization and some results have been reported in this research area [9–14]. The authors in [9] investigated lag synchronization between unidirectional coupled delayed Ikeda systems by using feedback control techniques. The authors in [10] considered lag synchronization problem with an application in secure communication. The authors in [11] studied lag synchronization of coupled chaotic delayed neural networks by using adaptive feedback control techniques.

In this paper, we investigate the lag synchronization of coupled systems with parameter mismatches and time delay in a general model. There are related works in the field of lag synchronization on the effects of parameter mismatch. The authors in [15] studied the effect of parameter mismatch on lag synchronization of chaotic systems with time delay in the framework of master-slave configuration. The authors in [16] investigated lag quasisynchronization of coupled time-delayed systems with parameter mismatch in general model. The authors in [17] studied the effect of parameter mismatch on lag synchronization of chaotic systems without time delay.

Inspired by above mentioned works, this paper examines the effects of the parameter mismatch on lag synchronization by periodically intermittent control. As a special discrete feedback control, intermittent control is activated during certain nonzero time intervals and sleeps during other time intervals, and has been extensively used in engineering control. Recently, intermittent control of nonlinear system has drawn increasing interests in stabilizing and synchronizing chaotic systems [18–20]. In this paper, by using Lyapunov stability theory and intermittent control technique, the intermittent controllers and corresponding parameter updating rules are designed to obtain lag synchronization of coupled systems with parameter mismatch, and then the error bound of lag synchronization is estimated in accordance with the parameter mismatch. Moreover, the error level can be smaller, compared with [15–17]. Therefore, the results in this paper improve the existing ones in [15–17]. Interestingly, we find that the error bound of lag synchronization depends on the switching rate *σ*. If we select a larger switching rate *σ*, the error level of lag synchronization can be smaller. Numerical simulations are reported to show their good agreement with the theoretical results.

The rest of the paper is organized as follows. In Sect. 2, we formulate the problem of lag synchronization of coupled systems with parameter mismatch. In Sect. 3, a general convergence criterion for the lag synchronization error of coupled systems is established by intermittent control method. Also, some corollaries and remarks are listed to show the advantage of this paper. In Sect. 4, numerical examples are given to verify the theoretical result. Conclusions are given in Sect. 5.

## 2 Problem formulation and preliminaries

*x*∈

*R*

^{m}denotes the state vector,

*C*

_{1},

*A*

_{1}, and

*B*

_{1}∈

*R*

^{m×m}are constant matrices,

*τ*is the time delay,

*f*,

*g*:

*R*

^{m}→

*R*

^{m}are nonlinear functions satisfying the Lipschitz condition, namely, there exist positive constants

*L*

_{f},

*L*

_{g}such that

*y*∈

*R*

^{m}denotes the state vector,

*C*

_{2},

*A*

_{2}, and

*B*

_{2}∈

*R*

^{m×m}are constant matrices, and

*u*(

*t*) is the intermittent linear state feedback control gain defined as follows:

*k*denotes the control strength, 0<

*σ*<1 denotes the switching rate,

*T*denotes the control period and

*θ*is the transmittal delay. In this paper, we consider that

*C*

_{1}≠

*C*

_{2},

*A*

_{1}≠

*A*

_{2}, and

*B*

_{1}≠

*B*

_{2}, namely, there exist parameter mismatches between chaotic system (1) and system (2).

*e*(

*t*)=

*y*(

*t*)−

*x*(

*t*−

*θ*), we have the following error dynamical system: where Δ

*C*=

*C*

_{2}−

*C*

_{1}, Δ

*A*=

*A*

_{2}−

*A*

_{1}and Δ

*B*=

*B*

_{2}−

*B*

_{1}represent the parameter mismatch errors.

*x*(

*t*) on [−

*θ*−

*τ*,−

*τ*] as

Then initial condition of system (5) is defined by \(e( t ) = \psi( t ) - \hat{\varphi} ( t - \theta), - \tau\le t \le0\).

It is obvious that the origin *e*=0 is not an equilibrium point of the error system (5), and the complete lag synchronization can not be happening. However, we found that it is possible to synchronize systems (1) and (2) up to considerably small error bound, which is dependent on the differences in the parameters between the two systems. In this paper, we investigate the lag synchronization of two delayed chaotic systems with error bound *ε* using the intermittent control.

To establish the main result of this paper, the following preliminaries are necessary.

### Definition 1

*φ*(

*t*),

*ψ*(

*t*)∈

*Ω*,

*t*∈[−

*τ*,0], there exist a small error level

*ε*>0 and a constant

*θ*>0, such that the lag synchronization error satisfies

### Lemma 1

[21]

*For any real matrices*

*Σ*

_{1},

*Σ*

_{2},

*Σ*

_{3},

*and a scalar*

*s*>0

*such that*\(0 < \varSigma_{3} = \varSigma_{3}^{T}\),

*the following inequality holds*:

### Lemma 2

[22]

### Lemma 3

[22]

*If function*

*y*(

*t*)

*is nonnegative when*

*t*∈(

*t*

_{0}−

*τ*,+∞)

*and satisfies the following*:

*where*

*a*,

*b*,

*ε*

*are positive*,

*and*

*a*>

*b*.

*Then*

*where*\(| y(t_{0}) |_{\tau } = \mathop{\max}_{t_{0} - \tau \le s \le t_{0}}| y(s) |\)

*and*

*r*

*is the unique root of the equation*

### Lemma 4

[22]

Throughout this paper, the vector norm is taken to be Euclidian, denoted by ∥ ∥. We use *P*^{T}, *λ*_{max}(*P*) to denote the transpose and the maximum eigenvalue of matrix *P*, respectively.

## 3 Main results

In this section, based on Lyapunov method and the intermittent control technique, we will present some sufficient conditions for lag quasi-synchronization of systems (1) and (2), and estimate the lag-synchronization error bound by stability analysis on the error dynamical system.

### Theorem 1

*Suppose that*

*Ω*={

*x*∈

*R*

^{m}|∥

*x*∥≤

*ω*

_{1}}

*is a region including the chaotic attractor of*(1)

*and the parameter*-

*mismatch satisfies*∥Δ

*C*∥+

*L*

_{f}∥Δ

*A*∥+

*L*

_{g}∥Δ

*B*∥≤

*ω*

_{2}.

*Moreover*,

*suppose that there exist positive scalars*

*h*

_{1}>0,

*h*

_{2}>0

*and*

*s*

_{i}>0 (

*i*=1,2,3,4,5,6)

*such that*

- (i)
\(C_{2} + C_{2}^{T} - 2kI + s_{1}A_{2}A_{2}^{T} + s_{1}^{ - 1}L_{f}^{2}I + s_{2}B_{2}B_{2}^{T} + s_{3}I + h_{1}I \le0\);

- (ii)
\(C_{2} + C_{2}^{T} + s_{4}A_{2}A_{2}^{T} + s_{4}^{ - 1}L_{f}^{2}I + s_{5}B_{2}B_{2}^{T} + s_{6}I - h_{2}I \le0\);

- (iii)
\(p = r(\sigma T - \tau) - (h_{2} + s_{5}^{ - 1}L_{g}^{2})(T - \sigma T) > 0\),

*where*

*r*

*is the unique root of the equation*: \(- r = - h_{1} + s_{2}^{ - 1}L_{g}^{2}\exp(r\tau)\).

*Then trajectory of error system*(5)

*converges exponentially to a small region*

*M*

*containing the origin under the intermittent control*,

*where*

*in which*

*and*

*i*.

*e*.,

*the master system*(1)

*and the slave system*(2)

*achieve lag quasisynchronized with an error level*

### Proof

*nT*≤

*t*<

*nT*+

*σT*, the derivative of (9) with respect to time

*t*along the trajectories of the first subsystem of the system (5) is calculated and estimated as follows:

*r*is the unique root of the equation: \(- r = - h_{1} + s_{2}^{ - 1}L_{g}^{2}\exp(r\tau)\).

*M*containing the origin, where

This implies that the master system (1) and the slave system (2) achieve lag quasi-synchronized with an error level \(\sqrt{\eta}\). The proof is thus completed. □

### Remark 1

In Theorem 1, we have a general criterion of lag quasi-synchronization associated with an error level. As *t* goes to infinity, for any arbitrary small positive number *ε*, the error bound \(\sqrt{| V(0) |_{\tau }\exp(p)} \exp( - \frac{pt}{2T}) + \sqrt{\eta}\) decrease and converges to the error level \(\varepsilon+ \sqrt{\eta}\).

### Remark 2

We obtain from Theorem 1 the following corollary.

### Corollary 1

*Suppose that*

*Ω*={

*x*∈

*R*

^{n}|∥

*x*∥≤

*ω*

_{1}}

*and the parameter mismatch satisfies*\(\| \Delta C \| + L_{f}\*\| \Delta A \| + L_{g}\| \Delta B \| \le\omega_{2}\).

*Moreover*,

*we suppose that there exist positive scalars*

*k*

*and*

*σ*

*satisfying*0<

*σ*<1

*such that*

*where*

*r*

*is the unique root of the equation*:

*Then the lag quasisynchronization between system*(1)

*and system*(2)

*is achieved with an error level*

*where*\(\alpha= \omega_{1}^{2}\omega_{2}^{2}/r\), \(\upsilon= (\alpha+ \beta)\exp((h_{2}^{*} + \sqrt{\lambda_{\max }(B_{2}B_{2}^{T})L_{g}} )(T - \sigma T)) - \beta\),

*and*\(\beta= \frac{\omega_{1}^{2}\omega_{2}^{2}}{h_{2}^{*} + \sqrt{\lambda_{\max }(B_{2}B_{2}^{T})L_{g}}}\).

### Remark 3

*T*is given, we can estimate the feasible region

*of control parameters (*

**D***k*,

*σ*) from Corollary 1. More concisely, let

*r*

^{∗}=(

*N*

_{C}+2

*N*

_{A}+2

*N*

_{B}+1)(

*T*−

*σT*)/(

*σT*−

*τ*)>0, where

*r*

^{∗}<

*r*.

*k*can be estimated as follows: Note that

*N*

_{C},

*N*

_{A}, and

*N*

_{B}are determined only by the system itself, and

*k*and

*σ*are control parameters. From (19), we can estimate the feasible region

*of control parameters (*

**D***k*,

*σ*),

## 4 Numerical example

In this section, we take Lu neural oscillator [23] as an example to show the effectiveness of the lag quasisynchronization scheme obtained in the previous section. The delay differential equations are solved numerically by the programs DDE23 in MATLAB.

### Example

*f*(

*x*(

*t*))=

*g*(

*x*(

*t*))=tanh(

*x*(

*t*)).

*f*(

*y*(

*t*))=

*g*(

*y*(

*t*))=tanh(

*y*(

*t*)).

In this example, we observe that *L*_{f}=*L*_{g}=1, *ω*_{1}=4 and *τ*=1. Note that parameter mismatches satisfy that ∥−Δ*C*∥+*L*_{f}∥Δ*A*∥+*L*_{g}∥Δ*B*∥≤*ω*_{2}=0.0211.

*T*is given, we can plot the feasible region

*of control parameters (*

**D***k*,

*σ*), as shown in Fig. 2. For numerical simulation, we take

*T*=8,

*σ*=0.9,

*θ*=0.01, and

*k*=30, and plot the lag quasisynchronization error curve, as shown in Figs. 3 and 4. Consequently, we estimate the error bound is

*M*={

*e*∈

*R*

^{m}|∥

*e*∥≤0.0343}. When a larger value is taken for the duration rate

*σ*, the lag quasisynchronization error level may become smaller, as shown in Fig. 5.

## 5 Conclusions

In this paper, we have formulated the lag quasisynchronization problem for chaotic systems with parameter mismatch by means of periodically intermittent control. Based on Lyapunov stability theory and intermittent control techniques, we establish several lag quasi-synchronization criteria and estimate the error bound. Numerical simulations have showed the validity of theoretical result.

## Acknowledgements

The work described in this paper was partially supported by the National Natural Science Foundation of China (Grant No. 60974020) and Natural Science Foundation Project of Chongqing CSTC (Grant No. cstc2011jjA0980), and the Foundation of Chongqing Education College (Grant No. KY201112A, KY201113B, KY201122C ).