Parameters Identification and Synchronization of Chaotic Delayed Systems Containing Uncertainties and Time-Varying Delay

Abstract

Time delays are ubiquitous in real world and are often sources of complex behaviors of dynamical systems. This paper addresses the problem of parameters identification and synchronization of uncertain chaotic delayed systems subject to time-varying delay. Firstly, a novel and systematic adaptive scheme of synchronization is proposed for delayed dynamical systems containing uncertainties based on Razumikhin condition and extended invariance principle for functional differential equations. Then, the proposed adaptive scheme is used to estimate the unknown parameters of nonlinear delayed systems from time series, and a sufficient condition is given by virtue of this scheme. The delayed system under consideration is a very generic one that includes almost all well-known delayed systems (neural network, complex networks, etc.). Two classical examples are used to demonstrate the effectiveness of the proposed adaptive scheme.

(ProQuest: ... denotes non-US-ASCII text omitted.)

Zhongkui Sun 1,2 and Xiaoli Yang 3

Recommended by Jerzy Warminski

1, Department of Applied Mathematics, Northwestern Polytechnical University, Xi'an 710072, China
2, Department of Physics, University of Potsdam, Postfach 601553, 14415 Potsdam, Germany
3, College of Mathematics and Information Science, Shaan'xi Normal University, Xi'an 710062, China

Received 30 December 2009; Revised 5 March 2010; Accepted 24 March 2010

1. Introduction

In recent years, delayed dynamical systems (so called, DDEs) have attracted lots of attention in the field of nonlinear dynamics, and the dynamical properties of DDEs have been extensively investigated. Strictly speaking, time delays are ubiquitous in real world due to the finite switching speed of amplifiers, finite signal propagation time in biological networks, finite chemical reaction times, memory effects, and so forth. Therefore, DDEs are used to model dynamical systems broadly in scientific and engineering areas, for instance, in population dynamics, biology, economy, neural networks, complex networks, and so on. It has been found that the presence of time delay(s) is often a source of complicated behaviors, for example, limit-cycle, loss of stability, bifurcation, and chaos [1-7]. Especially, one-dimensional delayed dynamical systems can generate high-dimensional chaos [8, 9]. Motivated by the study of chaotic phenomena, an increasing interest has been devoted to the study of chaos synchronization in delayed dynamical systems, for example, anticipating synchronization [10-15], generalized synchronization [16], phase synchronization [17], and complete synchronization [18-21]. Many applications of chaos synchronization in delayed dynamical systems have been found in many different areas including in secure communication, information science, optic systems, neural networks, and so forth. However, in most of the previous works, the considered systems are often specific, also the controllers are sometimes of limitation or too complicated to implement in nature; particularly, the controllers cannot be applied to delayed dynamical systems with time-varying time delay.

Very recently, many research of chaos synchronization has been devoted to neural networks or complex networks with time delay(s) and some theoretical and numerical results are obtained [22-27]. In [22], the authors considered the synchronization of two kinds of dynamical complex networks utilizing special matrix measure with constant linear coupling delays in the nodes of the whole networks. In [23], mathematical analysis is presented on the synchronization phenomena of linearly coupled systems described by ordinary differential equations with a single linear coupling delay. Reference [24] presented several delay-dependent conditions for continuous- and discrete-time complex dynamical network model with a single linear coupling delay using linear matrix inequalities. In [25], a strategy for synchronization of complex dynamical networks with a single linear coupling time-delay is proposed based on linear state feedback controllers. In [26], the authors studied the synchronization of neural networks subject to time-varying delays and sector nonlinearity based on the drive-response concept, where a complicated controller was designed to achieve synchronization. Cao et al. considered the synchronization of coupled identical neural networks with time-varying delay using a simple adaptive feedback scheme based on the invariant principle of functional differential equations, and a good result was obtained in [27]. But, to design the adaptive scheme, a strict constraint is added to the time-varying delay, and the state equation under consideration must be of specific form. However, there have been no previous reports of adaptive chaos synchronization of delayed dynamical systems with uncertain parameters and nonlinear time-varying delay.

An interesting application of chaos synchronization is to estimate parameters of a chaotic system from time series when partial information about the system is available [28-31]. In contrast to the large number of research papers on the parameter identification of chaotic systems without delay, only limited attention has been given to the parameter identification of delayed dynamical systems [32-38]. Among these works, most of them are limited to linear delayed system with constant delay. Actually, how to identify the unknown parameters from time series is still an open problem.

Motivated by the above discussion, in the present paper, we study parameters identification and synchronization of uncertain chaotic delayed systems with time-varying delay based on the famous Razumikhin condition and invariance principle of functional differential equations in the framework of Krasovskii-Lyapunov theory [39]. The system under consideration is a very general one that includes almost all well-known delayed systems, for example, Ikeda system [40], Mackey-Glass system [1], delayed Duffing system [7], Hopfield neural network [27], BAM neural network [41], cellular neural network [42], complex networks [22-24], and so on. The adaptive feedback controller utilized here is very simple, which is constructed by combination of adaptive scheme and linear feedback with the updated feedback strength.

This paper is organized as follows: problem statement and some preliminaries knowledge, including one lemma and two assumptions, are given in Section 2. Our main theoretical result is described in Section 3. In Section 4, two numerical examples, the delayed Rössler model [43] and delayed Hopfield neural network system [27], are employed to illustrate the effectiveness of the proposed adaptive scheme. Finally, conclusions and remarks are drawn in Section 5.

2. Problem Statement and Preliminaries

In this paper, we consider the general continuous-time delayed dynamical system with time-varying delay described by the following delayed differential equation: [figure omitted; refer to PDF] where x(t)=(_x1 (t),_x2 (t),...,_xn (t)^)T ∈^Rn is the state vector, A=_{(aij )n×n} and B=_{(bij )n×n} are unknown constant matrices representing the linear parts of the system, C=_{(cij )n×n} and D=_{(dij )n×n} are unknown constant matrices representing the nonlinear parts of the system, τ=τ(t)≥0 is the time-varying delay, f(x(t))=[_f1 (x(t)),_f2 (x(t)),...,_fn (x(t))^]T denotes the nonlinear function without delay, and g(x(t))=[_g1 (x(t-τ)),_g2 (x(t-τ)),...,_gn (x(t-τ))^]T denotes the nonlinear function with delay. Without loss of generality, we assume that the structure of the nonlinear dynamical system (2.1) is known and, furthermore, time series for all variables are available as the output of system (2.1). Let Ω∈^Rn be a chaotic bounded set.

For the nonlinear vector functions f(x) and g(x) and time-varying delay τ(t) , we have the following two assumptions.

Assumption.

For any x=(_x1 ,_x2 ,...,_xn^)T and y=(_y1 ,_y2 ,...,_yn^)T ∈Ω , there exist constants ^li1 >0 and ^li2 >0 (n=1,2,...,n ) such that [figure omitted; refer to PDF]

The above condition is the so-called uniform Lipschitz condition; ^li1 >0 and ^li2 >0 (n=1,2,...,n ) refer to the uniform Lipschitz constants.

Assumption.

τ=τ(t)≥0 is the smooth function of time t , and its derivative is bounded, that is, there exists some positive number M>0 such that |τ...(t)|≤M .

Clearly, this assumption is certainly ensured if the delay τ(t) is constant.

Remark 2.3.

Assumption 2.1 is a very loose constraint added to nonlinear vector functions f(x) and g(x) . One can easily check that a wide variety of delayed dynamical systems satisfy the above condition; particularly, condition (2.2) will hold as long as the partial differential ∂_fi /∂_xj and ∂_gi /∂_xj (i,j=1,2,...,n) are bounded in Ω∈^Rn . Therefore, the class of systems in the form of (2.1) includes almost all well-known delayed systems, for example, Ikeda system [40], Mackey and Glass system [1], delayed Duffing system [7], Hopfield neural network [27], BAM neural network [41], cellular neural network [42], complex networks [22-24], and so forth.

Remark 2.4.

Assumption 2.2 is a limitation of time-varying delay τ(t) . Comparing to that of time-varying delay τ(t) in [27], this limitation is very loose.

We refer to system (2.1) as the drive system. An auxiliary system of variables y(t)=(_y1 (t),_y2 (t),...,_yn (t)^)T ∈^Rn is introduced as the response system, whose evolution equations have identical form to system (2.1) [figure omitted; refer to PDF] where A...=(_a...ij)n×n , B...=(_b...ij)n×n , C...=(_c...ij)n×n , and D...=(_d...ij)n×n are the estimates of the unknown parameters matrices A=(_aij)n×n , B=(_bij)n×n , C=(_cij)n×n , and D=(_dij)n×n , respectively; u(t) is a simple adaptive-feedback controller, u(t)=K(y(t)-x(t)) with K updated adaptively according to some updated law.

Defining the synchronization error as e(t)=y(t)-x(t) and subtracting (2.1) from (2.3) yield the error system as follows: [figure omitted; refer to PDF] Therefore, the task of this paper is to design a suitable adaptive scheme [figure omitted; refer to PDF] such that y(t) can track x(t) , that is, ||e(t)||[arrow right]0 as t[arrow right]∞ .

Furthermore, we introduce a lemma [44], which is needed in the proof of the main results.

Lemma 2.5.

For any vectors _x1 ,_x2 ∈^Rn and any positive definite matrix Q∈_Rn×n , the following inequality holds: [figure omitted; refer to PDF]

3. Adaptive Synchronization Scheme

In this section, we investigate the adaptive synchronization between the drive system (2.1) and the response system (2.3) in the framework of Krasovskii-Lyapunov theory [39]. The main result is described in the following theorem.

Theorem 3.1.

Under Assumption 2.1 and Assumption 2.2, system (2.3) can synchronize with system (2.1) if one designs A... , B... , C... , and D... as [figure omitted; refer to PDF] with the coupling strength K=diag (_k1 ,_k2 ,...,_kn^)T updated by [figure omitted; refer to PDF] where _αij ,_βij ,_pij ,_qij , and _γi (i,j=1,2,...,n ) are arbitrary positive constants.

Proof.

Construct a Lyapunov function of the form [figure omitted; refer to PDF] where G(e)=g(y)-g(x)=g(e+x)-g(x) .

By differentiating the function V with respect to time along the trajectory of (2.4), one can obtain [figure omitted; refer to PDF] where F(e)=f(y)-f(x)=f(e+x)-f(x) .

Recalling Assumption 2.1, we have the following two inequalities: [figure omitted; refer to PDF] here _h1 =max {^l11 ,^l21 ,...,^ln1 } and _h2 =max {^l12 ,^l22 ,...,^ln2 } . Substituting inequalities (3.5) into the right side of the above equality and applying Lemma 2.5, one can obtain [figure omitted; refer to PDF] Applying the classical Razumikhin condition to inequality (3.6) and choosing [figure omitted; refer to PDF] one can obtain [figure omitted; refer to PDF] Clearly, V...(e(t))=0 if and only if e(t)=0 . Therefore, E={e=0,A...=_A0 ,B...=_B0 ,C...=_C0 ,D...=_D0 ,K=_K0 ∈^Rn } is the largest invariant set containing in {(e(t),A...,B...,C...,D...,K)|"V...=0} . According to the well-known invariant principle of functional differential equations [39], the trajectory of the argument system converges asymptotically to the largest invariant set E starting from any initial value as time tends to infinity, where the converged parameters _A0 ,_B0 ,_C0 ,_D0 , and _K0 depends on initial values. This completes the proof.

Remark 3.2.

Note that this theorem only guarantees that e(t)[arrow right]0 , that is, y(t)[arrow right]x(t) , and _a......ij [arrow right]0 , _b......ij [arrow right]0 , _c......ij [arrow right]0 , _d......ij [arrow right]0 as t[arrow right]∞ ; therefore, it is not necessary that _A0 =A , _B0 =B , _C0 =C and _D0 =D since it is possible that Ax(t)+Bx(t-τ(t))+Cf(x(t))+Dg(x(t-τ))=_A0 x(t)+_B0 x(t-τ(t))+_C0 f(x(t))+_D0 g(x(t-τ)) when _A0 ≠A , _B0 ≠B , _C0 ≠C , and _D0 ≠D . Another possible reason is that several sets of parameters A,B,C,D generate the same trajectory in response system (2.3), and A...,B...,C...,D... converge to only one such set. Both of the above two reasons will make _A0 ,_B0 ,_C0 ,_D0 in the largest invariant set E not unique in general except for some special delayed systems.

Therefore, we have the following corollary.

Corollary 3.3.

The unknown parameters A,B,C,D of concerned delayed system are identifiable and _A0 =A , _B0 =B , _C0 =C , _D0 =D if _A0 ,_B0 ,_C0 ,_D0 unique in the largest invariant set E .

This corollary can be derived directly from Theorem 3.1.

4. Numerical Illustrations

In this section, two chaotic nonlinear systems with time-varying delay, that is, delayed Rössler system [43] and delayed Hopfield neural networks system [27], are employed to demonstrate the effectiveness of the proposed adaptive synchronization scheme.

4.1. Delayed Rössler System

A delayed Rössler system is studied in [43] whose evolution equation is as follows (when _τ1 =_τ2 in [43]): [figure omitted; refer to PDF] When α=0.5, a=b=0.2, c=5.7 , and τ=_τ0 +A sin Ωt , the delayed Rössler system is chaotic; see Figure 1.

Figure 1: Chaotic attractor of delayed Rössler system for A=1, Ω=1, _τ0 =1 .

[figure omitted; refer to PDF]

Here we refer to system (4.1) with the parameters a,b,c being unknown as the drive system. The response system can be described as [figure omitted; refer to PDF] Clearly Assumption 2.1 and Assumption 2.2 are satisfied.

Defining the error state e=[_e1 ,_e2 ,_e3^]T =[_x2 -_x1 ,_y2 -_y1 ,_z2 -_z1^]T and following the procedure proposed in the Section 2, we can design the adaptive scheme as follows: [figure omitted; refer to PDF] with the feedback strength K=diag (_k1 ,_k2 ,_k3 ) updated by [figure omitted; refer to PDF] where _γ1 ,_γ2 ,_γ3 ,_α22 ,_α33 , and _p32 are arbitrary positive constants.

In the following numerical simulations, we take [_x1 (t),_y1 (t),_z1 (t)^]T =[5,5,5^]T , [_x2 (t),_y2 (t),_z2 (t)^]T =[-5,-5,-5^]T , [_k1 (t),_k2 (t),_k3 (t)^]T =[0,0,0^]T , and [a...(t),b...(t),c...(t)^]T =[0,0,0^]T for t∈[-2,0] . The results are shown in Figures 2, 3, 4, 5, and 6.

Numerically simulated time series _x1 ,_y1 ,_z1 (dotted line) and _x2 ,_y2 ,_z2 (black line).

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

(c) [figure omitted; refer to PDF]

Numerically simulated error e(t) .

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

(c) [figure omitted; refer to PDF]

The temporal trajectory of a...(t), b...(t) , and c...(t) .

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

(c) [figure omitted; refer to PDF]

The evolution of feedback strength k .

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

(c) [figure omitted; refer to PDF]

The evolution of adaptive controller u(t) .

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

(c) [figure omitted; refer to PDF]

4.2. Hopfield Neural Network Model

This subsection considers a typical delayed Hopfield neural network system [27] [figure omitted; refer to PDF] where x=[_x1 ,_x2^]T and τ=^et /(1+^et ) . When A=(-100-1) , B=(0000) , C=(2.0-0.1-5.03.2) , and D=(-1.6-0.1-0.18-2.4) , the delayed Hopfield neural network system (4.5) is chaotic.

We refer to system (4.5) with [figure omitted; refer to PDF] as the drive system, where parameters a,c,d are unknown. The response system has the following form: [figure omitted; refer to PDF] where y=[_y1 ,_y2^]T , u=[_u1 ,_u2^]T , and A=(a...00-1) , B=(0000) , C=(2.0c...-5.03.2) , and D=(-1.6-0.1-0.18d...) .

It is easy to verify that Assumption 2.1 and Assumption 2.2 are satisfied. Defining e=[_e1 ,_e2^]T =[_y1 -_x1 ,_y2 -_x1^]T and following the procedure proposed in the Section 2, the adaptive scheme can be designed as follows: [figure omitted; refer to PDF] with [figure omitted; refer to PDF]

The initial values are chosen as [_x1 (t),_x2 (t)^]T =[0.2,0.5^]T , [_y1 (t),_y2 (t)^]T =[-1.3,2.1^]T , [_k1 (t),_k2 (t)^]T =[0,0^]T and [a...(t),c...(t),d...(t)^]T =[0,0,0^]T . The results are depicted in Figures 7-9.

Numerically simulated time series _x1 ,_x2 (dotted line) and _y1 ,_y2 (black line) and error e=[_e1 ,_e2^]T .

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

(c) [figure omitted; refer to PDF]

(d) [figure omitted; refer to PDF]

The temporal trajectory of a...(t),c...(t) , and d...(t) .

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

(c) [figure omitted; refer to PDF]

The evolution of feedback strength k and adaptive controller u(t) .

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

(c) [figure omitted; refer to PDF]

(d) [figure omitted; refer to PDF]

By comparing Figures 4 and 8, one can see that in Figure 8 the estimates of the unknown parameters, a...,c...,d... , converge to the true value of a=-1, c=-0.1, and d=-2.4 ; however, for the delayed Rössler system, the estimates of unknown parameters, a...,b...,c... , converge to _a0 =-1.85, _b0 =1.4559 , and _c0 =1.3655 which are not equal to the true value of a,b,c . It is reasonable because for Hopfield neural network system, a...,c...,d... in the largest invariant set unique, which is equal to the true value of a,c,d ; but for delayed Rössler system, a...,b...,c... in the largest invariant set cannot be proven to be unique, which means what values a...,b...,c... do converge to depend on the initial value of a...,b...,c... .

5. Concluding Remarks

The present paper dealt with the problem of parameters identification and synchronization of chaotic delayed systems containing uncertainties and time-varying delay. A simple but efficient adaptive regime was designed firstly to synchronize the chaotic dynamics between two coupled identical time-varying delayed chaotic systems, which is seriously proved in the framework of Krasovskii-Lyapunov theory [39] based on the famous Razumikhin condition and extended invariance principle for functional differential equations. Then, the proposed technique was utilized to estimate the unknown parameters containing in model. By virtue of the synchronization-based method, the unknown parameters of Rössler system were estimated exactly. But, because of the limitation of invariant principle, that is, it only guarantees the estimates of the unknown parameters to converge to the largest invariant set containing in {V...=0} , hence, the synchronization-based adaptive schemes may fail if the largest invariant set containing in {(e(t),A...,B...,C...,D...,K)|"V...=0} has more than one element. It is worth emphasizing that the largest invariant set depends tightly on the configuration of the system under consideration, which can be calculated by following some classical steps as given in [39] (or other literatures related to invariant principle theorem). The regime proposed here is rigorous and global. The effectiveness of the proposed scheme on synchronization and parameters identification is well demonstrated by the numerical examples.

Acknowledgments

This work is partially supported by the National Natural Science Foundation of China (Grants no. 10902085 and 10902062), the NSF of Shaanxi Province (Grant no. 2009JQ1002), and Youth for NPU Teachers Scientific and Technological Innovation Foundation.

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