Research Article | Open Access
Parameters Identification and Synchronization of Chaotic Delayed Systems Containing Uncertainties and Time-Varying Delay
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.
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 , phase synchronization , 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 , 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 , mathematical analysis is presented on the synchronization phenomena of linearly coupled systems described by ordinary differential equations with a single linear coupling delay. Reference  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 , a strategy for synchronization of complex dynamical networks with a single linear coupling time-delay is proposed based on linear state feedback controllers. In , 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 . 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 . The system under consideration is a very general one that includes almost all well-known delayed systems, for example, Ikeda system , Mackey-Glass system , delayed Duffing system , Hopfield neural network , BAM neural network , cellular neural network , 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  and delayed Hopfield neural network system , 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: where is the state vector, and are unknown constant matrices representing the linear parts of the system, and are unknown constant matrices representing the nonlinear parts of the system, is the time-varying delay, denotes the nonlinear function without delay, and 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 be a chaotic bounded set.
For the nonlinear vector functions and and time-varying delay , we have the following two assumptions.
Assumption. For any and , there exist constants and () such that
The above condition is the so-called uniform Lipschitz condition; and () refer to the uniform Lipschitz constants.
Assumption. is the smooth function of time , and its derivative is bounded, that is, there exists some positive number such that .
Clearly, this assumption is certainly ensured if the delay is constant.
Remark 2.3. Assumption 2.1 is a very loose constraint added to nonlinear vector functions and . 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 and are bounded in . Therefore, the class of systems in the form of (2.1) includes almost all well-known delayed systems, for example, Ikeda system , Mackey and Glass system , delayed Duffing system , Hopfield neural network , BAM neural network , cellular neural network , complex networks [22–24], and so forth.
We refer to system (2.1) as the drive system. An auxiliary system of variables is introduced as the response system, whose evolution equations have identical form to system (2.1) where , , , and are the estimates of the unknown parameters matrices , , , and , respectively; is a simple adaptive-feedback controller, with updated adaptively according to some updated law.
Defining the synchronization error as and subtracting (2.1) from (2.3) yield the error system as follows: Therefore, the task of this paper is to design a suitable adaptive scheme such that can track , that is, as .
Furthermore, we introduce a lemma , which is needed in the proof of the main results.
Lemma 2.5. For any vectors and any positive definite matrix , the following inequality holds:
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 . 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 , , , and as with the coupling strength updated by where , and () are arbitrary positive constants.
Proof. Construct a Lyapunov function of the form
By differentiating the function with respect to time along the trajectory of (2.4), one can obtain where .
Recalling Assumption 2.1, we have the following two inequalities: here and . Substituting inequalities (3.5) into the right side of the above equality and applying Lemma 2.5, one can obtain Applying the classical Razumikhin condition to inequality (3.6) and choosing one can obtain Clearly, if and only if . Therefore, is the largest invariant set containing in . According to the well-known invariant principle of functional differential equations , the trajectory of the argument system converges asymptotically to the largest invariant set starting from any initial value as time tends to infinity, where the converged parameters depends on initial values. This completes the proof.
Remark 3.2. Note that this theorem only guarantees that , that is, , and , , , as ; therefore, it is not necessary that , , and since it is possible that when , , , and . Another possible reason is that several sets of parameters generate the same trajectory in response system (2.3), and converge to only one such set. Both of the above two reasons will make in the largest invariant set not unique in general except for some special delayed systems.
Therefore, we have the following corollary.
Corollary 3.3. The unknown parameters of concerned delayed system are identifiable and , , , if unique in the largest invariant set .
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  and delayed Hopfield neural networks system , are employed to demonstrate the effectiveness of the proposed adaptive synchronization scheme.
4.1. Delayed Rössler System
Defining the error state and following the procedure proposed in the Section 2, we can design the adaptive scheme as follows: with the feedback strength updated by where , and are arbitrary positive constants.
4.2. Hopfield Neural Network Model
We refer to system (4.5) with as the drive system, where parameters are unknown. The response system has the following form: where , , and , , , and .
By comparing Figures 4 and 8, one can see that in Figure 8 the estimates of the unknown parameters, , converge to the true value of ; however, for the delayed Rössler system, the estimates of unknown parameters, , converge to , which are not equal to the true value of . It is reasonable because for Hopfield neural network system, in the largest invariant set unique, which is equal to the true value of ; but for delayed Rössler system, in the largest invariant set cannot be proven to be unique, which means what values do converge to depend on the initial value of .
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  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 , hence, the synchronization-based adaptive schemes may fail if the largest invariant set containing in 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  (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.
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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