Modeling, Analysis, and Applications of Complex SystemsView this Special Issue
Research Article | Open Access
Synchronization of Chaotic Delayed Fuzzy Neural Networks under Impulsive and Stochastic Perturbations
The synchronization problem of chaotic fuzzy cellular neural networks with mixed delays is investigated. By an impulsive integrodifferential inequality and the Itô's formula, some sufficient criteria to synchronize the networks under both impulsive and stochastic perturbations are obtained. The example and simulations are given to demonstrate the efficiency and advantages of the proposed results.
Fuzzy cellular neural network (FCNN), which integrated fuzzy logic into the structure of a traditional cellular neural networks (CNNs) and maintained local connectivity among cells, was first introduced by T. Yang and L. Yang  to deal with some complexity, uncertainty, or vagueness in CNNs. Lots of studies have illustrated that FCNNs are a useful paradigm for image processing and pattern recognition . So far, many important results on stability analysis and state estimation of FCNNs have been reported (see [3–12] and the references therein).
Recently, it has been revealed that if the network’s parameters and time delays are appropriately chosen, then neural networks can exhibit some complicated dynamics and even chaotic behaviors [13, 14]. The chaotic system exhibits unpredictable and irregular dynamics, and it has been found in many fields. Since the drive-response concept was proposed by Pecora and Carroll  in 1990 for constructing the synchronization of coupled chaotic systems, the control and synchronization problems of chaotic systems have been extensively investigated. In recent years, various synchronization schemes for chaotic neural networks have derived and demonstrated potential applications in many areas such as secure communication, image processing and harmonic oscillation generation; see [16–32].
Synchronization procedures and schemes are rather sensitive to the unavoidable channel disturbances which are usually presented in two forms: impulse and random noise. However, in [25–27], authors provided some new schemes to synchronize the chaotic systems without considering both impulse and random noise. In [28, 29], under the condition of no channel disturbance, Yu et al. and Xing and Peng studied the lag synchronization problems of FCNNs, respectively. In [30, 31], authors studied the synchronization of impulsive fuzzy cellular neural networks (IFCNNs) with delays. In , authors derived some synchronization schemes for FCNNs with random noise. In fact, in real system, it is more reasonable that the two perturbations coexist simultaneously.
The criteria proposed in [25–32] are valid only for FCNNs with discrete delays. For example, in [25, 28, 30, 31], the involved delays are constants. In [26, 27, 32], the involved delays are time-varying delays which are continuously differentiable, and the corresponding derivatives are required to be finite or not greater than 1. In , Xing and Peng provided some new criteria on lag synchronization problem of FCNNs but they only considered the case for bounded time-varying delays. In fact, time delays may occur in an irregular fashion, and sometimes they may be not continuously differentiable. Besides this, distribution delays may also exist when neural networks have a spatial extent due to the presence of a multitude of parallel pathways with a variety of axon sizes and lengths.
Some conditions imposed on the impulsive perturbations are too strong. For instance, Feng et al.  required the magnitude of jumps not to be smaller than 0 and not greater than 2. However, the disturbance in the real environment may be very intense.
Therefore, it is of great theoretical and practical significance to investigate synchronization problems of IFCNNs with mixed delays and random noise. However, up to now, to the best of our knowledge, no result for synchronization of IFNNs with mixed delays and random noise has been reported.
Inspired by the above discussion, this paper addresses the exponential synchronization problem of IFCNNs with mixed delays and random noise. Based on the properties of nonsingular -matrix and the It’s formula, we design some synchronization schemes with a state feedback controller to ensure the exponential synchronization control. Our method does not resort to complicated Lyapunov-Krasovskii functional which is widely used. The proposed synchronization schemes are novel and improve some of the previous literature.
This paper is organized as follows. In Section 2, we introduce the drive-response models and some preliminaries. In Section 3, some synchronization criteria for FCNNs with mixed delays are derived. In Section 4, an example and its simulations are given to illustrate the effectiveness of theoretical results. Finally, conclusions are drawn in Section 5.
2. Model Description and Preliminaries
Let be the space of -dimensional real column vectors, and let represent the class of matrices with real components. denotes the Euclidean norm in . The inequality “” (“”) between matrices or vectors such as () means that each pair of corresponding elements of and satisfies the inequality “” (“”). is called a nonnegative matrix if , and is called a positive vector if . The transpose of or is denoted by or . Let denote the unit matrix with appropriate dimensions. , and , .
is continuous and bounded for all but at most countable points and at these points, and exist, . Here, is an interval; and denote the right-hand and left-hand limits of the function , respectively. Especially with the norm for .
is piecewise continuous and satisfies for some constant .
For and , we denote that and denotes the upper-right derivative of at time .
Consider IFCNNs with mixed delays as follows: where , denotes the number of units in the neural network. represents the state variable. is the activation function of the th neuron. represents the passive decay rate to the state of th neuron at time . and are elements of the fuzzy feedback MIN template. and are elements of the fuzzy feedback MAX template. and are elements of fuzzy feed-forward MIN template and fuzzy feed-forward MAX template, respectively. and are elements of feedback and feed-forward template, respectively. and denote the fuzzy AND and fuzzy OR operations, respectively. and denote input and bias of the th neuron, respectively. For any , corresponding to the transmission delay satisfies , and is the feedback kernel. For any , represents the impulsive perturbation, and denotes impulsive moment satisfying , .
We make the following assumptions throughout this paper. is globally Lipschitz continuous, that is, for any , there exists nonnegative constant such that For any , there is a nonnegative constant such that
Let (2) be the drive system, and let the response system with random noise be described by where is an -dimensional standard Brownian motion defined on a complete probability space with a natural filtration generated by and satisfying the usual conditions (i.e., it is right continuous, and contains all -null sets). The initial value which denotes the family of all bounded -measurable and -valued random variables with the norm , where denotes the expectation of stochastic process. is the state feedback controller designed by where , are the controller gain matrices to be scheduled. The diffusion coefficient matrix (or noise intensity matrix) satisfies the local Lipschitz condition and the linear growth condition (see ). In addition, for , there exist nonnegative constants , such that where .
For convenience, we use the following notations: , , , , with , for , , , , , , and .
The following definition and lemmas will be employed.
Definition 1. The systems (2) and (5) are called to be globally exponentially synchronized in -moment, if there exist positive constants , such that
It is said especially to be globally exponentially synchronized in mean square when .
For any nonsingular -matrix (see ), we define that
Lemma 2 (see ). For a nonsingular -matrix , is a nonempty cone without conical surface.
Lemma 3 (see ). For , , and , The sign of equality holds if and only if for all .
Lemma 5 (see ). For the integer and , there exists a positive constant such that
Lemma 6. For , assume that satisfies in which, with for , , , and , . is a nonsingular -matrix.Then, there must exist and such that provided that the initial value satisfies where , , and , can be determined by
Proof. By condition and Lemma 2, we can find such that , namely, . By the continuity, there must be some positive constant satisfying
Noting that , we can find a constant such that . Denote that . Obviously, and satisfy (16) and (17).
Let for , . For any small enough , (16) implies that , . Next, we claim that for any , If inequality (19) is not true, then there must exist some and such that On the other hand, (14) together with (17) and (20) leads to which contradicts (21). Therefore, (19) holds. Letting in (19), we get Suppose that for , the following inequalities hold For , from (14) and (24), we have Recalling , it follows from (24) and (25) that Repeating the proof similar to (19) can yield By the mathematical induction, we derive that for any , The proof is completed.
3. Exponential Synchronization
Theorem 7. Assume that – hold and for , is a nonsingular , where , , , with the impulsive perturbations satisfy where , and is determined by Then, drive-response systems (2) and (5) are globally exponential synchronization in -moment.
Proof. Since is a nonsingular -matrix, by Lemma 2 and the continuity, there must be a constant vector and a constant such that (31) holds.
We denote by the solution of error dynamical system (8) with the initial value and let Calculating the time derivative of along the trajectory of error system (8) and by the Itô’s formula , we get for any , where is given by By and Lemma 4, we have Using Lemma 3 and (), it is easy to get Thus, we have It follows from () and (32) that Substituting (38) into (33) gives Integrating and taking the expectations on both sides of (39) lead to where is small enough such that for .
By the continuity of , we conclude that which implies that for , , Meanwhile, it follows from () and (32) that for and , which means that Obviously, (42) and (44) indicate that satisfies inequality (14) in Lemma 6.
On the other hand, noting that in (8) and by a simple calculation, we get for any , which means for and . Recalling the definition of in (32) and , we conclude that for which further indicates that where . This implies that condition (16) in Lemma 6 holds.
Therefore, by Lemma 6, we derive that with . Meanwhile, (30) implies that there is a small enough constant such that Thus, inequality (48) together with (49) shows that for and for any , By Lemma 5, we get where . The proof is complete.
Remark 8. In [25–32], the authors established some useful criteria for ensuring synchronization of FCNNs with delays, respectively. However, once the unbounded distributed delays are involved, all results in [25–32] will be invalid. Hence, in this sense, the proposed Theorem 7 has a wider range of applications than those in previous papers.
Remark 9. In synchronization scheme, we take both impulsive perturbations and random noise into account. Comparing with the results in [25–32], Theorem 7 can reflect a more realistic dynamical behavior and synchronization procedure.
If the random noise has not been considered, which means in (5), then the response system reduces to In this case, the following globally exponential synchronization scheme for drive-response IFCNNs (2) and (53) can be derived.
Theorem 10. Assume that and hold and is a nonsingular -matrix, where , , the impulsive perturbations satisfy where , and is determined by Then, the drive-response systems (2) and (53) are globally exponential synchronization.
Proof. Let . Calculating the time derivative of along with the trajectory of error system can give The rest proof is similar to Theorem 7 and omitted here. We complete the proof.
Remark 11. In , Feng et al. derived some criteria on the globally exponential synchronization for a special case of (2) and (53) with , and for , . In order to achieve synchronous control, the conditions as for , have been imposed on the impulsive perturbations. However, Theorem 10 drops these restrictions.
4. Illustrative Example
In this section, a numerical example and its simulations are given to illustrate the effectiveness of our results.
Example 1. Consider the following 2-dimensional IFCNNs with mixed delays as the drive system
where , . for . For the simplicity of computer simulations, we choose for , for . The system parameters are as follows:
We can choose and such that and holds, respectively. Obviously, holds with , .
Choosing the initial value for , the drive system (57) possesses a chaotic behavior as shown in Figure 1.
Case 1. The response system without random noise is given by