Synchronization of Chaotic Delayed Fuzzy Neural Networks under Impulsive and Stochastic Perturbations

and Applied Analysis 3 (A 1 ) f i is globally Lipschitz continuous, that is, for any i ∈ N, there exists nonnegative constant L i such that 󵄨󵄨󵄨fi (u) − fi (v) 󵄨󵄨󵄨 ≤ L i |u − v| for u, v ∈ R. (3) (A 2 ) For any k ∈ N, there is a nonnegative constant η k such that 󵄨󵄨󵄨u + Iik (u) − v − Iik (v) 󵄨󵄨󵄨 ≤ ηk |u − v| for u, v ∈ R, i ∈ N. (4) Let (2) be the drive system, and let the response system with random noise be described by dy i (t) = [ [ −c i y i + n ∑ j=1 a ij f j (y j ) + n ∑


Introduction
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 [1] 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 [2].So far, many important results on stability analysis and state estimation of FCNNs have been reported (see [3][4][5][6][7][8][9][10][11][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 [15] 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][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32].
(1) 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][26][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 [32], 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.
(2) The criteria proposed in [25][26][27][28][29][30][31][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 [29], 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.
(3) Some conditions imposed on the impulsive perturbations are too strong.For instance, Feng et al. [31] 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 M-matrix and the Itô'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.

Model Description and Preliminaries
Let R  be the space of -dimensional real column vectors, and let R × represent the class of  ×  matrices with real components.| ⋅ | denotes the Euclidean norm in R  .The inequality "≤" (">") between matrices or vectors such as  ≤  ( > ) means that each pair of corresponding elements of  and  satisfies the inequality "≤" (">"). ∈ R × is called a nonnegative matrix if  ≥ 0, and  ∈ R  is called a positive vector if  > 0. The transpose of  ∈ R × or  ∈ R  is denoted by   or   .Let  denote the unit matrix with appropriate dimensions.N := {1, 2, . . ., }, and For ,  ∈ R × and  : R → R  , we denote that where and  + () denotes the upper-right derivative of () at time .
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  = 2.
For any nonsingular M-matrix  (see [34]), we define that Lemma 2 (see [35]).For a nonsingular M-matrix , M  is a nonempty cone without conical surface.

Exponential Synchronization
In this section, by using Lemma 6, we will obtain some sufficient criteria to synchronize the drive-response systems (2) and ( 5).
We denote by  = ( 1 , . . .,   )  the solution of error dynamical system (8) with the initial value  −  ∈ Calculating the time derivative of   (()) along the trajectory of error system (8) and by the Itô's formula [33], we get for any  ∈ N, where L  (()) is given by It follows from ( 4 ) and ( 32) that Substituting ( 38) into (33) gives Integrating and taking the expectations on both sides of (39) lead to where  > 0 is small enough such that ,  +  ∈ [ −1 ,   ) for  ∈ N. By the continuity of E  (()), we conclude that which implies that for  ̸ =   ,  ∈ N, Meanwhile, it follows from ( 2 ) and ( 32) that for  ∈ N and  ∈ N, which means that which further indicates that where  = (‖ − ‖  F /min ∈N {  }).This implies that condition (16) in Lemma 6 holds.
If the random noise has not been considered, which means  ≡ 0 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.
The rest proof is similar to Theorem 7 and omitted here.We complete the proof.
Case 1.The response system without random noise is given by The control gain matrices  and  can be chosen as Remark 12.It is worth noting that the impulsive perturbations   > 2, which are not a satisfied condition (2) in [31].That is to say, even in the absence of the unbounded distributed delays, the results in [31] still cannot be applied to the synchronization problem of (57) and (59).
is a nonsingular M-matrix, which implies that ( 4 ) holds.Meanwhile, we can choose  = 0.1 and  = (1, 3)  ∈ M D such that ( 5 ) holds.Hence, by Theorem 7, the driveresponse systems (57) and (62) are globally exponentially synchronized in mean square.The simulation result based on Euler-Maruyama method is illustrated in Figure 3.

Conclusion
In this paper, we investigate the synchronization problem of IFCNNs with mixed delays.Based on the properties of nonsingularM-matrix and some stochastic analysis approaches, some useful synchronization criteria under both impulse and random noise are obtained.The methods used in this paper are novel and can be extended to many other types of neural networks.These problems will be considered in near future.
1 (), . . .,   ())  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 ,  ∈ N,   () corresponding to the transmission delay satisfies 0 ≤   () ≤ , and   ∈ L  is the feedback kernel.For any  ∈ N,   (⋅) represents the impulsive perturbation, and   denotes impulsive moment satisfying   <  +1 , lim  → +∞   = +∞.)   is globally Lipschitz continuous, that is, for any  ∈ N, there exists nonnegative constant   such that       () −   ()     ≤   | − | for ,  ∈ R.