Abstract
This paper focuses on the global exponential almost periodic synchronization of quaternion-valued neural networks with time-varying delays. By virtue of the exponential dichotomy of linear differential equations, Banach’s fixed point theorem, Lyapunov functional method, and differential inequality technique, some sufficient conditions are established for assuring the existence and global exponential synchronization of almost periodic solutions of the delayed quaternion-valued neural networks, which are completely new. Finally, we give one example with simulation to show the applicability and effectiveness of our main results.
1. Introduction
It is well known that the quaternion is a mathematical concept discovered by Hamilton in 1843, which did not arouse much attention for quite a long time, let alone real world applications. Due to the noncommutativity of quaternion multiplication, the study of quaternion is much more difficult than that of plurality, which is one of the reasons for the slow development of quaternion. Fortunately, over the past two decades, the quaternion theory has achieved a rapid development, especially in algebra, and found many applications in the real world, like attitude control, quantum mechanics, robotics, computer graphics, and so on [1–5]. For example, in the application of color image compression technology, one can apply the quaternion theory to encode and improve the color image; see [5]. In recent years, the quaternion-valued neural networks, as an extension of the real-valued neural networks and the complex-valued neural networks [6, 7], research has become a hot topic. It should be pointed out that, at present, almost all the investigations on quaternion-valued neural networks are mainly dealing with the stability, robustness, or dissipation of the equilibrium of the neural networks; see [8–17].
On the one hand, since the concept of drive-response synchronization for coupled chaotic systems was proposed by Pecora and Carroll in [18], chaos synchronization has become a hot research topic due to its potential applications in secure communication, automatic control, biological systems, and information science; see, for instance, [19–26].
On the other hand, the almost periodicity is a generalization of periodicity and is much more common than the periodicity, and so, almost periodic oscillation is a very important dynamic phenomenon for nonautonomous neural networks [27–30]. However, so far, few results have been available for the almost periodicity of complex-valued neural networks [31] and quaternion-valued neural networks. Moreover, in the past few years, although the problem of periodic synchronization for real-valued systems has been extensively studied by many scholars (see, for instance, [32–43]), there are few articles dealing with the almost periodic synchronization for real-valued systems, let alone dealing with the almost periodic synchronization for complex-valued or quaternion-valued neural networks. Very recently, some authors study the synchronization for complex-valued neural networks with time delays [44, 45]. Based on the LMI method and mathematical analysis technique, they obtain some sufficient conditions for synchronization of complex-valued neural networks. However, to the best of our knowledge, there has been no paper published on the almost periodic synchronization of the quaternion-valued neural networks, which remains as an open challenge. So, it is necessary to investigate the synchronization of quaternion-valued neural networks.
Motivated by the above statement, the main purpose of this paper is to study the synchronization of quaternion-valued neural networks. The main contributions of this paper can be summarized as follows: this is the first time to investigate the almost periodic synchronization of neural networks. Compared with other results, the results of this paper are the ones about quaternion value and with time delays. Therefore, the results are less conservative and more general. Our methods used in this paper can easily apply to study the almost periodic synchronization for other types of neural networks with delays.
This paper is organized as follows: In Section 2, we give the model description and introduce some definitions and preliminary lemmas and transform the quaternion-valued system (9) into an equivalent real-valued system. In Section 3, we establish some sufficient conditions for the existence and global exponential synchronization of almost periodic solutions of (9). In Section 4, we give an example to demonstrate the feasibility of our results. This paper ends with a brief conclusion in Section 5.
2. Model Description and Preliminaries
In this section, we shall first recall some fundamental definitions and lemmas which are used in what follows.
First, we give some notations of this paper. Let and stand for the real field and the skew field of quaternions, respectively. , denote the set of all real-valued and quaternion-valued matrices, respectively. The skew field of quaternion is denoted by where , , , are real numbers and the elements , and obey Hamilton’s multiplication rules:
Definition 1 (see [46], let ). Function is said to be almost periodic if, for any , it is possible to find a real number , for any interval with length ; there exists a number in this interval such that , for all .
We denote by the set of all almost periodic functions from to .
Definition 2. Quaternion-valued function is an almost periodic function if and only if , , and are almost periodic functions.
Consider the following linear homogenous systemand linear nonhomogenous systemwhere is an almost periodic matrix function and is an almost periodic vector function.
Definition 3 (see [46]). System (3) is said to admit an exponential dichotomy if there exist a projection and positive constants such that the fundamental solution matrix satisfies
Lemma 4 (see [46]). If the linear system (3) admits an exponential dichotomy, then system (4) has a unique almost periodic solution that can be expressed aswhere is the fundamental solution matrix of (3).
Lemma 5 (see [46]). Let be an almost periodic function on andThen the linear system admits an exponential dichotomy on .
Consider the following quaternion-valued neural networks with time-varying delays:where and is the state of the th neuron at time ; is the self-feedback connection weight; and are the connection weight and the delay connection weight from neuron to neuron at time , respectively; is an external input on the th unit at time ; and denote the activation functions without and with time-varying delays, respectively; represents the time-varying delay and satisfies .
The initial value of (9) is given by where .
The response system of (9) is designed aswhere denotes the state of the th neuron at time of the response system and is a controller.
The initial condition associated with (11) is of the form where .
In order to overcome the inconvenience of the noncommutativity of quaternion multiplication, in the following, we always assume the following:Let ,. Then and can be expressed as where , , .
According to , system (9) can be transformed into the following real-valued system:where , , and It follows from (14) thatwhere The initial condition associated with (16) is of the form where .
Similarly, the response system (11) can be transformed into the following real-valued system:where .
The initial condition associated with (19) is of the form where .
Let , . Then, from (16) and (19), we obtain the following error system:In order to realize the almost periodic synchronization of the drive-response system, we choose the controllerwhere .
Definition 6. The response system (19) and the drive system (16) can be globally exponentially synchronized, if there exist positive constants , such that where , .
Let with the norm ; then is a Banach space.
Throughout this paper, we assume that the following conditions hold:Function with , , and with are almost periodic, where .There exist positive constants such that for all , and , where , .There exists a positive constant such that where There exists a positive constant such that for ,
3. Main Results
In this section, we will study the existence of almost periodic solutions of system (16).
Theorem 7. Let – hold. Then system (16) has a unique almost periodic solution in the region .
Proof. For any , we consider the linear almost periodic differential equation systemCombining and Lemma 5, we can conclude that the linear system admits an exponential dichotomy on ; furthermore, by Lemma 4, we conclude that system (28) has a unique almost periodic solution that can be expressed as where Now, we define an operator as follows:First, we show that, for any , we have . From (32), we have In a similar way, we can get By , we have which implies that , so the mapping is a self-mapping from to . Next, we shall prove that is a contraction mapping. In fact, for any , we have Similarly, we can get By , we have Hence, is a contraction mapping. Therefore, system (16) has a unique almost periodic solution in the region . The proof is complete.
Theorem 8. Suppose that – hold. Then the drive system (16) and the response system (19) are globally exponentially synchronized based on the controller (22).
Proof. In view of the error system (21), we have In a similar way, we can obtain Now, we consider the Lyapunov function as follows: where Computing the derivative of along the solutions of the error system (21), we can get Similarly, we can get From , we obtain which implies that for all .
On the other hand, we have Similarly, we can get It is obvious that ; thus, for , we have where Therefore, the drive system (16) and the response system (19) are globally exponentially synchronized based on the controller (22). The proof is complete.
4. An Example
In this section, we give an example to illustrate the feasibility and effectiveness of our results obtained in Section 3.
Example 1. Consider the following quaternion-valued neural networks with time-varying delay:and its response systemThe coefficients are taken as follows:By a simple calculation, we haveTake ; then we have , , Moreover, take , , , , , , , , , , We can verify that all the assumptions of Theorems 7 and 8 are satisfied. Therefore, the drive system (50) and its response system (51) are globally exponential synchronized (see Figures 1–3).
5. Conclusion
In this paper, we consider the global exponential almost periodic synchronization of quaternion-valued neural networks with time-varying delays. By means of the exponential dichotomy of linear differential equations, Banach’s fixed point theorem, Lyapunov functional method, and differential inequality technique, we establish the existence and global exponential synchronization of almost periodic solutions for system (9). To the best of our knowledge, this is the first time to study the almost periodic synchronization problem for neural networks. Furthermore, the method of this paper can be used to study other types of neural networks.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work is supported by the National Natural Sciences Foundation of People’s Republic of China under Grant 11361072.