Complexity
Volume 2018 (2018), Article ID 6504590, 13 pages
https://doi.org/10.1155/2018/6504590

Research Article

Almost Periodic Synchronization for Quaternion-Valued Neural Networks with Time-Varying Delays

Yongkun Li,¹ Xiaofang Meng,¹ and Yuan Ye²

¹Department of Mathematics, Yunnan University, Kunming, Yunnan 650091, China
²Graduate School, Yunnan University, Kunming, Yunnan 650091, China

Correspondence should be addressed to Yuan Ye; yye_yd@aliyun.com

Received 19 November 2017; Accepted 13 March 2018; Published 29 April 2018

Academic Editor: Arturo Buscarino

Copyright © 2018 Yongkun Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 system

and linear nonhomogenous system

where 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 as

where is the fundamental solution matrix of (3).

Lemma 5 (see [46]). Let be an almost periodic function on and

Then 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 as

where 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) that

where

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 controller

where .

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 system

Combining 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 system

The coefficients are taken as follows:

By a simple calculation, we have

Take ; 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).

Figure 1: The states of four parts of and .

Figure 2: The states of four parts of and .

Figure 3: Synchronization.

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.

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