Date Published: June 7, 2018
Publisher: Public Library of Science
Author(s): Yongkun Li, Huimei Wang, Takashi Nishikawa.
This paper studies the drive-response synchronization for quaternion-valued shunting inhibitory cellular neural networks (QVSICNNs) with mixed delays. First, QVSICNN is decomposed into an equivalent real-valued system in order to avoid the non-commutativity of the multiplicity. Then, the existence of almost periodic solutions is obtained based on the Banach fixed point theorem. An novel state-feedback controller is designed to ensure the global exponential almost periodic synchronization. At the end of the paper, an example is given to illustrate the effectiveness of the obtained results.
Quaternion was first proposed by Hamilton  in 1853. However, because of the non-commutativity of quaternion multiplicity, the development on quaternion was quite slow. Fortunately, with the development of modern science, the quaternion has been widely used in attitude control, quantum mechanics, computer graphics and so on, see [2–5] and references therein. In recent years, quaternion has attracted scholars from many fields, especially, the scholars in the field of neural network research. The quaternion-valued neural networks (QVNNs), as an special case of Clifford-valued neural networks , can be thought of as an extension of complex-valued neural networks (CVNNs) and real-valued neural networks (RVNNs). In fact, QVNNs can be applied to engineering and science. A great deal of studies have shown that, for the three dimensional data including color images and body images, via direct coding, QVNNs can do the process with high-efficiency . Indeed, based on the three primary colors and Hamilton rules of quaternion, one can realize the color face recognition efficiently. The quaternion representation treats the color image and dictionary in a holistic manner, while the real representation can only treat the three colors channels separately [8, 9]. Since all of these applications strongly rely on the dynamics of QVNNs, many researchers have studied some dynamical behaviours of QVNNs ([10–15]) recently.
We denote the skew field of quaternion by
where xR, xI, xJ, xK are real numbers and the elements i, j, k obey the Hamilton’s multiplication rules:
In this section, we establish the sufficient conditions for the existence of almost periodic solutions of system (1), and the sufficient conditions for the global exponential synchronization of the drive system (1) and the response system (9).
In this section, an example is shown for the effectiveness of the proposed method in this paper.
In this paper, a class of QVSICNNs with mixed delays is studied. To the best of our knowledge, this is the first on studying the problem. Since QVSICNNs include RVSICNNs and CVSICNNs as special cases, our method of this paper can be applied to study the almost periodic synchronization problem of other types of neural networks including RVNNs and CVNNs.