Novel methods to finite-time Mittag-Leffler synchronization problem of fractional-order quaternion-valued neural networks

Abstract

This paper proposes two methods to investigate the problem of finite-time Mittag-Leffler synchronization for the systems of fractional-order quaternion-valued neural networks (FQVNNs) with two kinds of activation functions, respectively. Generally, the first method mainly reflects in the new establishment of Lyapunov-Krasovskii functionals (LKFs) and the novel application of a new fractional-order derivative inequality which contains and exploits the wider coefficients with more values. Meanwhile, the second one is embodied in the comprehensive development of both the norm comparison rules and the generalized Gronwall-Bellman inequality with the help of Laplace transform of Mittag-Leffler function. Thanks to the above two methods, the flexible synchronization criteria are easily and separately obtained for the studied four systems of FQVNNs with general activation functions and linear threshold ones. Finally, two numerical simulations are given to demonstrate the feasibility and effectiveness of the newly proposed approaches.

Introduction

During the recent decade, the problem of chaos phenomenon and chaos synchronization has gotten more and more heated attention in many scientific research fields. There have been many kinds of solutions about the chaos synchronization in different dynamical systems, such as real-valued neural networks [3], [4], [10], complex-valued neural networks [2], [11], [19], quaternion-valued neural networks [12], [13], [14], [15], [16], memristive neural networks [5], [26], feedback control [5], [33], sampling control [38], sliding mode control [17], event-triggered control [40] and other nonlinear dynamics [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], [43]. Therefore, the research has become more popular in the wider applications and some potential development of chaos synchronization.

It is well-known that the nonlinear systems could be more accurately modeled and more generally applied by fractional-order systems than integer-order ones. On the basis of combination fractional derivative with nonlinear dynamical systems, the studied systems can have their superiorities such as more degrees of freedom and infinite memory owe to the essential characteristic of fractional derivative such as being nonlocal and owning weakly singular kernel [20]. In the past few years, many researchers have introduced fractional derivative to the dynamic study of various nonlinear systems. Particularly, by combining fractional derivative with neural networks, researchers have investigated fractional-order neural networks (FNNs) who-se memory terms are also participated. During the recent years, more and more researchers have paid an increasing attention to deal with the dynamical behaviours of FNNs such as stability [10], periodic solution [4], synchronization [10], [18], [26] and so on. Among the above dynamics analysis, the problem of synchronization for FNNs has been a hotpot. Up to now, there are numerous researches on various types of synchronization of FNNs such as Mittag-Leffler synchronization [5], [18], finite-time synchronization [26] and so on.

However, the above study are all based on real-valued neural networks (RVNNs). RVNNs have really received extensive concern because they can be widely applied in pattern recognition, static image treatment, combinatorial optimization and signal processing and so on. However, it is difficult to solve the problem of two-dimensional affine transforms in RVNNs. Fortunately, the above problem can be efficiently solved in complex-valued neural networks (CVNNs) [11]. However, as three-dime-nsional affine transforms such as body image develop, the construction of multidimension-valued neural networks becomes popular. Therefore, some researchers have introduced quaternion to neural networks to construct quaternion-valued neural networks (QVNNs) during the very recent years [12], [13], [14], [15], [16]. In [14], [15], authors studied the global $\mu -$ stability and exponential stability for QVNNs by creatively employing the separation method, respectively. In [6], the stability analysis was studied for QVNNs with linear threshold neurons both in discrete-time case and in continuous-time case. Moreover, authors in [12] investigated the problem of boundedness and periodicity for delayed QVNNs with linear threshold neurons. Furthermore, without decomposition, the global dissipativity problem was analyzed for QVNNs with time delays in [25]. In [30], authors realized the fixed-time synchronization for delayed and memristive QVNNs. Very recently, some researchers introduced complex or quaternion to FNNs to construct multidimension-valued FNNs such as fractional-order complex-valued neural networks (FCVNNs) [2], [19] and FQVNNs [33], [34], [36]. Authors in [19] discussed the global robust synchronization for FCVNNs. In [36], the problem of Mittag-Leffler stability and synchronization was studied for the system of FQVNNs. Further, authors in [33] introduced memristor and uncertain parameters to FQVNNs to realize the globally asymptotic synchronization. Although the problem of synchronization has been discussed for FQVNNs in [33], [36], the main disadvantage lies in that the synchronization time is infinite and can not settled. In practice, it is so necessary that the chaos synchronization could be realized in finite time.

Moreover, the design of the activation functions is very important. As authors in [6], [33], [36] claimed, the introduction of the linear threshold activation functions can help reduce unnecessary complexity and realize the linearization of system. In practice, the general activation functions are commonly adopted in neural networks. To complement the application of activation functions, the activation functions could be considered to be two cases such as general ones and linear threshold ones.

Motivated by the above analysis, this paper studies the problem of finite-time Mittag-Leffler synchronization for the system of FQVNNs with two cases of activation functions. We will mainly adopt two new methods to solve the problem of finite-time Mittag-Leffler synchronization of the specific four kinds of FQVNNs. On one hand, on basis of Lyapunov theory, the two criteria will be obtained owe to the new LKFs and the novel fractional-order derivative inequality with general coefficients. Moreover, the derived criteria could be adjusted as the relevant coefficients vary properly. That is, the criteria are not unique but alterable according to the suitable values of coefficients. On the other hand, the methods in [33], [36] are so ordinary and the proofs of synchronization criteria for FQVNNs in [33], [36] are too long to be understood. To improve the study method and reduce the trifling derivation of criteria, we will mainly employ both the norm comparison rules and the generalized Gronwall-Bellman inequality to obtain the simple criteria in norm form with the help of Laplace transform of Mittag-Leffler functions. Generally, the above criteria will be flexible and they could be easily verified. The processes of the derivation of the four criteria are concise and clear. This great advantage must help to improve the research of Mittag-Leffler synchronization for FQVNNs. The main progress of this work is summarized as follows:

• Based on the present systems such as [6], [33], [34], [36], the two kinds of activation functions are both considered in the studied systems and the solutions to deal with them are comprehensive and general.

• On basis of fractional-order Lyapunov theory, a novel fractional-order derivative inequality is applied and the new LKFs such as $V\left(t\right)={\sum }_{Q=R,I}^{J,K}{\sum }_{p=1}^n\left|{\widehat{\mu }}_p^Q{\widehat{z}}_p^Q\left(t\right)\right|$ is constructed by adding adjustable coefficients ${\widehat{\mu }}_p^Q$ with more values. Inspired by the previous work [4], the newer fractional-order derivative inequality in [34] is an extension of the derived one in [4] because the more general coefficients can be valued to be positive or negative, not limited to be 1.

• The Laplace transform of Mittag-Leffler function is firstly applied in the analysis on the dynamics of FQVNNs by combining the generalized Gronwall-Bellman inequality with norm comparison technique.

• Based on Lyapunov theory and norm knowledge, the flexible criteria are easily derived for the studied problem of the four cases of FQVNNs.