Mutiple $\psi$-type stability of fractional-order quaternion-valued neural networks

Highlights

• In this paper, the problem of multiple $\psi$-type stability of fractional-order quaternion-valued neural networks was investigated. To the best of our knowledge, this is the first time to discuss the multiple $\psi$-type stability problem for quaternion networks.

• A quaternion-valued matrix is uniquely decomposed into four real valued matrices and mapped into two similarly sized systems this method will help to avoid many redundant computations.

• Compared with the complex-valued neural networks quaternion valued neural networks have $3^{4n}$ equilibrium points. Among them $2^{4n}$ equilibrium points are exponentially stable.

Abstract

The multiple $\psi -$type stability of fractional-order quaternion-valued neural networks (FQVNNs) was investigated in this paper. Some new conditions ensuring the existence of multiple equilibrium points of the considered FQVNNs are provided. Meanwhile, the $\psi -$type stability for the proposed neural networks is studied by employing the fractional calculus theory and fractional derivative techniques into the system dynamics. Finally, an numerical simulation is given to show the effectiveness of the theoretical results.

Introduction

In recent years, real-valued neural networks (RVNNs) and complex valued neural networks (CVNNs) have been widely investigated for their successful applications in signal processing, pattern recognition, associative memory, engineering calculation and so on. In CVNNs, their states, connection weights, non linear activation functions, and external input vector are present in the complex domain ${\mathbb{C}}^n$. In RVNNs the activation functions are chosen to be smooth and bounded, but in CVNNs it is quite restricted to choose the actions because if we choose an activation to be smooth and bounded, it will be reduced to a constant, i.e., according to Liouvile’s theorem, the complex valued activation functions cannot be both bounded and analytic. The quaternion numbers are non-commutative division algebra [1], [2], [3], i.e, does not commute with multiplication, and so the study of quaternion numbers is much more complicated than that of real numbers as well as complex numbers which is the main reason for the slow improvement of the quaternion numbers. In many science and engineering applications, the multi-dimensional data is commonly encountered in general class of systems, and it can be solved using many real valued neurons. However, to some extent, this procedure is unusual, because some information can not be handled separately. Recently, researchers introduced the quaternion into neural networks to naturally describe multi dimensional informations, for example colour and 3-D coordinate via a quaternion valued neurons. QVNNs, is a generic extension of the RVNNs and CVNNs [4], [5], [6], [7]. QVNNs are much harder than that of CVNNs as well as RVNNs. In QVNNs stringent CRF (Cauchy–Riemann) condition and GCR (generalized Cauchy Riemann) conditions are provided to guarantee that the global analytical quaternion functions are only linear and constants respectively, which ensure that the quaternion functions are analytic. Therefore, recently many authors have considered the local alternative to the given CRF conditions, i.e, LAC (local analyticity condition) can allow use standard activations like tanh functions. Thus the QVNNs has received more increasing interest recently, due to its applications in information processing, optimization and automatic control [8], [9], [10], [11], [12].

Fractional calculus is a part of mathematics that plays an important role in derivatives and integral in the form of non-integer order. The development of fractional-order calculus have found extensive applications in different fields of those possessing dynamical nature and uncertain behaviors. Such type of applications includes biological modeling, control theory, engineering, etc., that motivates the scientists and researchers to concentrate more on the analysis of fractional-order characteristics into the systems. Fractional order derivatives also provide an outstanding instrument for describing the memory and hereditary properties of different materials and their processes. Also this system has infinite memory. Based on these advantages in the dynamic neural network like systems, the incorporation of the fractional-order calculus has gained desired better results than the integer-order ones [13], [14], [15], [16], [17]. As is well known, there are several advantages between the integer order neural networks and the corresponding fractional-order neural networks. However, the key difference is that the fractional-order systems are more precise than integer order systems, that is there are more degrees of freedom in the fractional-order systems. In addition, compared with existing integer-order systems, the infinite memory characterizes the fractional-order systems. Considering all the aforementioned explanations, the incorporation of a memory term into a neural network model is an unavoidable one.

Moreover, recently many dynamical behaviors of the neural network systems were discussed such as, the stability analysis, synchronization and bifurcations [18], [19], [20], [21], [22], [23], [24], [25], [26]. That is, the considered QVNNs tends to converge to an equilibrium point or a periodic orbit or a chaotic trajectory [19], [25]. As one of the classical phenomenon of fractional neural networks, the multi stability analysis has been extensively studied in [2],[27], [28], [29], [30], [31], [32], [33], [34], [35], [36]. It is known that, the multiple stability of the designed neural networks is the great requirement in various applications, such as optimal computations, associative memory, and pattern recognition [37], [38], [39]. Especially in pattern recognition, the system can converge to a certain stable equilibrium point for the process of memory attainment where the pattern is stored as binary vectors. Thus, it is more important to analyze the existence of multiple equilibra and their stable points. In [40], the sigmoid functions are employed in a class of recurrent neural networks and the existence of ${\left[(2K_i+1)\right]}^n$ equilibrium points and Mittag-Leffler stability of ${\left[(K_i+1)\right]}^n$ points are discussed. In [41] a class of integer-order recurrent neural networks with unbounded time varying delays were introduced, and it was shown that the addressed system have exactly ${(2K_i+1)}^n$ equilibrium points, of which $(K_i+1)$ were locally asymptotically stable while others are unstable by using the geometrical properties of non-monotonic activation functions. As a novelty, this work contributes to extend the results of multiple $\psi -$type stability of integer-order RVNNs or CVNNs to those of FQVNNs. To show that there exists more eqilibrium points in QVNNs than RVNNs, most existing works were done with integer-order ones. The difference between quaternion-valued neural networks and general neural networks is that QVNNs is the generic extended form of the general RVNNs with a four-dimensional hyper complex number system introduced by Hamilton. All the QVNNs parameters are quaternion valued parameters. However, stability problems in complex domain is used in some several traditional methods but it cannot be applied directly in quaternion field. Therefore discussing the dynamical studies of QVNNs with different approach is still open and challenging.

Motivated by the above discussion, we explore the study of multiple $\psi -$type stability results on FQVNNs. First, an $n$-dimensional QVNNs is converted into the $4n$-dimensional RVNNs system by using the decomposition and non commutativity properties of quaternions. Then some sufficient conditions are derived for the existence of equilibrium points. Then by selecting novel functions and parameters, we derive the multiple $\psi -$type stability of FQVNNs. Finally, a numerical example is given to demonstrate the effectiveness of the theoretical results. This paper’s main contributions can be addressed in brief as follows:

• To the best of our knowledge, the multiple $\psi -$type stability of FQVNNs is firstly studied;

• A quaternion-valued matrix is uniquely decomposed into four real valued matrices and mapped into two similarly sized systems, i.e, the quaternion valued matrix can be uniquely expressed as sum of four real valued matrices, this method will help us to avoid many redundant computations;

• Therefore, we study the multiple $\psi -$type stability of the FQVNNs by using a plural decomposition method of quaternions;

One of the main difficulty of this paper is the non-commutativity of quaternion multiplication. Hopefully, we can transform quaternion-valued systems into real-valued systems with the plural decomposition approach of quaternions by Hamilton rules. Then the QVNNs is decomposed into four real valued systems which are equivalent to the original QVNNs. The rest of this paper is arranged as follows: In Section 2, the QVNNs model description and basic preliminaries are provided. In Section 3, some sufficient conditions for the multiple $\psi -$type stability of FQVNNs are derived by the method of decomposing quaternion into four real valued systems. In Section 4, one numerical example is given to show that the effectiveness of the our theoretical results. Finally, Section 5 concludes this paper.

Notations: In this paper, $\mathbb{R}$ and $\mathbb{Q}$ represents the set of real and quaternion numbers, respectively. ${\mathbb{Q}}^n$ is the $n-$dimensional quaternion-valued vector space. ${\mathcal{C}}^m([0,+\infty ),\mathbb{R})$ is the Banach space of all continuous and $m$-differentiable functions. $D^{\nu }$ is the Caputo fractional derivative operator of order $\nu .$