Asymptotical synchronization analysis of fractional-order complex neural networks with non-delayed and delayed couplings

Abstract

In this paper, the asymptotical synchronization of fractional-order complex neural networks with non-delayed and delayed couplings is investigated. By employing the Kronecker product technique and weighted norm, two Lyapunov functions are constructed. Based on the fractional-order Lyapunov direct method and Kronecker product method, together with the Laplace transform and the comparison principle, several novel sufficient conditions on synchronization of fractional-order complex neural networks with non-delayed and delayed couplings, respectively, are proposed. Finally, three numerical examples are given to show the effectiveness of our proposed theoretical results and the difference between our main result and some existing results in the literature.

Introduction

Neural networks have been extensively researched over the past few decades because of their wide applications in pattern recognition, signal processing, electrical circuits, knowledge engineering, optimization combination, robot control problems and so on [1], [2]. The applications of neural networks heavily depend on their dynamical behaviors. Stability and synchronization are two kinds of important dynamical behaviors of neural networks. Various kinds of stability problems of neural networks have been extensively investigated [3], [4], [5]. The essence of synchronization control is to force a system to track another one by imposing a suitable controller. Up to now, synchronization of neural networks has been successfully applied to secure communication, public channel cryptography, neural cryptography and many other areas. Synchronization of chaotic system and neural networks has become an important area of study. Since Pecora and Carroll [6] proposed the concept of synchronization for the drive-response systems, various kinds of synchronization including phase synchronization, anomalous synchronization, complete synchronization, exponential synchronization and robust synchronization have been widely studied. Besides the classical drive-response mechanism for chaos synchronization, a wide variety of synchronization approaches such as adaptive feedback controlling method, impulsive control method and adaptive design control are developed. Basing on parameter identification and via output or state coupling, Lou et al. [7] study the synchronization of neural networks with time-varying delay. Based on Lyapunov theory, some new sufficient conditions are derived to guarantee the global synchronization of the weighted cellular neural network in [8]. Synchronization problem of chaotic neural networks with mixed time delays is investigated in [9]. A sufficient condition for global asymptotical synchronization of master–slave chaotic neural networks via output feedback impulsive control is established in [10]. In 2020, by constructing Lyapunov functionals and applying some inequality analysis techniques, several delay-dependent exponential synchronization criteria and polynomial synchronization criteria of the drive-response systems are obtained in [11]. Many results on the synchronization of systems of integer order have been obtained.

It is well known that a network may be constituted by two or more circuits. If the current or voltage in one circuit changes, it can affect other circuits and make other circuits to have similar changes. This network is a coupling circuit. Since the capacitor has the characteristic of isolating direct current and alternating current. It can couple the AC signal to the next stage amplifier. Time delays are inevitable due to the limited switching speed of the amplifiers. So coupling exists widely in circuits. Coupling may transfer the energy from one circuit to other circuits. However, there are not many results on synchronization of the coupled neural networks [12], [13].

As a special class of complex neural networks, the coupled neural networks may exhibit more complicated and unpredictable behaviors. Synchronization of coupled neural networks can achieve state consistency for all nodes of networks. Synchronization can play a positive role to avoid negative effects. Since synchronization of a coupled dynamical network can well explain many natural phenomena observed, the theoretical research and practical applications of synchronization for the coupled neural networks is with profound significance. By employing Lyapunov functional method and Kronecker product technique, sufficient conditions are derived to guarantee the global synchronization for static neural networks with hybrid couplings and time delays in [13]. In order to cope with a large number of highly interconnected dynamical units in coupled neural networks, basing on the similar idea in [13], we will construct an appropriate Lyapunov function by using the Kronecker product of matrices and investigate the synchronization of coupled neural networks.

As a generalization of the integer order integration and differentiation to arbitrary real order, fractional calculus was first proposed by Leibniz in 17th century, it can deal with the differential and integral of any arbitrary order. Naturally, in order to better describe the neurons’ dynamical behaviors, a class of fractional-order neural networks is formed when the fractional calculus is incorporated into neural networks. The main characteristics of fractional-order systems are with infinite memories, hereditary and more degrees of freedom [14], [15], [16]. So fractional-order neural networks have more complex dynamic behaviors [17], [18], [19] than the classical integer order systems. Compared with the traditional integer order calculus, fractional-order calculus is a good tool for describing memory and genetic characteristics in processing [20], [21].

One more fundamental difference between fractional-order neural network and integer order neural network is non-locality. The classical differential operators are local operators, the fractional derivatives are not. In order to calculate the n-th order derivative $f^{\left(n\right)}\left(t\right)$, it is sufficient to know $f\left(t\right)$ in an arbitrarily small neighbourhood of t. However, to calculate fractional-order derivative ${}^C{}_0{D}_t^{\alpha }f\left(t\right)$ for $\alpha \notin N$, we need to know $f\left(t\right)$ throughout the entire interval $[0,t]$. Fractional-order neural network can make use of information of all states from the initial time to the current time. So the reality can be accurately described by fractional-order neural networks due to the fractional-order derivatives’ memory and heredity. Different from the classical Leibniz’ formula and the chain rule, Leibniz’ formula and the chain rule for the fractional differential operator correspond a infinite sum, respectively. Since the classical Leibniz’ formula and the chain rule fail to work, more difficulties will lie in the construction of Lyapunov functions and the estimation for their fractional-order derivatives. So most of results on synchronization of complex neural networks focus on the integer order case in the existing literature.

Recently, in [22], out lag synchronization criterion of fractional-order complex neural networks with non-delayed and delayed couplings (FOCNNNDC) is obtained by the Kronecker product knowledge. Cheng et al. [23] study the memristor-based fractional-order neural networks by using Lyapunov method. Some new sufficient conditions on the global Mittag–Leffler stability and synchronization are established in [23]. By taking information on activation functions into account, a fractional-order-dependent Lyapunov function and a new quadratic Lyapunov function are constructed [24]. Based on a fractional-order differential inequality, a Mittag–Leffler synchronization criterion in terms of LMIs is presented for drive-response fractional-order Hopfield neural networks under linear control. Adaptive pinning synchronization in fractional-order uncertain complex dynamical networks with delay is investigated in [25]. Gu et al. [26] study the projective synchronization of fractional-order memristive neural networks with switching jumps mismatch. So far, many excellent results on stability and synchronization of fractional differential systems have been obtained in [27], [28], [29], [30]. Obviously, fractional-order Lyapunov direct method is one of important tools for the dynamic behaviors analysis of complex dynamical networks. How to construct a suitable Lyapunov function is the key to carry out synchronization analysis of fractional-order complex dynamical networks. Time delay widely exists in real networks due to finite information transmission and processing speeds among the units of networks. It has significant effect on dynamical behavior of neural networks. It may cause the degradation of the system performance, even leads to instability [31], [32], [33], [34]. Time delay systems are very important both in theoretical research and practical design of neural networks. Both internal delay and coupling delay should be considered in dynamical networks. Recently, by applying fractional Razumikhin theorem, the global synchronization of fractional complex networks with non-delayed and delayed couplings is dealt in [35]. Zhang et al. [36] discuss the synchronization stability problem of the Riemann–Liouville fractional coupled complex interconnected delayed neural networks. Several synchronization stability criteria are established in terms of Lyapunov functional approach and linear matrix inequality technique. As far as we know, in Caputo sense, how to obtain useful algebraic criteria on synchronization for a fractional nonlinear delayed system still faces serious challenge.

Motivated by the above discussion, this paper aims to investigate Caputo fractional complex networks with non-delayed and delayed couplings. By applying the fractional-order Lyapunov direct method, Kronecker product method, Laplace transform and the comparison principle, several novel sufficient conditions on synchronization are proposed. The main contributions of this paper are summarized as follows:

• A suitable Lyapunov function is constructed by using Kronecker product technique. The fractional-order differential inequality for convex function is used to estimate the fractional-order derivative of Lyapunov function, which can avoid directly computing the Caputo fractional-order derivative of Lyapunov function.

• Different from method used in [36], by using Laplace transform and characteristic equation, together with Kronecker product method, two novel synchronization stability conditions for FOCNNNDC are obtained in terms of matrix inequalities.

• Based on the fractional-order Lyapunov direct method and the comparison principle, a weighted Lyapunov function and a new sufficient condition are also presented to achieve the asymptotical synchronization between fractional-order drive system and the response system.

The rest of the paper is organized as follows. In Section 2, some main definitions such as Caputo fractional calculus and Laplace transform are presented. Some useful lemmas are given. In Section 3, the fractional-order complex neural network model with non-delayed and delayed couplings is introduced. A necessary assumption is given. In Section 4, three asymptotical synchronization criteria for FOCNNNDC are given. In Section 5, three numerical examples are presented to verify the validity of the proposed theorems. The paper is concluded in Section 6.

Notations: In this paper, R is the set of real numbers. $R^{+}$ denotes the set of positive real numbers. $N^{+}$ denotes the set of positive integer numbers. $R^n$ is the n-dimensional Euclidean space. $R^{m\times n}$ denotes the space of $m\times n$ real matrices. $C^n\left(\right[0,+\infty ),R)$ denotes Banach space of all continuous and n-order differentiable functions. $L_1[a,b]$ denotes the set of absolute integrable functions on $[a,b]$. The superscript ^T stands for the transpose of matrix or vector. $A\otimes B$ stands for the Kronecker product of matrices A and B. For a vector $z={(z_1,z_2,\dots ,z_n)}^T\in R^n,\Vert z{\Vert }_1$ and $\Vert z{\Vert }_2$ denote 1-norm and 2-norm of z, respectively. $\Vert z{\Vert }_1=\left|z_1\right|+\left|z_2\right|+\cdots +\left|z_n\right|$, $\Vert z{\Vert }_2=\sqrt{z_1^2+z_2^2+\cdots +z_n^2}$. For a matrix $Q\in R^{n\times n},\Vert Q{\Vert }_1$ denotes 1-norm of matrix Q, $\Vert Q{\Vert }_1=\mathit{sup}\{\Vert \mathit{Qx}{\Vert }_1:\forall x\in R^n,|\left|x\right|{|}_1\le 1\}$. $Q>0$ means that matrix Q is symmetric and negative definite. $Q⩽0$ represents that matrix Q is negative semi-definite. $I_n$ represents the n-dimensional identity matrix. $\mathit{diag}\{k_1,k_2,\dots ,k_n\}$denotes a diagonal matrix with elements $k_1,k_2,\dots ,k_n$. The symmetric term in a symmetric matrix is denoted by $\ast$.