Synchronization of delayed fractional-order complex-valued neural networks with leakage delay

https://doi.org/10.1016/j.physa.2020.124710Get rights and content

Highlights

  • The original FOCVNNs is no longer required to be separated into two FORVNNs. In this paper, the system is investigated as an entirety.

  • Both leakage delays and discrete delays are taken into account for FOCVNNs, with the goal of extending and improving existing results.

  • A new fractional inequalities is constructed, which plays a key role in the discussion on synchronization of FOCVNNs.

Abstract

This paper talks about the global synchronization of delayed fractional-order complex valued neural networks (FOCVNNs) with leakage delay. A new fractional differential inequality is proposed, which offers an important tool in the investigation of synchronization about FOCVNNs. Through constructing appropriate Lyapunov function and using the fractional order comparison theory, some new synchronization conditions are established. A numerical example is given to demonstrate the feasibility of the proposed method.

Introduction

Over the past decades, the fractional order dynamical systems have received a heated research in various fields. The fractional order systems offer infinite memory and more degrees of freedom compared to integer order systems. They can also offer more accurate instrument for the description of memory and genetic properties. Recently, many researchers have explored that fractional order derivatives can be incorporated into neural networks, forming fractional order neural networks (FONNs). Numerous interesting results on dynamical behaviors for FONNs have been reported in [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12].

All previous works considered the fractional order real valued neural networks (FORVNNs). Although real value neural networks (RVNNs) can be applied in many fields, they also have limitations. For example, in signal processing, the data processed are complex values and therefore could not be solved by RVNN. Spontaneously, the model of complex valued neural networks (CVNNs), utilizing complex valued variables and parameters, was proposed. Hence, CVNNs can directly deal with data in a complex domain, giving more attentions on studying the dynamical behaviors of fractional order complex valued neural networks (FOCVNNs) [13], [14].

Delays are inescapable in the real world, and many researchers have considered delayed FOCVNNs and produced interesting results on dynamical behaviors, including bifurcations, dissipativity and stability analysis and synchronization, as shown in [15], [16], [17], [18], [19], [20], [21], [22], [23]. The fundamental goal in all previous studies was that the original CVNNs was first separated into two RVNNs, hence studying the analysis on stability and synchronization. The method is reasonable, but encounters two problems: (1) The dimensions of the model are double that of CVNNs, increasing complexity in analysis and calculation. (2) The complex signal carries both amplitude and phase of the information signal, and therefore processing with two real-value systems may cause losing some information, which is intrinsically related to the amplitude and phase of the signal. In order to solve these two problems, a new idea is proposed, which directly discusses the synchronization of FOCVNNs using the theory of complex valued field.

On the other hand, the leakage delay tremendously affects the dynamical behaviors of systems. It was pointed out that the leakage delay has a great destabilizing impact on systems [24], [25]. It revealed that the leakage delay cannot be ignored in real systems. So far, there exist few studies considering FOCVNNs with leakage delay [26], [27], [28]. For example, in Ref. [26], authors discussed the stability of FOCVNNs with leakage delay based on the contracting mapping principle. Ref. [27] studied the finite-time stability of memristor FOCVNNs with leakage delay. It is important to emphasize that the synchronization analysis of delayed FOCVNNs with leakage delay has not been studied. Furthermore, due to the technical complications, it is difficult to handle these two delays. Therefore, this constitutes the object of this paper.

In comparison with previous studies, the novelties of this paper can be highlighted in the following aspects: (1) The original FOCVNNs is no longer required to be separated into two FORVNNs. In this paper, the system is investigated as an entirety. (2) Both leakage delay and discrete delay are taken into account for FOCVNNs, with the goal of extending and improving existing results. (3) A new fractional inequality is constructed, which plays a key role in the discussion on synchronization of FOCVNNs.

Notations

Throughout this paper, let R and C be the real value and complex value, Cn and Cn×m denote n-dimensional complex vectors and n×m complex matrices. x̄ denotes the conjugate complex value of x, A stands for the conjugate transpose of the matrix A.

Section snippets

Preliminaries

Throughout this paper, Caputo derivative is adopted to discuss the FOCVNNs.

Definition 1

[29]

For a function g(t), its Caputo fractional derivative of order α is given as follows:

Dαg(t)=1Γ(nα)0t(tτ)nα1g(n)(τ)dτ,

where t0, n1<α<n with n is a positive integer, Γ(m)=0tm1etdt is the Gamma function.

We will consider a class of delayed FOCVNNs with leakage delay: Dαxj(t)=cjxj(tσ)+k=1najkfk(xk(t))+k=1nbjkgk(xk(tτ))+Ij(t),j=1,2,,n or equivalently Dαx(t)=Cx(tσ)+Af(x(t))+Bg(x(tτ))+I(t),where α(0,1), x(t)=

Synchronization results

This section addresses the synchronization of delayed FOCVNNs with leakage delay.

We define the error vector between (1), (3) as follows: e(t)=y(t)x(t),where e(t)=(e1(t),,en(t))TCn,ej(t)=yj(t)xj(t),j=1,2,,n.

By combining (1), (3), (10), the above error system is given by: Dαej(t)=cjej(tσ)+k=1najk[fk(yk(t))fj(xk(t))]+k=1nbjk[gk(yk(tτ))gk(xk(tτ))]+Uj(t), or equivalently Dαe(t)=Ce(tσ)+A[f(y(t))f(x(t))]+B[g(y(tτ))g(x(tτ))]+U(t). In the meantime, the control function Uj(t) is

Numerical simulations

The following delayed FOCVNN with leakage delay is considered as the master system: Dαx(t)=Cx(tσ)+Af(x(t))+Bg(x(tτ))+I(t),where x(t)=(x1(t),x2(t))T, xj(t)=uj(t)+ivj(t),j=1,2, and α=0.98,σ=1,τ=0.5.

C=1+2i001+2i, A=1i1+ii1i, B=12i1+i1ii, I(t)=(1+2i,i)T, f(x(t))=(f1(x1(t)),f2(x2(t)))T, g(x(t))=(g1(x1(t)),g2(x2(t)))T, and fj(xj)=1euj1+euj+i11+evj, gj(xj)=1evj1+evj+i11+euj, for j=1,2.

The controlled slave system is given by: Dαy(t)=Cy(tσ)+Af(y(t))+Bg(y(tτ))+I(t)+U(t),where y(t)=(y

Conclusions

Leakage delays affect dynamical behaviors of neural networks, and therefore cannot be ignored. Synchronization of delayed FOCVNNs with leakage delay is investigated. The synchronization analysis is presented in the complex valued field. Constructing a new fractional inequality, sufficient conditions are derived. Numerical simulations show the effectiveness of the method. It should be pointed that the method can be applied to investigate the stability and synchronization of fractional order

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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    Project supported by the National Natural Science Foundation of China (Grant No 61573096), the Natural Science Foundation of Anhui Province (Grant No 1908085MA01), the Natural Science Foundation of the Higher Education Institutions of Anhui Province (Grant No KJ2018A0365, Grant No KJ2019A0573, Grant No KJ2019A0556) and the Special Foundation for Young Scientists of Anhui Province (Grant No gxyq2019048).

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