Zhuang Yao ,Ziye Zhang ,Zhen Wang ,Chong Lin and Jian Chen

1 College of Mathematics and Systems Science,Shandong University of Science and Technology,Qingdao 266590,China

2 Institute of Complexity Science,Qingdao University,Qingdao 266071,China

3 College of Automation Engineering,Qingdao University of Technology,Qingdao 266555,China

Abstract This paper investigates the polynomial synchronization(PS)problem of complex-valued inertial neural networks with multi-proportional delays.It is analyzed based on the non-separation method.Firstly,an exponential transformation is applied and an appropriate controller is designed.Then,a new sufficient criterion for PS of the considered system is derived by the Lyapunov function approach and some inequalities techniques.In the end,a numerical example is given to illustrate the effectiveness of the obtained result.

Keywords: complex-valued inertial neural networks,polynomial synchronization,multiproportional delays,non-separation approach

1.Introduction

Neural networks (NNs) have always possessed a wide range of applications in associative memory,image processing,dynamic optimization,and machine learning [1–3] over the years,which have aroused many scholars’ interest and produced a large number of research results [4–6].In 1986,the inertial neural networks(INNs)model was produced by introducing an inductor into an artificial neural network circuit[7,8].Different from the artificial neural networks model described by the first-order differential equation,the INNs model obeys the second-order one and the second-order term is called the inertial term.The introduction of the inertia term can effectively improve the disordered search performance of the network,and it is also the key factor to produce complex behaviors such as bifurcation and chaos.Moreover,the inertia term itself has a strong biological background.For example,the cell membrane of some organisms can be realized by the circuit system containing inductance,and the axons of squid and penguin can be simulated by the inductance in the circuit[9].Compared with the first-order NNs,INNs show more complex dynamic behaviors.So,it shows the extremely important theoretical significance and practical application value in studying the dynamical behaviors of INNs deeply [10,11].

The above research results are all on the basis of real-valued neural networks with real-valued activation functions and state variables.With the involvement of complex signals,complexvalued neural networks (CVNNs) models entered the vision of scholars[12–19].It is not only a simple extension of real-valued NNs but also has the characteristics of fast calculation and strong processing ability,and it can be used to deal with some problems that cannot be solved in the real number field.Naturally,complexvalued inertial neural networks (CVINNs) have also aroused widespread concern,and the development of dynamic behavior has increased rapidly in recent years.When developing dynamic behaviors,most of the work is based on the separation method by separating the real part and imaginary part of the considered systems,such as stability analysis[20–22],stabilization[23,24],and synchronization[25–29].Later,driven by related theories in a complex number field,the non-separation method,that is,regarding the considered systems as a whole,is used fully to study various dynamics,including state estimation [30],exponential convergence [22],finite/fixed-time synchronization [28,29,31],anti-synchronization [23,24,32],and exponential and adaptive synchronization [33].Compared with the separation method,the non-separation method can reduce conservatism and computational complexity.

Synchronization has been applied to some practical fields such as image encryption,secure communication,and brainlike intelligence,which makes it a very active research topic.Besides those synchronization issues mentioned above,there are others,for example,polynomial synchronization (PS).The concept of PS originated from polynomial stability and was first proposed in [34],which is a kind of dynamic behavior in some special cases of stochastic differential equations and wave equations [35,36].As stated in[37,38],its basic characteristic is that its top Lyapunov exponent equals 0.It is called polynomial stability because its definition containst-λ(λ>0),which is similar to the structure of polynomials.It is more general than exponential stability,which is between exponential stability and asymptotic stability.Its convergence speed is slower than that of exponential stability,but both of them can degenerate into asymptotic stability.Thus,based on these outstanding characteristics,polynomial stability is also a very typical dynamic behavior.For PS of the drive-response systems,that is,polynomial stability of the error systems,there has been no relevant result about CVINNs up to now.This fact promotes our research.

It should be pointed out that the time delay cannot be ignored in the whole design process of CVINNs.As we all know,due to the inherent delay time of inertial neurons and the limited speed of information transmission,signal acquisition,and processing between neurons,the time delay of the inertial neural system is inevitable.It is found that time delay has a great impact on the dynamic behaviors of neural networks,which may lead to instability,oscillation,and other poor performance.Therefore,it is very important to study the dynamic characteristics of delayed neural networks.As one of many delay types,the proportional delay is time-varying and unbounded with τ(t)=(1-q)t(0

Considering the clear feature of multi-proportional delays[38,46],such as more generality and reality in contrast to single-proportional delays,we will focus on the PS problem of CVINNs with multi-proportional delays in this paper.The main contributions are summarized as follows:

(1) Considering the importance of PS and proportional delay,the PS problem of CVINNs with multi-proportional delays is studied for the first time.

(2) To maintain the originality of the considered system and reduce computational pressure,our work is carried out by a non-separation approach instead of decomposition.

(3) To achieve the PS of the addressed CVINNs with multiproportional delays,by applying the exponential transformation and designing an appropriate controller,a new criterion for PS is proposed based on the Lyapunov function approach and some inequalities techniques.

2.Preliminaries

In this paper,we consider the complex-valued inertial neural networks described in the following form:

wherej=1,2,…,n,t≥t0≥1,uj(t)?Crepresents the state of thejth neuron at timet;indicates the inertial term;ajandbjare the positive constants;cjrandmjrare the delayed connection weight coefficients;fr(·),gr(·)?Care complex-valued activation functions;pr,qrare proportional delays factors withpr,qr?(0,1];prt=t-(1-pr)t,qrt=t-(1-qr)t,in which(1-pr)tand(1-qr)tare the transmission delay functions;Ijis the external input.

Remark 1.On one hand,as a special type of delay,the proportional delay is a time-varying and unbounded delay,which is different from constant delay,distributed delay,and so on.In many fields,such as control science,and physical and biological systems,it can be found that proportional delay plays an important role.On the other hand,as a special neural network,CVINNs have attracted extensive attention because of their rich biological and engineering background.Compared with firstorder neural networks,the dynamic behaviors of the secondorder CVINNs are more complex and it is more difficult to deal with them.So far,there have been few reports on CVINNs with proportional delays [21,31].Here,we will continue to study deeply the CVINNs model with multi-proportional delays and develop the relative achievement for the PS problem.

The initial conditions of system (1)

wherevj(t) represents the state of the response system andUj(t) represents the control input.The initial conditions of system (3)

By introducing the following variable transformation:

Remark 2.In realizing the PS of the addressed model,the main difficulty and challenge are how to obtain the power exponential termt-λaccording to the definitions of polynomial stability and synchronization.Here,firstly,the exponential transformation is involved [34].Then,by designing a suitable controller and constructing an appropriate Lyapunov function,our purpose is achieved.

Assumption 1.In this paper,activation functionsfr(·)andgr(·)are assumed to satisfy the following Lipschitz conditions:

3.Main result

In this section,we will develop a sufficient condition to ascertain the PS for systems (1) and (3).First,we design the following control input

where positive numberskj,ηj,ρjand σjare control gains.

Remark 3.Here,the above controller(9)is designed to guarantee the PS performance for CVINNs with multi-proportional delays.Instead of the decomposition used in [20–29],this controller is designed based on the direct method.So,it is very concise.

Theorem 1.Suppose Assumption 1 holds,if there exist positive numberskj,ηj,jρ,jσ,andε>0such that the following inequalities hold:

then systems(1)and(3)are globally polynomially synchronized based on the controller(9).

Proof.Construct the Lyapunov function as below

Calculating the derivative ofV(t),we can obtain

According to Assumption 1,we have

Submitting (13)-(15) into (12),we can obtain

From (10),one has

which impliesV(t)≤V(logt0),fort≥logt0.

Moreover,from (11),we can obtain

and

where

Thus,we have

Remark 4.Sufficient conditions for polynomial synchronization based on CVINNs systems with multi-proportional delays are established in theorem 1.In our work,by constructing Lyapunov functions and making full use of inequality techniques,the expected results are obtained.To the best of our knowledge,this is the first time that the problem of polynomial synchronization has been discussed under CVINNs.

Remark 5.At present,the methods for dealing with CVINNs are mainly divided into two types,namely the separation method[20,21,23–27] and the non-separation method [31–33].The former can increase the number of variables and dimensions of systems whereas the latter can keep the originality of systems and reduce the limitations.Obviously,the advantages of the latter are more typical,which inspires us to fully utilize the nonseparation method to achieve the goal.

Remark 6.As we know,it is the first time that the PS problem of CVINNs with multi-proportional delays is studied.Here,we use the non-separation method which can simplify the process and result.Moreover,we also consider the multi-proportional delays which own the easier controllability.Therefore,our work is very important in theory and application value.

Remark 7.There are many types of time delays,such as timevarying delays,proportional delays,distributed delays,and so on.As we know,a large number of achievements involve time-varying delays,for example,proportional-integral synchronization [47] and finite-time synchronization of Markovian coupled neural networks [48].Different from these results,this paper focuses on multi-proportional delays,which are unbounded delays and widely used in the light absorption of stellar matter,nonlinear dynamic systems,and many other fields.

4.Numerical example

In this section,we will provide an example and some simulations to illustrate the effectiveness of the theoretical result.

Example 1.Consider the following two-dimension drive system

and the response system is given by

Figure 1.The curves of the variables u1,v1,u2,v2 without the controller.

Figure 2.The curves of the variables u1,v1,u2,v2 with the controller.

Then,we will consider the PS of systems (22) and (23)under the controller (9).It is easy to verify that Assumption 1 holds andforj=1,2.Through calculating(10)of theorem 1,we can obtain thatk1>1.7,k2>1.7,η1>10.9060,η2>13.0343,ρ1>7.5345,ρ2>10.776,and σ1>12.7906,σ2>9.0780.So we choosek1=1.71,k2=1.71,η1=10.91,η2=13.04,ρ1=7.54,ρ2=10.78,and σ1=12.80,σ2=9.08.From theorem 1,systems(22)and(23)can reach the PS under the controller (9),which can be also seen in figure 2.

5.Conclusion

In this paper,we have studied the PS problem of CVINNs with multi-proportional delays by the non-separation method.In order to realize the PS of the addressed system,we have involved the exponential transformation,obtained the power exponential term via fully utilizing some inequality techniques,and solved the main difficulty based on the designed suitable controller and Lyapunov function method.Then,we obtained a simple sufficient condition for PS.Finally,we have provided a numerical simulation example to verify the validity of our work.In the following work,we will focus on other synchronization phenomena of CVINNs,such as module-phase synchronization,projective synchronization,combination synchronization,and so on.

Acknowledgments

This work was supported in part by the National Natural Science Foundation of China (61503222,62173214),in part by the Natural Science Foundation of Shandong Province of China(ZR2021MF100),in part by the Research Fund for the Taishan Scholar Project of Shandong Province of China,in part by the Science and Technology Support Plan for Youth Innovation of Colleges and Universities of Shandong Province of China (2019KJI005),and in part by the SDUST Research Fund.