# Function Projective Synchronization of Two Complex Networks with Unknown Sector Nonlinear Input and Multiple Time-Varying Delay Couplings

Authors: Na Fang; Da Wei; Nan-nan Yin; Dan-ying Xu; Hua Liu; Jie Fang
Journal: Advances in Mathematical Physics
 (2022)

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## Abstract

This paper deals with the function projective synchronization of two complex dynamic networks with unknown sector nonlinear input, multiple time-varying delay couplings, model uncertainty, and external interferences. Based on Lyapunov stability theory and inequality transformation method, the robust adaptive synchronization controller is designed, by which the drive and response systems can achieve synchronization according to the function scaling factor. Different from some existing studies on nonlinear system with sector nonlinear input, this paper studies the synchronization of two complex dynamic networks when the boundary of sector nonlinear input is unknown. The controller does not include the boundary value of the sector nonlinear input and the time delay term, so it is more practical and relatively easy to implement. The corresponding simulation examples demonstrate the effectiveness of the proposed scheme.


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## Body

## 1. Introduction

There are all kinds of complex systems in nature world. These complex systems can be seen as networks, such as Internet, power grid, communication networks, transportation networks, ecological networks, and social networks. The dynamic behavior of complex networks affects almost every aspect of our lives. Among many researches on complex networks, synchronization research is one of the most important branches. So far, many types of synchronization have been investigated, such as complete synchronization [1, 2], antisynchronization [3], exponential synchronization [4–6], quasisynchronization [7], lag synchronization [8, 9], combined synchronization [10], projection synchronization [11], and function projection synchronization [12–14]. Function projection synchronization is a general synchronization form, which means that the driving system and the response system can be synchronized according to a certain function proportional relationship. The complete synchronization, antisynchronization, and projection synchronization are all its exceptional cases. Function projection synchronization has attracted widespread attention because of its implied application in information science and secure communication [15, 16].It is well known that various time delays are unavoidable in actual engineering applications. The time delay may destroy the dynamic characteristics and decrease the stability of the system, which is extremely detrimental to the control system [17–19]. Multiple time-varying delay couplings mean that multiple different time-varying delays exist in the complex network. The description of multiple time-varying delay couplings is a general description of time delay, and the constant time-delay couplings and single time-varying delay couplings are its special circumstances. The synchronization researches of complex networks with multiple time-varying delay couplings are more realistic and representative [20, 21]. Zhang et al. [22] researched the synchronization of uncertain complex networks with time-varying node delay and multiple time-varying coupling delays via the adaptive control. In [23], the authors researched the synchronization in nonlinear complex networks with multiple time-varying delays. Wang et al. [24] studied the lag synchronization between two coupled complex networks with multiple time-varying delays via the adaptive pinning control. Zhao et al. [25] studied the synchronization issue of uncertain complex networks with multiple time-varying delays. Lu et al. [26] established a robust adaptive synchronization scheme for general complex networks with multiple time-varying coupling delays and uncertainties. Guan et al. [27] studied the synchronization of complex networks with system delay and multiple time-varying coupling delays via impulsive distributed control.In the actual control system, the backlash, friction, dead zone, and hysteresis will cause the nonlinearity of the control input, which lead to system instability or control performance degeneration [28–34]. Therefore, the synchronization researches of complex networks with input nonlinearity are meaningful. Sector nonlinear input is one type of the nonlinear input, which means that the system input is in a fan-shaped area. Sector nonlinear input represents a large type of input nonlinearity. Many scholars have studied the control of nonlinear systems with sector nonlinear input. Boulkroune and Msaad [35] researched the adaptive variable-structure control of uncertain chaotic MIMO systems with both sector nonlinearities and dead-zones. Fang et al. [36] researched the modified projective synchronization of chaotic systems with sector nonlinearities input. Boubellouta et al. [37] achieved synchronization for a class of fractional-order chaotic systems with sector nonlinearities. Wang and Liu [38] researched the sliding mode control of the master-slave chaotic systems with sector nonlinear input. Yang et al. [39] addressed an adaptive two-stage sliding mode control to realize the synchronization for a class of n-dimensional nonlinear systems with sector nonlinearity input. Although the researches on sector nonlinear input have achieved certain results, most existing studies mainly focus on a single system rather than complex networks. Recently, Fang et al. [40] studied the modified function projective synchronization of complex dynamic networks with sector nonlinear input. In the controller design, it is assumed that the range of the sector nonlinear input is known. However, it is difficult to determine the exact boundary value of the sector nonlinear input. Once the restricted boundary of the control input is unknown, the controller designed in [40] is no longer applicable. How to realize function projective synchronization of complex dynamic networks under unknown sector nonlinear input is a challenging research topic.Based on the results of previous researches, the function projective synchronization for a class of complex dynamic networks with unknown sector nonlinear input, multiple time-varying delay couplings, model uncertainty, and external interferences is studied in this paper. Through the designed adaptive controller, two complex dynamic networks can realize synchronization according to the corresponding function scaling factor. Compared with the existing research results, the contributions of this paper are (a) the complex network model includes the input nonlinearity, multiple time-varying delay couplings, model uncertainty, and external interferences, which is a more general model. (b) Many of the existing studies are concerned with synchronization between complex networks and single systems. This paper studies the synchronization between two complex networks, which is more complex and general. (c) Different from known sector nonlinear inputs in previous studies, this paper investigates the function projective synchronization of complex dynamic networks with unknown sector inputs. The boundary value of the sector nonlinear input and the delay term is not needed in controller design, so it is relatively easy to implement in practical engineering. (d) Function projective synchronization is a more general synchronization form. The controller in this paper can also realize complete synchronization, antisynchronization, and projective synchronization of complex dynamic networks.


## 2. Model Description

In this article, a type of complex dynamic networks with unknown sector nonlinear input, multiple time-varying delay couplings, model uncertainty, and external interferences is described as the drive system:(1)ẋit=fixit+Fixitθi+∑l=0m−1clt∑j=1NaijlΓlxjt−τlt+divt=fixit+Fixitθi+c0t∑j=1Naij0Γ0xjt−τ0t+c1t∑j=1Naij1Γ1xjt−τ1t+⋯+cm−1t∑j=1Naijm−1Γm−1xjt−τm−1t+divt,the corresponding response system is
(2)ẏit=giyit+Giyitηi+∑l=0m−1clt∑j=1NaijlΓlyjt−τlt+dist+ϕiuit=giyit+Giyitηi+c0t∑j=1Naij0Γ0yjt−τ0t+c1t∑j=1Naij1Γ1yjt−τ1t+⋯+cm−1t∑j=1Naijm−1Γm−1yjt−τm−1t+dist+ϕiuit,where xit=xi1t,xi2t,⋯,xintT,i=1,2,⋯,N is the state vector of the ith node in the drive system, yit=yi1t,yi2t,⋯,yintT,i=1,2,⋯,N is the state vector of the ith node in the response system. fi×,gi×I^Rn are the continuous nonlinear function vectors, Rn denotes the n-dimensional vector space on the real number field R, θi,ηi∈Rw are unknown w-dimensional constant parameter vector, Fi⋅,Gi⋅∈Rn×w are the continuous nonlinear function matrices, Rn×w denotes the n×w order matrix on the real number field R. divt and dist are the disturbances. The complex network is divided into subnetworks by τlt, τlt≥0,l=0,1,⋯,m−1 is the different time-varying delays, and especially τ0t=0 means that the coupling delay is 0; clt is the coupling strength; Γl is the inner coupling matrix; Al=aijlN×N is weight configuration matrix, representing the topological structure of the network. If nodes i and jj≠i are connected, then, aijl≠0. If nodes i and jj≠i have no connection, then. aijl=ajil=0. The diagonal elements of the matrix Al are defined as aiil=−∑j=1,j≠iNaijl,i,j=1,2,⋯,N.ϕiuit=ϕi1ui1t,ϕi2ui2t,⋯,ϕinuintTϕi0=0 is the control input. ϕikuikt is in a sector puikt,quikt, where p and q are two positive numbers and satisfy p≤ϕikuikt/uikt≤q when uikt≠0. The sector nonlinear input is shown in Figure 1.Definition 1 (see [15]).
For the complex dynamic networks (1) and (2), if Eq. (3) holds, the complex network (1) and (2) will realize function projective synchronization when
(3)limt⟶∞eit=limt⟶∞yit−htxit=0,i=1,2,⋯,N.where eit=ei1t,ei2t,⋯,eintT,i=1,2,⋯,N,⋅ denotes the Euclidean norm of a vector. ht≠0 is function scaling factor, which is a continuously differentiable and bounded function.Assumption 2.
External disturbancesdivt and dist are bounded, and there exist positive constants αiv,αis, such that divt≤αiv,dist≤αis.Corollary 3.
Becauseht is a continuously differentiable and bounded function, there exists a positive constant ℏ and satisfies ht≤ℏ. Under Assumption 2, there exists a positive constant αi≥αis+ℏαiw, such that
(4)dist−htdivt≤dist+htdivt≤dist+htdivt≤αis+ℏαiv≤αi.Assumption 4.
The time-varying coupling strengthclt is bounded, and there exists a positive constant c, such that
(5)clt≤c.Assumption 5.
The time-varying delayτlt,l=0,1,⋯,m−1 is a continuously differentiable function and satisfies 0≤τ̇lt≤ε<1, so it is easy to get
(6)1−τ̇lt21−ε≥1−ε21−ε=12,where 0<ε<1 is positive constant. This assumption is still satisfied if τlt is zero or some other constants.Lemma 6 (see [9]).
For any vectorsX,Y∈Rn and a positive definite matrix Q∈Rn×n (Rn denotes the n-dimensional vector space on the real number field R, Rn×n denotes the n×n order matrix on the real number field R), the following matrix inequality holds:2XTQY≤XTQQTX+YTY.Proof.
LetA=a1t,a2t,⋯,antT,B=b1t,b2t,⋯,bntT.
It is easy to getATB,BTA∈R and ATB=BTA,ATA=a12+a22+⋯+an2,BTB=b12+b22+⋯+bn2,ATB=BTA=a1×b1+a2×b2+⋯+an×bn.
Becausea12+a22+⋯+an2+b12+b22+⋯+bn2−2×a1×b1+a2×b2+⋯+an×bn=a1−b12+a2−b22+⋯+an−bn2≥0,then, a12+a22+⋯+an2+b12+b22+⋯+bn2≥2×a1×b1+a2×b2+⋯+an×bn, i.e., ATA+BTB≥2ATB.
LetA=QTX,B=Y, then, we can get 2XTQY≤XTQQTX+YTY.
This completes the proof.


## 3. Controller Design

To realize function projective synchronization, the controller and parameter adaptive laws are designed as follows:(7)uikt=−γgikyit−htfikxit−ḣtxikt+Gikyitη^i−htFikxitθ^i+1γν^eiktϖ^+ψ^iksgneikt,i=1,2,⋯,Nk=1,2,⋯,n.(8)θ^̇i=−FiTxithteit,(9)η^̇i=GiTyiteit,(10)ϖ^̇=∑i=1NeiTtgiyit−htfixit−ḣtxit+Giyitη^i−htFixitθ^i+ν^eit,(11)ψ^̇i=eit,(12)ν^̇=∑i=1NeiTteit,where γ,ν are positive constants and satisfy γ>1,ν>0.ϖ=1/p>0,ψi=αi/p>0.ϖ^,ψ^i,θ^i,η^i,ν^ is the estimated parameter for ϖ,ψi,θi,ηi,ν, respectively. Fik⋅,Gik⋅∈R1×w are the kth row of the function matrices Fi⋅,Gi⋅∈Rn×w.Remark 7.
Letuikt=−γμiksgneikt, then, μik=gikyit−htfikxit−ḣtxikt+Gikyitη^i−htFikxitθ^i+1/γν^eiktϖ^+ψ^ik.Becauseγ>1,ν>0,ϖ=1/p>0,ψi=αi/p>0, we can getμik>0.Lemma 8 (see [40]).
Letuikt=−γμiksgneikt,μik>0, we can get ∑i=1Neiktϕikuikt≤∑i=1N−pγμikeikt, i.e., eiTtϕiuit≤−piγμiTeit.Proof.
It can be known fromp≤ϕikuikt/uikt≤q that puik2t≤uiktϕikuikt≤quik2t.
Wheneikt=0, the equation obviously holds, that is, eiktϕikuikt=−pγμikeikt.
Wheneikt≠0, substituting uikt=−γμiksgneikt into puik2t≤uiktϕikuikt≤quik2t, we can get pγ2μik2sgn2eikt≤−γμiksgneiktϕikuikt.
Usingeikt/eikt instead of sgneikt, we can get pγ2μik2eikteikt/eikteikt≤−γμikϕikμikteikt/eikt.
Multiplying both sides of the inequality byeik2t, we can get pγ2μik2eikt2≤−γμikeikteiktϕikuikt. Dividing both sides by γμikeikt, we can get eiktϕikuikt≤−pγμikeikt.
It is easy to get∑i=1Neiktϕikuikt≤∑i=1N−pγμikeikt, then, eiTtϕiuit≤−piγμiTeit.
This completes the proof.Theorem 9.
If Assumptions2–5 are satisfied, the drive system (1) and the response system (2) can realize function projective synchronization with the controller (7) and adaptive laws (8)–(12).Proof.
From Definition1, we have the error term:
(13)eit=yit−htxit.
The time derivative ofeit is
(14)ėit=ẏit−htẋit−ḣtxit=giyit+Giyitηi+∑l=0m−1clt∑j=1NaijlΓlyjt−τlt+dist+ϕiuit−htfixit−htFixitθi−ht∑l=0m−1clt∑j=1NaijlΓlxjt−τlt−htdivt−ḣtxit.
Choosing Lyapunov function as(15)Vt=12p∑i=1NeiTteit+pψ∧i−ψi2+12ϖ∧−ϖ2+12pν∧−ν∗2+12p∑i=1Nθ∧i−θi2+∑i=1Nη∧i−ηi2+11−ε∫t−τltt∑l=1m−1∑i=1NeiTδeiδdδ,where ϖ^,ψ^i is the estimated parameter for ϖ,ψi. ν∗ is the positive constant to be designed.
Taking the derivative of the Lyapunov function, we can get(16)V̇t=∑i=1N1peiTtėit+ψ∧i−ψiTψ^̇i+ϖ^−ϖϖ^̇+1pν^−ν∗ν^̇+1p∑i=1Nθ∧i−θiTθ^̇i+1p∑i=1Nη∧i−ηiTη^̇i+∑l=1m−1∑i=1NeiTteit2p1−ε−1−τ̇lt2p1−ε∑l=1m−1∑i=1NeiTt−τlteit−τlt.
Substituting (8), (9), and (14) into (16), we can get
(17)V̇t=1p∑i=1NeiTtgiyit+Giyitηi+dist+∑l=0m−1clt∑j=1NaijlΓlyjt−τlt+ϕiuit−htfixit−htFixitθi−ht∑l=0m−1clt∑j=1NaijlΓlxjt−τlt−htdivt−ḣtxit+∑i=1Nψ∧i−ψiTψ^̇i+ϖ^−ϖϖ^̇+1pν^−ν∗ν^̇−1p∑i=1Nθ∧i−θiTFiTxithteit+1p∑i=1Nη∧i−ηiTGiTyiteit+12p1−ε∑l=1m−1∑i=1NeiTteit−1−τ̇lt2p1−ε∑l=1m−1∑i=1NeiTt−τlteit−τlt.
Because∑i=1NeiTthtFixitθi=∑i=1NθiTFiTxithteit,∑i=1NeiTtGiyitηi=∑i=1NηTGiTyiteit, we can get
(18)−1p∑i=1NeiTthtFixitθi−1p∑i=1Nθ∧i−θiTFiTxithteit=−1p∑i=1Nθ∧iTFiTxithteit,(19)1p∑i=1NeiTtGiyitηi+1p∑i=1Nη∧i−ηiTGiTyiteit=1p∑i=1Nη∧iTGiTyiteit,so
(20)V̇t=1p∑i=1NeiTtgiyit+∑l=0m−1clt∑j=1NaijlΓlyjt−τlt+ϕiuit+dist−htfixit−ht∑l=0m−1clt∑j=1NaijlΓlxjt−τlt−htdiwt−ḣtxit+∑i=1Nψ∧i−ψiTψ^̇i+ϖ^−ϖϖ^̇−1p∑i=1Nθ∧iTFiTxithteit+1p∑i=1Nη∧iTGiTyiteit+1pν^−ν∗ν^̇+12p1−ε∑l=1m−1∑i=1NeiTteit−1−τ̇lt2p1−ε∑l=1m−1∑i=1NeiTt−τlteit−τlt.
Substituting Lemma8 and Corollary 3 into (20), we can get
(21)V̇t≤1p∑i=1NeiTtgiyit−htfixit−ḣtxit+αi−γμiTeit+∑i=1Nψ∧i−ψiTψ^̇i+ϖ^−ϖϖ^̇−1p∑i=1Nθ∧iTFiTxithteit+1p∑i=1Nη∧iTGiTyiteit+1pν^−ν∗ν^̇+1p∑i=1NeiTtcl∑j=1Naij0Γ0ejt+cl∑l=1m−1∑j=1NaijlΓlejt−τlt+12p1−ε∑l=1m−1∑i=1NeiTteit−1−τ̇lt2p1−ε∑l=1m−1∑i=1NeiTt−τlteit−τlt.
Substitutingψi=αi/p and ϖ=1/p into (21), because η∧iTGiTyiei−θ∧iTFiTxihei=η∧iTGiTyi−θ∧iTFiTxihei≤η∧iTGiTyi−θ∧iTFiTxihei=η∧iTGiTyiei−θ∧iTFiTxihei,where ei=ei1,ei2,⋯,einT, we can get
(22)V̇t≤∑i=1NϖeiTtgiyit−htfixit−ḣtxit+eiTtψi−γμiTeit+ψ∧i−ψiTeit+ϖ^−ϖϖ^̇+ϖi∑i=1Nη∧iTGiTyiei−θ∧iTFiTxihei+ϖν^−ν∗ν^̇+∑i=1NϖieiTtcl∑j=1Naij0Γ0ejt+cl∑l=1m−1∑j=1NaijlΓlejt−τlt+ϖi21−ε∑l=1m−1∑i=1NeiTteit−1−τ̇lt21−εϖi∑l=1m−1∑i=1NeiTt−τlteit−τlt.
Substituting (10) and (11) into (22), because ∑i=1NeTithFixθ^i=θ∧TFTxhe,∑i=1NeTitGiyη^i=η∧TGTye, then, the above formula can be simplified as
(23)V̇t≤∑i=1NeiTtψi−γeiTtgiyit−htfixit−ḣtxit+Giyη^i−htFixθ^i+1γν^eitϖ^+ψ^i+ψ∧i−ψiTeit+ϖ^∑i=1NeiTtgiyit−htfixit−ḣtxit+Giyη^i−htFixθ^i+∑i=1NϖeiTtcl∑j=1Naij0Γ0ejt+cl∑l=1m−1∑j=1NaijlΓlejt−τlt−ϖν∗∑i=1NeiTteit+ϖ21−ε∑l=1m−1∑i=1NeiTteit−1−τ̇lt21−εϖ∑l=1m−1∑i=1NeiTt−τlteit−τlt.
In order to simplify the proof process,V̇t is decomposed into two parts V̇1t and V̇2t:
(24)V̇1t=∑i=1N1peiTtcl∑j=1Naij0Γ0ejt+cl∑l=1m−1∑j=1NaijlΓlejt−τlt+12p1−ε∑l=1m−1∑i=1NeiTteit−1−τ̇lt2p1−ε∑l=1m−1∑i=1NeiTt−τlteit−τlt−ν∗p∑i=1NeiTteit,(25)V̇2t=∑i=1NeiTtψi−γeiTtgiyit−htfixit−ḣtxit+Giyη^i−htFixθ^iϖ^+ψ^i+ψ∧iTeit+ϖ^∑i=1NeiTtgiyit−htfixit−ḣtxit+Giyη^i−htFixθ^i.
Letet=e1Tt,e2Tt,⋯,eNTtT∈RN×1,B0=A0⊗Γ0,B1=A1⊗Γ1,⋯,Bl=Al⊗Γl, where ⊗ represents the Kronecker product, then, we can get
(26)V̇1t≤1pcleTtB0et+cl∑l=1m−1eTtBlet−τlt+121−ε∑l=1m−1eTtet−12∑l=1m−1eTt−τltet−τlt−ν∗eTtet.
Based on Lemma6, it is
(27)cleTtBlet−τlt≤12cl2eTtBlBlTet+12eTt−τltet−τlt,so we can get
(28)V̇1t≤1peTtclB0+12cl2∑l=1m−1BlBlTet+∑l=1m−1121−εeTtet−ν∗eTtet≤1pλmaxclB0+12cl2∑l=1m−1BlBlT+m−121−ε−ν∗eTtet.
Becausep>0, if ν∗≥λmaxclB0+1/2cl2∑l=1m−1BlBlT+m−1/21−ε, we can get V̇1t≤0, where λmaxQ is the maximum eigenvalue of the matrix Q.
Making a simple equation transformation toV̇2t, we can get
(29)V̇2t≤∑i=1N1−γeiTtgiyit−htfixit−ḣtxit+Giyη^i−htFixθ^iϖ^+ψ^i≤∑i=1N1−γeiTtμi.
Becauseγ>1,ϖ=1/p>0,ψi=αi/p>0, then
(30)V̇2t≤∑i=1N1−γeiTtμi≤0.
Based on the above analysis, we can get thatV̇t≤V̇1t+V̇2t≤0 if ν∗≥λmaxclB0+1/2cl2∑l=1m−1BlBlT+m−1/21−ε. According to Lyapunov stability theory, we can obtain eit⟶0 as t⟶∞, which means that the function projective synchronization between the drive system (1) and the response system (2) is achieved. This completes the proof.Remark 10.
In the proof of Theorem9, based on Lyapunov stability theory and inequality transformation method, by introducing Lemma 6 and 8 and some reasonable Assumptions, the controller is designed flexibly without the boundary value (p and q) of the sector nonlinear input and the delay term τlt.Remark 11.
Whenτlt,l=0,1,2,⋯,m−1 is constant or τ0t=τ1t=⋯=τm−1t=τt, the multiple time-varying delay couplings are transformed into constant time-delay coupling or single time-varying delay coupling. When τlt,l=0,1,⋯,m−1 is constant or τ0t=τ1t=⋯=τm−1t=τt, Assumption 5 is also satisfied, and the control method in this article is also applicable to constant time-delay coupling or single time-varying delay coupling.Remark 12.
Ifht is a constant, the function projection synchronization is transformed into the projection synchronization. In particular, when ht=1 or ht=−1, the function projection synchronization turns into complete synchronization or antisynchronization.


## 4. Numerical Simulation

In order to verify the correctness of the theoretical analysis, we select communication network with chaotic nodes as simulation examples.(31)ẋi1tẋi2tẋi3t=0−xi1txi3t−xi2txi1txi2t+−xi1t+xi2t000xi1t000−xi3t×θi1θi2θi3+di1vtdi2vtdi3vt+c0t∑j=14aij0Γ0xjt+c1t∑j=14aij1Γ1xjt−τ1t+c2t∑j=14aij2Γ2xjt−τ2t.i=1,2,3,4.Example 1.
Considering a communication network withN=4,n=3, the drive system is composed of four Lorenz chaotic systems with two different time-varying delay couplings.The response system is composed of four Chen chaotic systems with two different time-varying delay couplings.(32)ẏi1tẏi2tẏi3t=0−yi1tyi3tyi1tyi2t+yi2t−yi1t00−yi1tyi2t+yi1t000−yi3t×ηi1ηi2ηi3+di1stdi2stdi3st+ϕi1ui1tϕi2ui2tϕi3ui3t+c0t∑j=14aij0Γ0yjt+c1t∑j=14aij1Γ1yjt−τ1t+c2t∑j=14aij2Γ2yjt−τ2t.i=1,2,3,4.In MATLAB numerical simulation, setc0=c1=c2=0.2,τ1t=t/3+t,τ2t=et/3+et,dit=0.2cost,Γ0=Γ1=Γ2=I3×3,ht=2cos2t. The nonlinear input is ϕiuit=ϕi1ui1t,ϕi2ui2t,ϕi3ui3tT=0.3+0.2cosui1tui1t,0.6+0.2sinui2tui2t,0.4−0.2cosui3tui3tT. The topological structure matrices A0,A1,A2 are as follows:
(33)A0=−21010−21111−20101−2,A1=−10010−11000−11010−1,A2=−10100−21111−20010−1.And the topology of the driver network and response network is shown in Figure2.Figure 1
The nonlinear inputϕikuikt within the sector.Figure 2
Topology of the multilink complex networks in Example1 with four nodes and three subnetworks.
(a)
A0(b)
A1(c)
A2Figure 3
The time evolution of synchronization betweeny1i and htx1i.
(a)
Synchronization statesy1i,htx1i(b)
Synchronization errorse1iFigure 4
The time evolution of synchronization betweeny2i and htx2i.
(a)
Synchronization statesy2i,htx2i(b)
Synchronization errorse2iFigure 5
The time evolution of synchronization betweeny3i and htx3i.
(a)
Synchronization statesy3i,htx3i(b)
Synchronization errorse3iFigure 6
The time evolution of synchronization betweeny4i and htx4i.
(a)
Synchronization statesy4i,htx4i(b)
Synchronization errorse4iFigure 7
Topology of the multilink complex networks in Example2 with eight nodes and four subnetworks.
(a)
A0(b)
A1(c)
A2(d)
A3Figure 8
The time evolution of synchronization betweeny1i and htx1i.
(a)
Synchronization statesy1i,htx1i(b)
Synchronization errorse1iFigure 9
The time evolution of synchronization betweeny2i and htx2i.
(a)
Synchronization statesy2i,htx2i(b)
Synchronization errorse2iFigure 10
The time evolution of synchronization betweeny3i and htx3i.
(a)
Synchronization statesy3i,htx3i(b)
Synchronization errorse3iFigure 11
The time evolution of synchronization betweeny4i and htx4i.
(a)
Synchronization statesy4i,htx4i(b)
Synchronization errorse4iFigure 12
The time evolution of synchronization betweeny5i and htx5i.
(a)
Synchronization statesy5i,htx5i(b)
Synchronization errorse5iFigure 13
The time evolution of synchronization betweeny6i and htx6i.
(a)
Synchronization statesy6i,htx6i(b)
Synchronization errorse6iFigure 14
The time evolution of synchronization betweeny7i and htx7i.
(a)
Synchronization statesy7i,htx7i(b)
Synchronization errorse7iFigure 15
The time evolution of synchronization betweeny8i and htx8i.
(a)
Synchronization statesy8i,htx8i(b)
Synchronization errorse8iThe MATLAB simulation results are shown in Figures3–6. It displays that the error signal between the drive system and the response system can stably approach to zero with the designed adaptive controller, that is, the function projection synchronization of the complex dynamic networks is realized.
(34)ẋi1tẋi2tẋi3tẋi4t=0−xi1txi3t−xi2txi1txi2t−xi1t+−xi1t+xi2t0xi4t00xi1t00000−xi3t−xi4t000×θi1θi2θi3θi4+di1vtdi2vtdi3vtdi4vt+c0t∑j=18aij0Γ0xjt+c1t∑j=18aij1Γ1xjt−τ1t+c2t∑j=18aij2Γ2xjt−τ2t+c3t∑j=18aij3Γ3xjt−τ3t.i=1,2,⋯,8.Example 2.
Considering a communication network withN=8,n=4, the drive system is composed of eight LS hyperchaotic systems with three different time-varying delay couplings.The response system is composed of four hyperchaotic systems with three different time-varying delay couplings.(35)ẏi1tẏi2tẏi3tẏi4t=0−yi1tyi3t+yi4tyi1tyi2t0+yi2t−yi1t0000yi1t0000−yi3t0000−yi1t×ηi1ηi2ηi3ηi4+di1stdi2stdi3stdi4st+ϕi1ui1tϕi2ui2tϕi3ui3tϕi4ui4t+c0t∑j=18aij0Γ0yjt+c1t∑j=18aij1Γ1yjt−τ1t+c2t∑j=18aij2Γ2yjt−τ2t+c3t∑j=18aij3Γ3yjt−τ3ti=1,2,⋯,8.To simplify numerical simulation, setc0=c1=c2=c3=0.1,τ1t=t/3+t,τ2t=et/3+et,τ3t=et/6+et,dit=0.2sint,Γ0=Γ1=Γ2=I4×4,ht=2sin2t. The nonlinear input is ϕiuit=ϕi1ui1t,ϕi2ui2t,ϕi3ui3t,ϕi4ui4tT=0.3+0.2cosui1tui1t,0.6+0.2sinui2tui2t,0.4−0.2cosui3tui3t,0.4+0.2sinui4tui4tT. The topological structure matrices A0,A1,A2,A3 are as follows:
(36)A0=−100000100−100010010−100000100−210000100−210000000000000000000011000−2,A1=−110000000−110000000−100010000000000100−100011000−200001001−200001000−1,A2=−200010010−110000010−100000001−210000000000000100−100000000001110000−3,A3=−100100000−100100000−110000101−310000010−100010000−210000000001010000−2.And the topology of the driver network and response network is shown in Figure7.The MATLAB simulation results are shown in Figures8–15. The function projection synchronization can still be achieved when the number of nodes and the system dimension of the complex network are increased, which further verifies the correctness of the theoretical analysis.


## 5. Conclusion

In this paper, the function projective synchronization of complex dynamic networks with unknown sector nonlinear input, multiple time-varying delay couplings, model uncertainty, and external interferences is studied. Based on Lyapunov stability theory, adaptive control theory, and inequality theory, the robust adaptive controller is formulated to make the drive and response systems synchronize by the function scaling factor. The controller designed in this paper can effectively overcome the effects of unknown sector input and multiple time-varying delays, so it is more general and easier to implement. Our future research work will focus on how to realize the complex network synchronization with other forms of input constraints and how to apply the research results of this paper to the fields of information security.



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*Source: 2656836-2022-08-08.xml*