Hong-Jun Shi(侍紅軍),Lian-Ying Miao(苗連英),Yong-Zheng Sun(孫永征), and Mao-Xing Liu(劉茂省)

1School of Mathematics,China University of Mining and Technology,Xuzhou 221008,China

2School of Science,North University of China,Taiyuan 030051,China

1 Introduction

Since being proposed by Pecora and Carroll,[1]synchronization in coupled systems has been extensively studied in many areas,such as biological systems,information processing,economical systems,etc.[2?11]Many kinds of synchronization have been explored,involving complete synchronization,[12]generalized synchronization,[13?14]projective synchronization,[15]and lag synchronization.[16]Due to its widely applications,different control techniques have been constructed,such as adaptive control,[17?19]pinning control,[20]finite-time control,[21?24]sliding mode control,[23,25]and so on.

As a hot research topic,the outer synchronization of complex networks has been extensively investigated.[14,19,26?27]Different from the inner synchronization happening inside a network,outer synchronization is referred as synchronization occurring between two or more networks.In Refs.[14,27],based on the LaSalle-type invariance principle for stochastic differential equations,the outer synchronization between two nonidentical networks with noise coupling was investigated.The adaptive outer synchronization between two complex delayed networks was discussed in Ref.[19].Sufficient criterion for outer synchronization between two coupled discrete-time networks were established in Ref.[26].

Most previous works on network synchronization can be divided into two classes:in finite-time synchronization and finite-time synchronization.Finite-time synchronization means that the trajectories of the response system can reach to those of the drive system in a finite horizon,which is very important and signi ficant in the practical engineering process.Because of the fact that the finite settling time is heavily dependent on the initial conditions of the system,it is necessary to obtain the initial states in advance.Nevertheless,the knowledge of initial conditions may be hard to achieve or even impossible to estimate in practical applications.To overcome this obstacle,a new approach named fixed-time control has been proposed.[20,28?34]

Fixed-time control implies that the system is globally finite-time stable and the convergence settling time is bounded and independent of the initial states.Thus,this new technique has important practical implications.For example,to ensure excellent power supply quality and avoid voltage collapse,we should eliminate the frequency deviation and stabilize the voltage of power systems to its nominal value within a limited time. In Ref.[29],for chaos suppression and voltage stabilization in some three-bus power systems,a fixed-time sliding mode control approach was proposed.To realize the finite-time and fixed-time cluster synchronization,Liu and Chen designed simple distributed protocols with or without pinning control.[30]For a class of delayed memristor-based recurrent neural networks,Cao and Li established sufficient conditions of fixed-time synchronization.[31]Honget al.found that fixed-time control protocols can guarantee any prescribed convergence time regardless of the initial states.[32]In Ref.[33],for fixed-time stabilization of singleinput and multi-input systems,Polyakov proposed two different types of nonlinear controller.Based on the Lyapunov methods,a fixed-time terminal sliding-mode control technique for the second-order nonlinear systems was investigated in Ref.[34].To our best knowledge,there are few results on the fixed-time outer synchronization of complex networks with noise coupling.Actually,noise may play an important role on the collective dynamics of complex systems,[35]which inspired us to study the effect of noise on the fixed-time outer synchronization.

In this paper,based on the theory of fixed-time stability and matrix inequalities,sufficient conditions for the if xed-time outer synchronization of complex networks with noise coupling are presented.The theoretical results show that the setting time can be adjusted to a desired value regardless of the initial states.To verify the effectiveness of the proposed synchronization scheme,numerical simulations are performed.

The rest of this paper is organized as follows.In Sec.2,the network modeling and some preliminaries are given.In Sec.3,sufficient conditions for fixed-time outer synchronization are established.In Sec.4,numerical simulations are given to verify the effectiveness of the proposed control schemes.Section 5 concludes this paper.

NotationsThroughout this paper unless speci fied webe Euclidean norm.IfAis a vector or matrix,its transpose is denoted byAT.LetE(·)denote the expected value of a random variable.

2 Network Modeling and Preliminaries

Consider a complex network consisting ofNnodes with linear couplings:

whereis the state vector of nodei,are the synchronization errors between networks(1)and(2),are the controllers to be designed.The noise term in system(2)is often used to describe the coupling process in fl uenced by many factors,such as inaccurate design of coupling strength and environmental lf uctuation.is the noisy intensity matrix,is ans-dimensional Brownian motion de fined on a complete probability spacewith a natural filtration

In order to get our main results in the next section,we state here some needed concepts,assumptions and lemmas.

De finition 1Networks(1)and(2)are said to achieve fixed-time complete synchronization if,for any initial statesxi(0),yi(0),there exists a setting timeT0,which is independent of initial states,such that

Remark 1Different from the finite-time synchronization,the settling time of the fixed-time synchronization is independent of the initial states.Thus,this new technique has important practical signi ficance.

To achieve the complete outer synchronization in fixed time,we design

whereis continuously differentiable,is the coupling con figuration matrix,which is similar to the de finition of matrixis the generalized synchronization error between networks(1)and(5),?:a given continuously differentiable map andThe controllersin Eq.(5)are to be designed.

Remark 2In this paper,the con figuration matricesCandDof networks(1)and(5)are not necessarily to be symmetric or irreducible,which means that networks can be directed or undirected networks including isolated nodes and clusters.

De finition 2Networks(1)and(5)are said to achieve if xed-time generalized synchronization if,for the given continuously differentiable map?(·)and any initial states,there exists a setting timeT0,which is independent of the initial states,such that

Since the speed of the environmental fl uctuations is much less than the change rate of concrete systems,[18,27,36?38]for the noise intensity function,we have the following assumption:

Assumption 2The noise intensity functionσi(ei(t))satis fies the Lipschitz condition,and there exists a positive constantqsuch that

(i)

(ii)For some,any solutionx(t)satis fied the inequality

then,V(x)will converge to zero in the fixed time and the setting time can be estimated as follows:

3 Main Results

3.1 Fixed-Time Complete Outer Synchronization

Theorem 1Suppose that Assumptions 1 and 2 hold.The fixed-time outer synchronization between complex networks(1)and(2)with noise coupling can be achieved under the controllers(3)if the following condition is satis fied:

The fixed settling timeT0can be estimated as follows:

From Lemma 1,we can get the conclusion and the fixed settling timeT0can be estimated as Eq.(8).This completes the proof of Theorem 1.

Remark 4Whenc2=0,using the similar proof of Theorem 1,we can get the finite-time outer synchronization of complex networks with noise coupling.However,the settling time of finite-time synchronization is heavily dependent on the initial states of the system,which may not be easily adjusted in real practice.Different from the finite-time synchronization,the settling time in this paper is regardless of the initial states and the upper bound can be estimated in advance.Thus,the fixed-time synchronization is more favorable and applicable.

According to the proof of Theorem 1,the fixed-time complete outer synchronization of complex networks can be achieved with the given control parametersθ1=1?1/γ,θ2=1+1/γ(γ>1).Based on Lemma 2,a more accurate estimation ofT0can be obtained.

Corollary 1Suppose that Assumptions 1 and 2 hold.The fixed-time complete outer synchronization between complex networks(1)and(2)with noise coupling can be achieved under the following controllers:

Remark 5From inequality Eq.(21),it is easy to see that the upper bound of the setting time is independent of initial states and can be determined by the network sizeN.Thus,the pre-speci fied setting time can be adjusted to a desired value by appropriately selecting control parameterγ.

If system(2)is free of noise perturbation(that is,σi(ei(t))≡0,i=1,2,...,N),from Theorem 1,we have the following corollary.

Corollary 2Let Assumption 1 hold.Ifsystem(2),the fixed-time outer synchronization between complex networks(1)and(2)can be achieved under the following control scheme:

whereandc2are coupling strength,and control parameterγ>1.The settling time can be estimated in Eq.(21).

3.2 Fixed-Time Generalized Outer Synchronization

The fixed-time complete outer synchronization can not be achieved if two networks have different dynamics.In the following,we consider the generalized outer synchronization between networks(1)and(5).For the given continuously differentiable map?(·),the dynamic behavior of generalized synchronization errorei(t)=yi(t)??(xi(t))between networks(1)and(5)can be described as follows:

The controllersui(t)are designed as follows:

Theorem 2Suppose that Assumptions 1 and 2 hold.The fixed-time generalized outer synchronization between complex networks(1)and(5)can be achieved under the controllers(23),with the control parameters satisfyingwhereThe settling timeT0can be estimated as follows:

Ifk>lg+q+λmax(Ds),we have

The rest is similar to the proof of Theorem 1 and is therefore omitted.

Remark 6In Theorem 2,we assume that the dimension of node dynamics in Eq.(5)is the same as that in Eq.(1).If they are differentusing the similar process and supposewe can also get the conclusion.

4 Numerical Simulations

In this section several numerical results are given to verify the effectiveness of the theoretical results.In the simulations,the Euler-Maruyama numerical scheme[42]for stochastic differential equations was used.Without loss of generality,we take the inner coupling matrix Γ as an identity matrix.The initial conditions of the nodes are randomly taken from the interval[?1,1].The total synchronization erroris used to measure the evolution process and the convergence indicatorT0is defined as:

4.1 Numerical Example of Complete Synchronization

We take the R?ssler-like system as the node dynamics of networks(1)and(2),which can be described as

As shown in Fig.1,the R?ssler-like system has a chaotic attractor whenδ=0.03,?=1.5,η=0.2,μ=1.5,λ=0.75,ξ=21.43,andκ=0.075.It is easy to verify that the continuously differentiable nonlinear vector functionf:Rn→Rnsatis fied the Assumption 1 withlf=0.4926.

Fig.1 Chaotic attractor generated by the system(29)when δ=0.03,?=1.5,η =0.2,μ =1.5,λ =0.75,ξ=21.43,and κ=0.075.

Letσi(ei(t))Then,σi(ei(t))satis fies the locally Lipschitz condition and the linear growth condition,i.e.,To verify the effectiveness of the proposed synchronization scheme in Theorem 1,we takeC=(cij)N×Nas the coupling con figurations of BA scale-free network[43]withm0=10,m1=5,wherem0is the number of starting vertices andm1is the degree of new vertex added at each time step.

For brevity,takingN=100 andσ0=1.9,we simulate the evolution of the networks according to the controllers de fined in Eq.(3)with the parametersk=0.6,c1=0.1,c2=0.2,θ1=0.5,θ2=1.5.Figure 2 shows the trajectories of synchronization erroreij(t)(i=1,2,...,100;j=1,2,3)and the total synchronization errorE(t).From Fig.2,one can find that the fixed-time outer synchronization is realized,and the simulation matches the theoretical results perfectly.

Fig.2 Trajectories of the synchronization error(a)and the total synchronization error(b)between networks(1)and(2)with N=100,σ0=1.9 and k=0.6,c1=0.1,c2=0.2,θ1=0.5,θ2=1.5.

Fig.3 Trajectories of the total synchronization error between networks(1)and(2)with finite-time control(c2=0)and fixed-time control(c2=0.2),the parameters N=100,σ0=1.9 and k=0.6,θ1=0.5,θ2=1.5.

Compared with finite-time synchronization,the settling time of fixed-time synchronization is bounded and independent of the initial states.Figure 3 gives the trajectories of the total synchronization errorE(t)with fixedtime control(c2=0.2)and finite-time control(c2=0).It is shown that the convergence rate of fixed-time synchronization is faster than that of finite-time synchronization.Compared to finite-time control, fixed-time control strategy shows more superiority.From inequality(21),we can see that the upper bound of the convergence time is determined by the network sizeN.Figure 4 shows that the more nodes of networks,the slower the convergence rate of complex networks is.

Fig.4 Trajectories of the total synchronization error between networks(1)and(2)when σ0=1.9,k=0.6,c1=0.1,c2=0.2,θ1=0.5,θ2=1.5,and N=100,200,400.

To obtain a more accurate estimation ofT0,we can appropriately adjust the parameterγof the controller(20).Figure 5 indicates that the convergence speed decreases with the increasing of the parameterγ.

Fig.5 Trajectories of the total synchronization error between networks(1)and(2)when σ0=1.9,k=0.6,c1=0.1,c2=0.2 and γ=2,4,8.

Actually,the fixed-time synchronization of complex networks may be achieved without controlling all the network nodes.In fact,many complex networks contain large number of nodes,taking all nodes under control is highcost and difficult to implement.Therefore,pinning control scheme,[16,20,30,44]which only needs a small fraction of nodes to be controlled,is more suitable to control complex networks to reduce computational burden and equipment resource.The effects of choosing different pinning schemes(high-degree,low-degree and random)on the convergence speed are exhibited in Fig.6,wherenDis the percentage of controlled nodes in all nodes.It is shown that the high-degree control can minimize the convergence timeT0,followed by random control,the worst is low-degree control.

Fig.6 The effect of different pinning schemes on the convergence indicator of Scale-free networks with N=400,σ0=1.9,k=0.6,c1=0.1,c2=0.2,θ1=0.5,θ2=1.5.

4.2 Numerical Example of Generalized Synchronization

To demonstrate the effectiveness of Theorem 2,we take the R?ssler-like system(29)and hyperchaotic Lüsystem as the node dynamics of networks(1)and(5),respectively.The hyperchaotic Lüsystem can be described as

For brevity,takingN=100 andσ0=1.9,we also takeD=(dij)N×Nas a new coupling con figurations of BA scale-free network withm0=10,m1=6,and simulate the evolution of the networks according to the controllers de fined in Eq.(23)with the parametersk=0.6,c1=0.1,c2=0.2,θ1=0.8,θ2=1.3.From Fig.7,one can find that the fixed-time generalized outer synchronization is realized,which con firms the theoretical analysis of Theorem 2.

Fig.7 Trajectories of the fixed-time generalized synchronization error(a)and the total generalized synchronization error(b)between networks(1)and(5)with N=100,σ0=1.9 and k=0.6,c1=0.1,c2=0.2,θ1=0.8,θ2=1.3.

5 Conclusion

In this paper,we have investigated the fixed-time outer synchronization between complex networks with noise coupling.Based on the fixed-time control method and inequality techniques,sufficient conditions of fixed-time outer synchronization are proposed.The theoretical analysis indicates that the upper bound of convergence time is determined by the network sizeN.To verify the effectiveness of the proposed synchronization scheme,for a given scale-free network,numerical simulations are performed and the effects of different pinning schemes on the convergence rate are analyzed.Compared with finite-time synchronization,the settling time of fixed-time synchronization is bounded and independent of the initial states,and thus,has a better performance.Note that time delay may in fl uence the dynamic behavior of complex networks,the fixed-time outer synchronization of time-delayed complex networks is our future direction.

We thank the anonymous referees for their helpful comments and suggestions.

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