nodes

Generalized function projective (lag,anticipated and complete)

synchronization between two different complex networks with

nonidentical nodes

Xiangjun Wu a ,b ,?,Hongtao Lu a

Department of Computer Science and Engineering,Shanghai Jiao Tong University,Shanghai 200240,PR China b Department of Computing Center,Institute of Complex Intelligent Network System,Henan University,Kaifeng 475004,PR China

a r t i c l e i n f o Article history:Received 15November 2010Received in revised form 17July 2011Accepted 31October 2011Available online 11November 2011Keywords:Complex networks Nonidentical nodes Generalized function projective (lag,anticipated and complete)synchronization Nonlinear feedback control

a b s t r a c t

Generalized function projective (lag,anticipated and complete)synchronization between

two different complex networks with nonidentical nodes is investigated in this paper.

Based on Barbalat’s lemma,some suf?cient synchronization criteria are derived by apply-

ing the nonlinear feedback control.Although previous work studied function projective

synchronization on complex dynamical networks,the dynamics of the nodes are coupled

partially linear chaotic systems.In our work,the dynamics of the nodes of the complex net-

works are any chaotic systems without the limitation of the partial linearity.In addition,

each network can be undirected or directed,connected or disconnected,and nodes in

either network may have identical or different dynamics.The proposed strategy is applica-

ble to almost all kinds of complex networks.Numerical simulations further verify the effec-

tiveness and feasibility of the proposed synchronization method.Numeric evidence shows

that the synchronization rate is sensitively in?uenced by the feedback strength,the time

delay,the network size and the network topological structure.

A complex network is a large set of interconnected nodes,where the nodes and connections can be https://www.360docs.net/doc/c8174642.html,plex networks widely exist in our life,from Internet to World Wide Web (WWW),from linguistic networks to social networks,from food webs to metabolic networks and so forth [1–6],and thus attract increasing attention of researchers from various ?elds of science and engineering.

In recent years,the dynamics of complex networks has been extensively investigated.As a typical kind of dynamics,syn-chronization in complex networks has received a great deal of interests.Synchronization is a fundamental phenomenon that enables coherent behavior in networks as a result of interactions.Pecora and Carroll [7]applied the master stability function method to determine the stability of the synchronous state in coupled systems.Wang and Chen [8,9]studied synchroniza-tion in two speci?c kinds of complex networks:small-world networks and scale-free networks.Gao et al.[10]presented sev-eral new delay-dependent conditions for a general complex dynamical network model with coupling delays,which guarantee the synchronized states to be asymptotically stable.Zhou et al.[11]introduced a general model of complex de-layed dynamical networks with impulsive effects,and further investigated its synchronization criteria.Yu et al.[12]inves-tigated pinning synchronization of a class of complex dynamical networks,and derived a general criterion for ensuring 1007-5704/$-see front matter ó2011Elsevier B.V.All rights reserved.

doi:10.1016/https://www.360docs.net/doc/c8174642.html,sns.2011.10.035

?Corresponding author at:Department of Computer Science and Engineering,Shanghai Jiao Tong University,Shanghai 200240,PR China.Tel.:+862115800632702;fax:+862134204879.

E-mail address:wuhsiang@https://www.360docs.net/doc/c8174642.html, (X.Wu).

3006X.Wu,H.Lu/Commun Nonlinear Sci Numer Simulat17(2012)3005–3021

network synchronization by the adaptive method and pinning technique.Wang et al.[13]studied synchronization in com-plex dynamical networks under‘‘successful’’but recoverable attacks by utilizing the framework of switching systems.

The above-mentioned studies on synchronization of networks mainly focused on the phenomenon that all nodes in a net-work achieve a coherent behavior,which was called inner synchronization[14],as it is a collective behavior within a net-work.This kind of synchronization reveals one aspect of the real world.In reality,synchronization between two or more networks regardless of synchronization of the inner network,which is called‘‘outer synchronization’’[14],always does exist in our lives.A representative illustration is predator–prey interactions in ecological communities[15],where predators and prey can in?uence one another’s evolution.For instance,plant-eating animals,such as mice and rabbits,would soon strip the land bare without the controlling effect of predators.Areas where large predators have been reduced through trapping, shooting,and other predator-control methods often develop large populations of mice and rabbits that can destroy the plants required by other wildlife species for both food and shelter.If the predators had been allowed to remain,the prey species probably would have been kept under control.In sum,the relationship between the network of predators and that of preys is of importance in maintaining balance among different animal species.Mankind has been trying many ways to maintain this balance.A great many examples about relationships between different networks are omitted here.Therefore, the investigation of the dynamics between two networks has theoretical and practical importance.Under the implicit assumption that all nodes of the networks are identical,Li et al.[14]pioneered in studying outer synchronization between two unidirectionally coupled networks and derived a criterion for the synchronization between two networks with identical topological structures.Shortly after,Tang et al.[16]studied the outer synchronization between two complex networks with different or identical topologies by using the adaptive method.Chen et al.[17]investigated adaptive synchronization be-tween two different complex networks with time-varying delay coupling.Based on the Lyapunov function method,Li and Xue[18]obtained the outer synchronization between two coupled networks by using arbitrary coupling strength for net-works with balanced structure topology.Wang et al.[19]studied the outer synchronization between two delay-coupled complex dynamical networks with different topological structures and noise perturbation.However,the above assumption is not realistic,since the nodes in a network such as Internet,the WWW,the protein network etc.,are in general different. Therefore,it is essential to study the outer synchronization between two complex networks with nonidentical nodes.

On the other hand,since the seminal work of Pecora and Carroll[20],chaos synchronization has become an active re-search subject in nonlinear science due to its great potential practical applications in secure communication,encryption, automatic control and biological systems[21–24].So far,a variety of different synchronization phenomena,such as complete synchronization(CS)[25],generalized synchronization(GS)[26],phase synchronization[27],lag synchronization(LS)[28], anticipated synchronization[29,30],projective synchronization(PS)[31–33]etc.,have been reported in the literature.CS means that the coupled chaotic systems remain in step with each other in the course of time.Only in coupled systems with identical elements(i.e.,each component having the same dynamics and parameter set),can we observe CS.In the study of nonidentical coupled systems,particularly in the drive-response systems,for example,using X(h,t)to drive Y(h0,t)where h and h0are different parameters,GS is observed under suf?ciently strong driving:the response system is a function of the driving system,i.e.,Y(t)=u(X(t)).Phase synchronization is de?ned as the entrainment of phases of chaotic oscillators, n U xàm U y=constant(n and m are integers),whereas their amplitude remains chaotic and uncorrelated.PS is the dynamical behavior in which the amplitude of the driver’s state variable and that of the response’s synchronize up to a constant scaling factor a(a proportional relation).Complete synchronization and anti-synchronization are special cases of the projective syn-chronization in cases of a=1and a=à1,respectively.LS means a coincidence of shifted-in-time states of two coupled sys-tems.The state variable of the drive system is delayed by positive s in comparison with that of the driven system,i.e., Y(t)=X(tàs)(s>0),whereas for anticipated synchronization,the driven system anticipates the driver,Y(t)=X(t+s) (s>0).Amongst all kinds of chaos synchronization,projective synchronization is one of the most interesting problems and has been especially extensively studied during recent years because of the proportionality between its synchronized dynamical states,which can be used to extend binary digital to M-nary digital communication for achieving fast communi-cation[34,35].Recently,Chen and Li[36]proposed a new type of projective synchronization method,called function pro-jective synchronization(FPS),where the drive and response systems can be synchronized up to a scaling function u(t)(–0)instead of a constant a(–0).Obviously,setting the scaling function to constant or unity will get projective synchro-nization or identical synchronization,respectively.So FPS is a more general de?nition of PS.FPS of chaotic systems is widely studied because the unpredictability of the scaling functions can additionally enhance the security of communication [37,38].

Hu et al.[39]introduced the drive-response dynamical networks with coupled partially linear chaotic systems,and then researched its projective synchronization,furthermore manipulated the scaling factor of the dynamical networks by adopt-ing pinning control technique.Zheng et al.[40]investigated adaptive projective synchronization(APS)between two complex networks with time-varying coupling delay by the adaptive control method.Wu and Lu[41]applied the adaptive technique to investigate generalized projective synchronization between two completely different complex dynamical networks with delayed coupling.Feng et al.[42]studied projective-anticipating,projective and projective-lag synchronization of time-de-layed chaotic systems on random networks based on the Lyapunov stability theory.More recently,Zhang et al.[43]consid-ered the function projective synchronization in the drive-response dynamical networks by assuming that the node dynamics are identical.However,the assumption of identical nodes is not realistic for many real-world networks.For example,the nodes in the metabolic networks,Internet,the WWW,etc.are in general distinct.The behavior of dynamical networks with non-identical nodes is much more complicated than that in the identical-node case[44].Very recently,Du[45]investigated

the function projective synchronization in the drive-response dynamical networks with non-identical nodes by employing the adaptive open-plus-closed-loop (AOPCL)method.However,to our best knowledge,at present,there are few theoretical results about generalized function projective (lag,anticipated and complete)synchronization between two different general complex dynamical networks with nonidentical nodes.

As is well known,time delay is ubiquitous in many physical systems due to the ?nite signal propagation time in biological networks,the ?nite switching speed of ampli?ers,traf?c congestions and so forth [46].Further,time delay in the interaction may in?uence the dynamical behavior of the https://www.360docs.net/doc/c8174642.html,rge-scale networks of oscillators with coupling delays have recently attracted much attention due to their tremendous application in different areas.Some numerical studies have regarded syn-chronization under delays for special cases such as globally coupled logistic maps [47]or carefully chosen delays [48].The in?uence of delays on the synchronization of delay coupled oscillators has also been investigated analytically and numerically

[10,11,17,40,41,46,49–51].In the aforementioned studies,the concept of time delay s is usually identi?ed with the time that the signal/information propagation takes to travel between the elements (nodes)of the network.In that sense,many work has been done in order to understand and identify different behaviors,and characterize the synchronization of the networks basi-cally due to the nature of these delays.However,so far,few papers in the literature focus on lag or anticipated synchronization between two complex dynamical networks.In our work,we will study generalized function projective (lag or anticipated)synchronization between two different complex dynamical networks.Here the time delay s does not mediate the interaction among the nodes of the network but instead just compares instantaneous states of the networks at different times.

Motivated by the above discussions,in this paper,we will study generalized function projective (lag,anticipated and complete)synchronization between two completely different complex dynamical networks with distinct nodes,where the nodes in one network and their counterparts in another network are asymptotically synchronized up to the desired scal-ing functions.By means of the nonlinear feedback control method,some synchronization criteria are derived based on Bar-balat’s lemma.Finally,some representative examples are performed to verify the effectiveness of the proposed scheme.In our method,the coupling con?guration matrix need not be symmetric or irreducible and no constraint is imposed on the inner coupling matrix.Different from the existing studies on function projective synchronization in the drive-response dynamical networks with coupled partially linear chaotic systems,our presented synchronization method can be applied to any complex dynamical networks (regular,random,small-world and scale-free).Numerical results demonstrate that the feedback strength,the time delay,the network size and the network topological structure can in?uence the network syn-chronization speed.

The rest of this paper is outlined as follows.Section 2gives the network models and some useful preliminaries.In Section 3,by using the nonlinear feedback control method,some suf?cient synchronization criteria are derived to guarantee general-ized function projective (lag,anticipated and complete)synchronization between two completely different complex dynam-ical networks with distinct nodes.Section 4uses some representative examples to validate the effectiveness of the proposed approach.Finally,some concluding remarks are drawn in Section 5.

https://www.360docs.net/doc/c8174642.html,work models and preliminaries

2.1.Mathematical notations

Some necessary mathematical notations that will be utilized throughout this paper are ?rst introduced as follows.Let A T (or x T )be the transpose of the matrix A (or vector x ).k x k means the 2-norm of the vector x ,i.e.,k x k ????????x T x p .Denote I n 2R n ?n as the identity matrix with dimension n . indicates the Kronecker product of two matrices.k max (á)represents the maximum eigenvalue of a matrix.

https://www.360docs.net/doc/c8174642.html,work models

Consider a general complex dynamical network consisting of N nonidentical nodes with linear couplings,which is de-scribed by

_x i et T?A i x i et Ttf i ex i et TTtX

N j ?1c ij Px j et T;i ?1;2;...;N ;e1T

where x i =(x i 1,x i 2,...,x in )T 2R n is the state vector of the i th node;A i 2R n ?n is a constant matrix and f i :R n ?R n is a smooth nonlinear vector ?eld;the nonlinear function /i (x i (t ))=A i x i (t )+f i (x i (t ))describes the dynamics of node i in the absence of interactions with other nodes;P 2R n ?n is an inner coupling matrix and C =(c ij )N ?N 2R N ?N is the coupling con?guration ma-trix representing the coupling strength and the topological structure of the network.The matrix C is de?ned as follows:if there exists a connection from node j to node i (i –j ),then c ij –0;otherwise c ij =0.The diagonal elements of matrix C are de?ned by

c ii ?àX

N j ?1;j –i c ij ;i ?1;2;...;N :

X.Wu,H.Lu /Commun Nonlinear Sci Numer Simulat 17(2012)3005–30213007

We take the network given by Eq.(1)as the drive network,and the response network with a nonlinear control scheme is given by

_y i et T?B i y i et Ttg i ey i et TTtX

N j ?1d ij Qy j et Ttu i et T;i ?1;2;...;N :e2T

Here y i =(y i 1,y i 2,...,y in )T 2R n is the response state vector of the i th node,and u i (t )2R n are the nonlinear feedback controllers.g i ,B i and Q have the same meanings as those of f i ,A i and P in Eq.(1),respectively.D =(d ij )N ?N 2R N ?N is the coupling con?g-uration matrix,which has the same meaning as that of matrix C .The following is a brief introduction of the nonlinear feed-back control method.

The drive system and the response system are de?ned as follows:

et T?F ex et TT;e3T_y et T?G ey et TTtu et T;e4Twhere x (t ),y (t )2R n are the state vectors;F ,G :R n ?R n are continuous nonlinear vector functions;u (t )is the nonlinear feed-back controller.The controller u (t )is further expressed in the following form:

u et T?u 1et Ttu 2et T;e5Twhere u 1(t )is the nonlinear part of u (t ),i.e.,u 1(t )=W (x ,y )(W (á)is a continuous nonlinear vector function),and u 2(t )is the linear feedback part,i.e.,u 2(t )=k (y àx ).As one of the effective methods for controlling chaos,the nonlinear feedback control strategy has attracted the interest of many researchers [52–54].

2.3.Preliminaries

In order to introduce our main results,the following de?nition and lemma are required.

De?nition 1.Let x i (t às (t ))be the lag (anticipated or current)state of node i in the drive network (1),and y i (t )be the current state of node i in the response network (2).For given the time-varying delay s (t ),if there exist the nonzero scaling functions a i (t )such that

lim t !1k y i et Tàa i et Tx i et às et TTk ?0;i ?1;2;...;N

then it is said that we achieve

(i)generalized function projective lag synchronization (GFPLS)between networks (1)and (2)with s (t )>0;

(ii)generalized function projective anticipated synchronization (GFPAS)between networks (1)and (2)with s (t )<0;(iii)generalized function projective synchronization (GFPS)between networks (1)and (2)with s (t )=0.

Remark 1.Obviously,if the scaling functions a i (t )(i =1,2,...,N )are taken as the nonzero constants h i ,GFPLS,GFPAS and GFPS are simpli?ed to generalized projective lag synchronization (GPLS),generalized projective anticipated synchronization (GPAS)and generalized projective synchronization (GPS),respectively.

Remark 2.In particular,if a i (t )=1(i =1,2,...,N ),GFPLS,GFPAS and GFPS are reduced to generalized lag synchronization (GLS),generalized anticipated synchronization (GAS)and generalized synchronization (GS),respectively.

Lemma 1(Barbalat’s lemma [55]).If u :R ?R +is a uniformly positive function for t P 0and if the limit of the integral lim t !1R t 0u ed Td d exists and is ?nite,then lim t ?1u (t)=0.

3.Synchronization criteria

In this section,we will study GFPLS (GFPAS or GFPS)between two completely different dynamical networks with non-identical nodes by the nonlinear feedback control method.

De?ne the synchronization errors as e i (t )=y i (t )àa i (t )x i (t às (t ))(i =1,2,...,N )where a i (t )is the nonzero scaling function.The time derivative of e i (t )is:

_e i et T?_y i et Tà_a i et Tx i et às et TTàa i et T_x i et às et TTe1à_s et TT;i ?1;2;...;N :e6T

3008X.Wu,H.Lu /Commun Nonlinear Sci Numer Simulat 17(2012)3005–3021

By substituting Eqs.(1)and(2)into Eq.(6),we can obtain the error dynamical system as follows:

etT?B i e ietTta ietTeB iàA iTx ietàsetTTtg iey ietTTàa ietTf iex ietàsetTTTà_a ietTx ietàsetTTta ietT_setT_x ietàsetTT

X N

j?1d ij Qe jetTàa ietT

X N

j?1

ec ij Ptd ij QTx jetàsetTTtu ietT;e7T

where i=1,2,...,N.

Theorem1.For given nonzero scaling functions a i(t),GFPLS(GFPAS or GFPS)between the drive network(1)and the response network(2)can be achieved by using the following nonlinear controllers:

u ietT?àa ietTeB iàA iTx ietàsetTTàg iey ietTTta ietTf iex ietàsetTTTt_a ietTx ietàsetTTàa ietT_setT_x ietàsetTTta ietT?

X N

j?1

ec ij Ptd ij QTx jetàsetTTàke ietT;e8Twhere i=1,2,...,N,and the feedback strength k is a suf?ciently large positive constant.

Proof.Let eetT?e T

1;e T

;...;e T

àáT

2R nN,and construct the Lyapunov candidate function as follows:

VetT?1

e TetTeetT?

X N

i?1

e T

etTe ietT:e9T

Obviously,V(t)P0.Taking the time derivative of V(t)along the trajectories of the error system(7),and by using Eq.(8),we have

_VetT?

X N

i?1e T

etT_e ietT?

X N

i?1

e T

etT

B i e ietTta ietTeB iàA iTx ietàsetTTtg iey ietTTàa ietTf iex ietàsetTTTà_a ietTx ietàsetTT

ta ietT_setT_x ietàsetTTt

X N

j?1d ij Qe jetTàa ietT

X N

j?1

ec ij Ptd ij QTx jetàsetTTtu ietT

X N

i?1e T

etTB i e ietTt

X N

j?1

d ij Q

e jetTàke ietT

X N

i?1e T

etTB i e ietTt

X N

i?1

X N

j?1

d ij

e T

etTQe jetTàk

X N

i?1

e T

etTe ietT?e TetTb B eetTte TetTb D eetTàk e TetTeetT

max b Btb B T

tk max

b Dtb D T

àk

e TetTeetT;

where b B?diageB1;B2;...;B NT2R nN?nN,and b D?D Q2R nN?nN.

Taking the feedback strength as k P k??k maxeeb Btb B TT=2Ttk maxeeb Dtb D TT=2Tt1,we can obtain _VetT6àe TetTeetT60:

So V(t)is uniformly continuous.Moreover,we have V(t)6V(0)eà2t.Thus lim t!1R t

VesTd s exists,which indicates that V(t)is

integrable on[0,+1).On the basis of Barbalat’s lemma,one has lim t?+1V(t)=0,i.e.,lim t?+1e i(t)=0(i=1,2,...,N).Therefore, GFPLS(GFPAS or GFPS)between two completely different dynamical networks with distinct nodes is asymptotically achieved by the nonlinear controllers(8).This completes the proof.h

Remark3.In our work,the coupling con?guration matrices C and D need not be symmetric or irreducible.In addition,there is not any constraint imposed on the inner coupling matrices P and Q.Furthermore,each node of the networks may have different local dynamics.The proposed approach is applicable to all kinds of complex dynamical networks.

Remark4.The feedback strength k can be chosen appropriately beforehand to adjust the synchronization rate.Theoreti-cally,a larger k will lead to the faster synchronization.However,the inequality k P k?is just only a suf?cient condition but not a necessary https://www.360docs.net/doc/c8174642.html,ter numerical examples will demonstrate that a smaller k can also be applied to achieve the required synchronization.

Based on Theorem1,the following corollaries can be easily derived.

Corollary1.For given nonzero scaling functions a i(t),if the drive network(1)and the response network(2)have identical topo-logical structures and inner coupling matrices,i.e.,C=D and P=Q,then the two networks can achieve GFPLS(GFPAS or GFPS) under the following controllers:

X.Wu,H.Lu/Commun Nonlinear Sci Numer Simulat17(2012)3005–30213009

u ietT?àa ietTeB iàA iTx ietàsetTTàg iey ietTTta ietTf iex ietàsetTTTt_a ietTx ietàsetTTàa ietT_setT_x ietàsetTTàke ietT;

i?1;2;...;N;e10Twhere k P k??k maxeeb Btb B TT=2Ttk maxeeb Dtb D TT=2Tt1.

Corollary2.For given nonzero scaling functions a i(t),if the ith nodes in networks(1)and(2)have the same dynamics,i.e.,A i=B i and f i=g i(i=1,2,...,N),then the two networks can achieve GFPLS(GFPAS or GFPS)by the following control scheme: u ietT?àf iey ietTTta ietTf iex ietàsetTTTt_a ietTx ietàsetTTàa ietT_setT_x ietàsetTTta ietT

X N

j?1

ec ij Ptd ij QTx jetàsetTTàke ietT;

i?1;2;...;N;e11Twhere k P k??k maxeeb Btb B TT=2Ttk maxeeb Dtb D TT=2Tt1.

Corollary3.Assume that the time delay s(t)is constant,i.e.,_setT?0.For given nonzero scaling functions a i(t),the drive network (1)and the response network(2)can achieve GFPLS(GFPAS or GFPS)under the controllers as follows:

u ietT?àa ietTeB iàA iTx ietàsetTTàg iey ietTTta ietTf iex ietàsetTTTt_a ietTx ietàsetTT

ta ietT

X N

j?1

ec ij Ptd ij QTx jetàsetTTàke ietT;i?1;2;...;N;e12Twhere k P k??k maxeeb Btb B TT=2Ttk maxeeb Dtb D TT=2Tt1.

Corollary4.Assume that the time delay s(t)is constant,i.e.,_setT?0.If the drive network(1)and the response network(2)have identical topological structures and inner coupling matrices,and the ith nodes in networks(1)and(2)have identical dynamics,i.e., C=D,P=Q,A i=B i and f i=g i(i=1,2,...,N).For given nonzero scaling functions a i(t),the two networks can realize GFPLS(GFPAS or GFPS)by the following controllers:

u ietT?àf iey ietTTta ietTf iex ietàsetTTTt_a ietTx ietàsetTTàke ietT;i?1;2;...;N;e13Twhere k P k??k maxeeb Btb B TT=2Ttk maxeeb Dtb D TT=2Tt1.

4.Illustrative examples

In this section,some representative examples are performed to verify the effectiveness of the proposed synchronization scheme in the previous section.The following quantity

EetT?

?????????????????????????????????X N

i?1

X n

j?1

etT=N v u

u t

is used to measure the quality of the synchronization process.It is obvious that when E(t)no longer increases,the drive and response networks achieve the desired synchronization globally.

4.1.GFPLS between two complex dynamical networks

4.1.1.GFPLS between two different complex dynamical networks with nonidentical nodes

In this subsection,we focus on studying GFPLS between two completely different general complex dynamical networks with nonidentical nodes.In the simulations,the drive network(1)is described by the following10different Lüchaotic sys-tems[56]

etT

_x i3etT

0 B@1

C A?

à36360

0e20t0:5?eià1TT0

00à3

C A

x i1etT

x i2etT

x i3etT

C At

àx i1etTx i3etT

x i1etTx i2etT

C A;e14T

For the response network(2),the?rst5nodes are described by the following Liu chaotic systems[57]

etT

_y i3etT

0 B@1

C A?

à10100

4000

00à2:5

C A

etT

C At

ày i1etTy i3etT

4y2

etT

C Ae15T

3010X.Wu,H.Lu/Commun Nonlinear Sci Numer Simulat17(2012)3005–3021

and the other5nodes are described by the following Chen chaotic systems[58] _y

etT

_y i3etT

0 B@1

C A?

à35350

à7280

00à3

C A

etT

C At

ày i1etTy i3etT

etTy i2etT

C A;e16T

The inner coupling matrices are taken arbitrarily as P=diag(0.1,0.5,0.2)and Q=diag(0.2,0.1,0.3).The coupling con?guration matrices are given respectively as follows:

à3010010010

0à210010000

10à51101001

100à2000100

0110à410010

00001à21000

100100à3001

0000001à210

00001010à31

011101001à5

B B

C C

C A

and

à1000010000

0à301000101

10à40101010

010à3010100

0010à200001

10010à30010

010010à3100

0010010à301

00010010à20

110010010à4

B B

C C

C A

It is easy to obtain k maxeeb Btb B TT=2T?30:98and k maxeeb Dtb D TT=2T?0:04.Thus we get k?=32.02.

We arbitrarily choose the scaling functions as a i1=2àcos t,a i2=1àeàt and a i3=1.5+0.5sin(2t)(i=1,2,...,10).The ini-tial values of x i(0)and y i(0)are selected randomly in(à5,5).According to Theorem1,we design the controllers by the control scheme(8).Fig.1displays the time evolution of the synchronization errors e i(t)and E(t)with different positive time delays s(t)and feedback strengths k,which indicates that GFPLS between the drive network(1)and the response network(2)is achieved.When k=25,the synchronization errors e i(t)and E(t)for different values of time delay s(t)are depicted in Figs. 1(a)and(b).Obviously,from Figs.1(a)and(b),the synchronization speed turns faster with the increase of the time delay s(t).On the other hand,let s(t)=0.1and only change the feedback strength k.As described in Figs.1(a)and(c),one can see that the larger feedback strength k,the faster the convergence to synchronization.In addition,numerical evidence shows that when we choose the feedback strength k

4.1.2.GFPLS between two identical complex dynamical networks

Consider the drive network(1)and the response network(2)have identical topological structures and inner coupling matrices,and each node in two networks has identical local dynamics,i.e.,C=D,P=Q,A i=B i and f i=g i(i=1,2,...,N).Accord-ing to Corollary4,we can easily derive the control scheme by Eq.(13).The network size is taken as N=100.We choose Chen chaotic system(16)as node dynamics in the drive and the response networks.For simplicity,the inner coupling matrices are selected as P=Q=diag(1,1,1).We arbitrarily choose the scaling functions as a i1=1.5+sin(0.1t),a i2=2cos(0.5t)à3and a i3=1+0.2sin(5t)+0.1cos(10t).The initial conditions for x i(0)and y i(0)are chosen randomly in[0,2].In the following, we will investigate GFPLS in the BA scale-free network and the WS small-world network,respectively.

Example1.Consider GFPLS in the BA scale-free network.The BA scale-free network is constructed with m=m0=5and N=100.The detailed generation algorithm of the BA scale-free network can be found in[59].With the parameters speci?ed above,we have k maxeeb Btb B TT=2T?37:97and k maxeeb Dtb D TT=2T?30:31.Thus k?=69.28.To obtain a critical value of the feedback strength k to achieve GFPLS,we set the time delay s(t)=0.1and only continuously increase the feedback strength k, from k=0,in steps of0.5.When k<45,no synchronous phenomenon is observed.When k=45,the synchronization error E(t)is shown in Fig.2,which indicates that GFPLS between two complex dynamical networks is achieved.It implies that when k

X.Wu,H.Lu/Commun Nonlinear Sci Numer Simulat17(2012)3005–30213011

synchronization.When s (t )=0.1,Fig.2displays the synchronization errors E (t )for different values of k .Fig.3depicts the synchronization errors E (t )for different values of k when s (t )=1.5.From Figs.2and 3,one can ?nd that the larger feedback strength k ,the faster is to achieve the desired GFPLS.Furthermore,comparing the results in Fig.2with those in Fig.3,one can surprisingly ?nd that the time delay s (t )has little in?uence on the synchronization

speed.

Fig.1.The GFPLS errors between the drive network (1)and the response network (2)with different s (t )and k .

Fig.2.The results of GFPLS in the BA scale-free network with s(t)=0.1and different k.

Fig.3.The results of GFPLS in the BA scale-free network with s(t)=1.5and different k.

Example2.Consider GFPLS in the WS small-world network.To construct the WS small-world network,the parameters are chosen as follows:the network size N=100,the number of the connected nearest neighbors K=5and the rewiring proba-bility p=0.2.We perform the similar simulations as those in the above example.We?nd that when k P45,GFPLS between two dynamical networks can only be realized.Fig.4(a)describes the results of GFPLS in the WS small-world networks as s(t)=0.1and s(t)=1.5,respectively.From Fig.4(a),it can be found that the synchronization speed becomes faster with the increase of the feedback strength.Moreover,for the same feedback strength k,the synchronization speed with s(t)=0.1is faster than that with s(t)=1.5.

For revealing the relation between the probability p and the synchronization effect,we only increase the probability p and keep other parameters unchanged.The results of GFPLS in the WS small-world network with p=0.6are shown in Fig.4(b). From the simulation results,we?nd that GFPLS in the WS small-world network can appear when k P45.As described in Fig.4(b),the increase of the feedback strength k will reduce the time for reaching GFPLS.In addition,comparing the results in Fig.4(b)with those in Fig.4(a),we can conclude that the synchronization speed turns slower with the increase of the probability p.For instance,in Fig.4(a),?x s(t)=0.1and p=0.2,when k=45,k=50and k=60,GFPLS can be achieved after t P46,t P19and t P7,respectively.However,in Fig.4(b),set s(t)=0.1and p=0.6,when k=45,k=50and k=60,GFPLS can be obtained after t P49.7,t P19.2and t P7.7,respectively.

From the numerical results described in the above two examples,one can further conclude that the WS small-world network is easier to achieve GFPLS using the proposed scheme than the BA scale-free network.

4.2.GFPAS between two complex dynamical networks

4.2.1.GFPAS between two identical complex dynamical networks with nonidentical nodes

In this subsection,we will investigate GFPAS between two identical complex dynamical networks with diverse nodes. Both the drive network(1)and the response network(2)have the same topological structures and uniform inner coupling matrices,i.e.,C=D,P=Q.Moreover,the i th nodes in networks(1)and(2)have identical local dynamics,namely,A i=B i and f i=g i(i=1,2,...,10).

In the following,we consider the drive network consisting of 10distinct Lüchaotic systems,which is written as follows:

i et T?A i x i et Ttf i ex i et TTtX 10j ?1c ij Px j et T;i ?1;2;...;10;e17Twhere A i ?à363600

e20t0:5?ei à1TT000à30

B @1

C A and f i ex i et TT?0àx i 1et Tx i 3et Tx i 1et Tx i 2et T

0B @1C A :

The corresponding response network is described by _y i et T?A i y i et Ttf i ey i et TTtX

10j ?1c ij Py j et Ttu i et T;i ?1;2;...;10:e18T

For brevity,the inner coupling matrix P ,the coupling con?guration matrix C and the scaling functions a i (t )are chosen as those in Section 4.1.1.We can easily derive k max eeb B

tb B T T=2T?29:46and k max eeb D tb D T T=2T?0:05,which gives k ?=30.51.The initial conditions for x i (0)and y i (0)(i =1,2,...,10)are chosen arbitrarily in (à5,5).The controllers can be derived by the control scheme (13)in Corollary 4.The time evolution of the GFPAS error E (t )with time delay s (t )=à0.5and different k is shown in Fig.5,from which it is easy to see that GFPAS between the drive network (17)and the response network (18)is realized.Further,as predicted in Remark 4,the larger feedback strength can accelerate the synchronization speed.To take a clearer view of the relationships between dynamics of nodes in two networks,the corresponding variables of node 6are plot-ted in Fig.6,where the solid line denotes the state variables x i (t )of node 6in the drive network (17),and the dashed line represents the state variables y i (t )of node 6in the response network (18).

4.2.2.GFPAS between two identical complex dynamical networks in the BA scale-free network

Here we consider the drive network and the response network have the same topological structures and uniform inner coupling matrices,i.e.,C =D ,P =Q ,and the i th nodes in the drive and response networks have identical dynamics,

namely,

Fig.4.The results of GFPLS in the WS small-world network.

A i=

B i and f i=g i(i=1,2,...,N).Take Lüchaotic system as node dynamics in the drive and response networks.The controllers are designed based on the control law(13)in Corollary4.Set the network size N=100.For brevity,we always take P=Q=diag(1,1,1)and select the scaling functions as those in Section4.1.2.The initial conditions for x i(0)and y i(0) (i=1,2,...,10)are taken randomly in[0,2].

Numerical results of GFPAS in the BA scale-free network with m=m0=6are shown in Fig.7.Fig.7(a)depicts the time evolution of the synchronization error E(t)with s(t)=à0.5and different k.From Fig.7(a),one can clearly see that the syn-chronization speed turns faster with the increase of the feedback strength k.We further investigate the effect of the time delay s(t)on the synchronization performance.We only increase the time delay s(t)and keep other parameters unchanged. Fig.7(b)displays the time evolution of the synchronization errors E(t)with different s(t)when k=45.As described by Fig.7(b),when s(t)=à0.1,s(t)=à0.5and s(t)=à1,GFPAS can be realized after t P5.79,t P5.17and t P5.47,respectively. It indicates that the time delay s(t)has little in?uence on the synchronization speed.

4.3.GFPS between two complex dynamical networks

4.3.1.GFPS between two different complex dynamical networks with nonidentical nodes

In this subsection,we will discuss GFPS(s(t)=0)between two completely different general complex dynamical networks with distinct nodes.Consider the drive network composed of10different Lüchaotic systems as given by Eq.(17)in Section 4.2.1,and the corresponding response network described by10different uni?ed chaotic systems[60]as follows:

_y i etT?B i y ietTtg iey ietTTt

X10

j?1

d ij Qy

etTtu ietT;i?1;2;...;10;e19Tbetween the drive network(17)and the response network(18)with

Fig.6.The time evolution of the variables of node6in networks(17)and(18)with s(t)=à0.5and k=35.

where

B i ?àe2:5?ei à1Tt10Te2:5?ei à1Tt10T0e28à3:5?ei à1TT

e2:9?ei à1Tà1T000àe0:1?i ?ei à1Tt8T=30

B @1

C A and g i ey i et TT?0ày i 1et Ty i 3et Ty i 1et Ty i 2et T0B @1

C A :

For simplicity,the inner coupling matrices P and Q ,the coupling con?guration matrices C and D ,and the scaling functions a i (t )are chosen as those in Section 4.1.1.Clearly,k max eeb B

tb B T T=2T?28:55and k max eeb D tb D T T=2T?0:04,thus we have k ?=29.59.

The initial values of x i (0)and y i (0)(i =1,2,...,10)are taken arbitrarily in (à5,5).By means of Theorem 1,we can easily design the controllers to ensure GFPS between the drive network (17)and the response network (19)by the control scheme

(8).Fig.8shows the time evolution of the synchronization errors e i (t )and E (t )with k =45.As depicted in Fig.8,GFPS be-tween the drive and response networks is obtained after t P 14.8.To show the synchronization process clearly,some sub-

Fig.7.The results of GFPAS in the BA scale-free network.

Fig.8.The GFPS errors between the drive network (17)and the response network (19)with k =45.

Fig.9.The time evolution of subvariables of nodes5and7in networks(17)and(19)with k=45.

Fig.10.(Left)a star coupled network;(right)a directed ring network.

Fig.11.The results of GFPS in the star coupled network with different k.

Example1.GFPS in the star coupled network.Simulation results show that only when k P20.2,can GFPS in the star coupled network occur.Fig.11plots the time evolution of the synchronization error E(t)with the feedback strength k increasing. From11we can easily conclude that the synchronization speed becomes faster with the increase of the feedback strength.

Fig.12.The results of GFPS in the directed ring network with different k.

13.The results of GFPS in the WS small-world network with with

Fig.14.The results of GFPS in the WS small-world network with different k.

Example 2.GFPS in the directed ring network.From the numerical results,it is found that GFPS in the directed ring network can be achieved when k P 20.2.The synchronization error E (t )with different k is displayed in Fig.12,which indicates that the synchronization rate turns faster with the increase of the feedback https://www.360docs.net/doc/c8174642.html,pared with the results in Fig.11,the time required to reach GFPS obviously decreases.In other words,for the same value of k ,GFPS in the directed ring network can be achieved faster than that in the star coupled network.

Example 3.GFPS in the WS small-world network.We generate the WS small-world network with the parameters as follows:the network size N =100,the number of the connected nearest neighbors K =5and the rewiring probability p =0.3.Numer-ical simulations display that GFPS can be obtained when k P 34.Fig.13depicts the synchronization error E (t )with k =34,which tends to zero after t P 87.6.Fig.14describes the time evolution of the synchronization error E (t )while the feedback strength k is increasing,from which we can see that the synchronization speed becomes faster with the increase of the feed-back strength.

We further investigate the effect of the probability p on the synchronization effect.We ?x k =45and only vary the prob-ability p .We ?nd that GFPS in the WS small-world network can be realized when 0.16p 61.We demonstrate this by Fig.15,which shows that the probability p has a signi?cant in?uence on the synchronization rate.The simulations show that when p =0.2,the synchronization rate is the fastest and the synchronization performance is the best;while p =0.8,the synchro-nization speed is the slowest and the synchronization effect is the worst.In addition,comparing the results in Figs.13–15with those in Figs.11and 12,we ?nd that the WS small-world network is more dif?cult to obtain GFPS than the star coupled network or the directed ring network.

5.Conclusions

Considering few studies on generalized function projective synchronization between two complex networks in the liter-ature,in this paper,we investigate generalized function projective (lag,anticipated and complete)synchronization between two completely different complex dynamical networks with nonidentical nodes.Based on Barbalat’s lemma,we

have Fig.15.The results of GFPS in the WS small-world network with k =45and different p .

3020X.Wu,H.Lu/Commun Nonlinear Sci Numer Simulat17(2012)3005–3021

proposed a nonlinear control scheme to achieve GFPLS(GFPAS or GFPS).The coupling con?guration matrix is not necessarily symmetric or irreducible.In addition,there is not any constraint imposed on the inner coupling matrix.The presented meth-od can be applied to a great variety of complex dynamical networks.Some representative examples are provided to show the effectiveness and feasibility of the proposed synchronization method.From the simulation results,we?nd that the network synchronization rate is ef?ciently affected by not only the feedback strength and the time delay but also by the network size and the network topological structure.

The presented synchronization scheme in our work can be applied to secure communication,encryption and automatic control.As we all know,synchronization between two different networks with nonidentical nodes is a hard problem and has not been resolved very well until now.How to design the effective and simple control schemes for GFPLS(GFPAS or GFPS) between two dynamical networks is an important issue and remains to be discussed in future.

Acknowledgements

The authors are very grateful to thank the anonymous reviewers and the editor for their valuable comments and sugges-tions.This research is jointly supported by the National Natural Science Foundation of China(Grant Nos.61004006and 60873133),Natural Science Foundation of Henan Province,China(Grant No.112300410009)and Natural Science Foundation of Educational Committee of He’nan Province,China(Grant No.2011A520004).

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