一類分?jǐn)?shù)階復(fù)雜網(wǎng)絡(luò)混沌系統(tǒng)的投影同步*

毛北行　李慶賓

(鄭州航空工業(yè)管理學(xué)院理學(xué)院, 鄭州　450015)

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一類分?jǐn)?shù)階復(fù)雜網(wǎng)絡(luò)混沌系統(tǒng)的投影同步*

毛北行?李慶賓

(鄭州航空工業(yè)管理學(xué)院理學(xué)院, 鄭州450015)

根據(jù)分?jǐn)?shù)階系統(tǒng)的相關(guān)理論研究了一類分?jǐn)?shù)階復(fù)雜網(wǎng)絡(luò)混沌系統(tǒng)的投影同步問題,給出了分?jǐn)?shù)階復(fù)雜網(wǎng)絡(luò)以及分?jǐn)?shù)階時滯復(fù)雜網(wǎng)絡(luò)系統(tǒng)實(shí)現(xiàn)投影同步的充分性條件,仿真結(jié)果表明了方法的正確性.

投影同步,分?jǐn)?shù)階系統(tǒng),復(fù)雜網(wǎng)絡(luò)

引言

定義1[14]：Caputo分?jǐn)?shù)階導(dǎo)數(shù)定義為:

n-1<α

1　分?jǐn)?shù)階復(fù)雜網(wǎng)絡(luò)系統(tǒng)的投影同步

考慮如下一類分?jǐn)?shù)階復(fù)雜網(wǎng)絡(luò)系統(tǒng)：

ui(t)(i=1,2,…,N),

(1)

A為適當(dāng)維數(shù)的常數(shù)矩陣,ui為控制輸入,Γ是網(wǎng)絡(luò)的內(nèi)部耦合矩陣,C=(cij)N×N是外部耦合矩陣,滿足cij=0,(i≠j),cij≥0(i≠j),對角線元素定義為：

(2)

假設(shè)1：復(fù)雜網(wǎng)絡(luò)的孤立節(jié)點(diǎn)的解滿足：

Dtqs(t)=As(t)+f(s(t))

(3)

s(t)可以是一個穩(wěn)定點(diǎn),或者周期解,也可以是混沌軌跡.

定義1： 對給定的分?jǐn)?shù)階系統(tǒng)(1),若存在一個非零矩陣Λ,使得

假設(shè)2：非線性函數(shù)滿足條件：

‖f(xi(t))-Λf(s(t))‖≤li‖xi(t)-Λs(t)‖,其中l(wèi)i為大于零的常數(shù).

定義系統(tǒng)誤差為：

ei(t)=xi(t)-Λs(t), (i=1,2,…,N),

則有：

Dtqei(t)=Aei(t)+f(xi(t))-Λf(s(t))+

引理1[15]：對于一般的分?jǐn)?shù)階自治非線性微分方程Dtαx(t)=f(x(t)),當(dāng)系統(tǒng)的階數(shù)0<α≤1時,如果存在實(shí)對稱正定矩陣P,使得J(x(t))=xT(t)PDtαx(t)<0,則上述分?jǐn)?shù)階系統(tǒng)漸近穩(wěn)定.

定理1：設(shè)計控制器ui(t)=-kiei(t),若滿足條件(A+(li-ki)IN)+C?Γ<0則分?jǐn)?shù)階復(fù)雜網(wǎng)絡(luò)系統(tǒng)(1)可以實(shí)現(xiàn)投影同步.

證明：由

Dtqei(t)=Aei(t)+f(xi(t))-Λf(s(t))+

C?Γ]ei(t)<0,

根據(jù)引理1,很容易得到定理1.

2　分?jǐn)?shù)階時滯復(fù)雜網(wǎng)絡(luò)系統(tǒng)的投影同步

考慮如下一類分?jǐn)?shù)階時滯復(fù)雜網(wǎng)絡(luò)系統(tǒng)：

Dtqxi(t)=Axi(t)+f(xi(t-τ))+

(i=1,2,…,N),

(5)

A為適當(dāng)維數(shù)的常數(shù)矩陣,ui為控制輸入,τ為時滯常數(shù),Γ是網(wǎng)絡(luò)的內(nèi)部耦合矩陣,C=(cij)N×N是外部耦合矩陣,滿足cij≥0(i≠j),同時對角線元素定義為：

(6)

假設(shè)3：復(fù)雜網(wǎng)絡(luò)的孤立節(jié)點(diǎn)的解滿足：

Dtqs(t)=As(t)+f(s(t-τ))

(7)

s(t)可以是一個穩(wěn)定點(diǎn),或者周期解,也可以是混沌軌跡.

假設(shè)4：非線性函數(shù)滿足條件：

‖f(xi(t-τ))-Λf(s(t-τ))‖≤

li‖xi(t-τ)-Λs(t-τ)‖

定義系統(tǒng)誤差為：

ei(t)=xi(t)-Λs(t),(i=1,2,…,N),

則有：

Dtqei(t)=Aei(t)+f(xi(t-τ))-Λf(s(t-τ))+

(8)

引理2[16]：分?jǐn)?shù)階時滯系統(tǒng)

Dtαx(t)=f(x(t),x(t-τ)),如果有正定的矩陣P和半正定矩陣Q滿足

xT(t)PDtαx(t)+xTQx(t)-

xT(t-τ)Qx(t-τ)≤0,

則上述分?jǐn)?shù)階時滯系統(tǒng)是Lyapunov穩(wěn)定的.

定理2：設(shè)計控制器ui(t)=-kiei(t),若滿足如下矩陣不等式(9),則分?jǐn)?shù)階復(fù)雜網(wǎng)絡(luò)系統(tǒng)(5)可以實(shí)現(xiàn)投影同步.

(9)

證明：根據(jù)引理2：

其中

e(t-τ)=[‖e1(t-τ)‖,‖e2(t-τ)‖,…,

‖eN(t-τ)‖]T.

3　數(shù)值仿真

選取分?jǐn)?shù)階Lorenz系統(tǒng)為例,系統(tǒng)描述為

Dtqx1=a(x2-x1)

Dtqx2=bx1-x1x3-x2

Dtqx3=x1x2-cx3,

Dtqs1=a(s2-s1)

Dtqs2=bs1-s1s3-s2

Dtqs3=s1s2-cs3

其中x1,x2,x3為狀態(tài)變量,a,b,c為系統(tǒng)參數(shù),當(dāng)q=0.93,a=10,b=28,c=8/3時系統(tǒng)處于混沌狀態(tài).為了方便,取含三個節(jié)點(diǎn)的網(wǎng)絡(luò)進(jìn)行仿真.

圖1　定理1中的系統(tǒng)誤差曲線Fig. 1　The system errors for Theorem 1

圖2　定理2中的系統(tǒng)誤差曲線Fig. 2　The system errors for Theorem 2

定理1中選取控制器ui(t)=-kiei(t),Λ=diag(-1,1,1),Γ=I3,li=1.2ki=1,從系統(tǒng)的誤差曲線如圖1所示, 定理2中選取控制器ui(t)=-kiei(t), Λ=diag(-1,1,1),Γ=I3,τ=0.5,li=1.5,ki=1.5,系統(tǒng)的誤差曲線如圖2所示.

4　結(jié)論

研究了一類分?jǐn)?shù)階復(fù)雜網(wǎng)絡(luò)混沌系統(tǒng)及其時滯系統(tǒng)的投影同步問題,基于Lyapunov穩(wěn)定性理論和分?jǐn)?shù)階微積分的相關(guān)理論,給出了分?jǐn)?shù)階復(fù)雜網(wǎng)絡(luò)以及分?jǐn)?shù)階時滯復(fù)雜網(wǎng)絡(luò)實(shí)現(xiàn)投影同步的充分性條件,將系統(tǒng)實(shí)現(xiàn)投影同步的充分性條件轉(zhuǎn)化為矩陣不等式,從而更容易MATLAB求解,仿真結(jié)果表明了方法的正確性.

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*The project supported by the National Natural Science Foundation of Tianyuan (11226337)

? Corresponding author E-mail: bxmao329@163.com

18 May 2015,revised 18 September 2015.

PROJECTIVE SYNCHRONIZATION OF A CLASS OF FRACTIONAL-ORDER COMPLEX NETWORK CHAOS SYSTEMS*

Mao Beixing?Li Qingbin

(DepartmentofMathematicsandPhysics,ZhengzhouInstituteofAeronauticalIndustryManagement,Zhengzhou450015,China)

The paper studied the projective synchronization problem of a class of fractional-order complex network chaos systems based on fractional order systems theory. The sufficient conditions for fractional-order complex network and its time-delayed systems to realize the projective synchronization was proposed. Numerical simulations of chaotic system verified the validity of the proposed method.

projective synchronization,fractional order systems,complex networks

E-mail: bxmao329@163.com

10.6052/1672-6553-2015-72

2015-05-18收到第1稿,2015-09-18收到修改稿.

*國家自然科學(xué)基金數(shù)學(xué)天元基金資助項(xiàng)目(11226337)