一类分数阶复杂网络混沌系统的投影同步*

毛北行　李庆宾

(郑州航空工业管理学院理学院, 郑州　450015)



一类分数阶复杂网络混沌系统的投影同步*

毛北行†李庆宾

(郑州航空工业管理学院理学院, 郑州450015)

根据分数阶系统的相关理论研究了一类分数阶复杂网络混沌系统的投影同步问题,给出了分数阶复杂网络以及分数阶时滞复杂网络系统实现投影同步的充分性条件,仿真结果表明了方法的正确性.

投影同步,分数阶系统,复杂网络

引言

定义1[14]：Caputo分数阶导数定义为:

n-1<α

1　分数阶复杂网络系统的投影同步

考虑如下一类分数阶复杂网络系统：

ui(t)(i=1,2,…,N),

(1)

A为适当维数的常数矩阵,ui为控制输入,Γ是网络的内部耦合矩阵,C=(cij)N×N是外部耦合矩阵,满足cij=0,(i≠j),cij≥0(i≠j),对角线元素定义为：

(2)

假设1：复杂网络的孤立节点的解满足：

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

(3)

s(t)可以是一个稳定点,或者周期解,也可以是混沌轨迹.

定义1： 对给定的分数阶系统(1),若存在一个非零矩阵Λ,使得

假设2：非线性函数满足条件：

‖f(xi(t))-Λf(s(t))‖≤li‖xi(t)-Λs(t)‖,其中li为大于零的常数.

定义系统误差为：

ei(t)=xi(t)-Λs(t), (i=1,2,…,N),

则有：

Dtqei(t)=Aei(t)+f(xi(t))-Λf(s(t))+

引理1[15]：对于一般的分数阶自治非线性微分方程Dtαx(t)=f(x(t)),当系统的阶数0<α≤1时,如果存在实对称正定矩阵P,使得J(x(t))=xT(t)PDtαx(t)<0,则上述分数阶系统渐近稳定.

定理1：设计控制器ui(t)=-kiei(t),若满足条件(A+(li-ki)IN)+C⊗Γ<0则分数阶复杂网络系统(1)可以实现投影同步.

证明：由

Dtqei(t)=Aei(t)+f(xi(t))-Λf(s(t))+

C⊗Γ]ei(t)<0,

根据引理1,很容易得到定理1.

2　分数阶时滞复杂网络系统的投影同步

考虑如下一类分数阶时滞复杂网络系统：

Dtqxi(t)=Axi(t)+f(xi(t-τ))+

(i=1,2,…,N),

(5)

A为适当维数的常数矩阵,ui为控制输入,τ为时滞常数,Γ是网络的内部耦合矩阵,C=(cij)N×N是外部耦合矩阵,满足cij≥0(i≠j),同时对角线元素定义为：

(6)

假设3：复杂网络的孤立节点的解满足：

Dtqs(t)=As(t)+f(s(t-τ))

(7)

s(t)可以是一个稳定点,或者周期解,也可以是混沌轨迹.

假设4：非线性函数满足条件：

‖f(xi(t-τ))-Λf(s(t-τ))‖≤

li‖xi(t-τ)-Λs(t-τ)‖

定义系统误差为：

ei(t)=xi(t)-Λs(t),(i=1,2,…,N),

则有：

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

(8)

引理2[16]：分数阶时滞系统

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

xT(t)PDtαx(t)+xTQx(t)-

xT(t-τ)Qx(t-τ)≤0,

则上述分数阶时滞系统是Lyapunov稳定的.

定理2：设计控制器ui(t)=-kiei(t),若满足如下矩阵不等式(9),则分数阶复杂网络系统(5)可以实现投影同步.

(9)

证明：根据引理2：

其中

e(t-τ)=[‖e1(t-τ)‖,‖e2(t-τ)‖,…,

‖eN(t-τ)‖]T.

3　数值仿真

选取分数阶Lorenz系统为例,系统描述为

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为状态变量,a,b,c为系统参数,当q=0.93,a=10,b=28,c=8/3时系统处于混沌状态.为了方便,取含三个节点的网络进行仿真.

图1　定理1中的系统误差曲线Fig. 1　The system errors for Theorem 1

图2　定理2中的系统误差曲线Fig. 2　The system errors for Theorem 2

定理1中选取控制器ui(t)=-kiei(t),Λ=diag(-1,1,1),Γ=I3,li=1.2ki=1,从系统的误差曲线如图1所示, 定理2中选取控制器ui(t)=-kiei(t), Λ=diag(-1,1,1),Γ=I3,τ=0.5,li=1.5,ki=1.5,系统的误差曲线如图2所示.

4　结论

研究了一类分数阶复杂网络混沌系统及其时滞系统的投影同步问题,基于Lyapunov稳定性理论和分数阶微积分的相关理论,给出了分数阶复杂网络以及分数阶时滞复杂网络实现投影同步的充分性条件,将系统实现投影同步的充分性条件转化为矩阵不等式,从而更容易MATLAB求解,仿真结果表明了方法的正确性.

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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.

*国家自然科学基金数学天元基金资助项目(11226337)