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  山东大学学报(工学版)  2017, Vol. 47 Issue (4): 43-49  DOI: 10.6040/j.issn.1672-3961.0.2016.122
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引用本文 

张玉婷, 李望, 王晨光, 刘友权, 侍红军. 不连续耦合的时滞复杂动态网络的同步[J]. 山东大学学报(工学版), 2017, 47(4): 43-49. DOI: 10.6040/j.issn.1672-3961.0.2016.122.
ZHANG Yuting, LI Wang, WANG Chenguang, LIU Youquan, SHI Hongjun. Synchronization of time-delayed complex dynamical networks with discontinuous coupling[J]. Journal of Shandong University (Engineering Science), 2017, 47(4): 43-49. DOI: 10.6040/j.issn.1672-3961.0.2016.122.

基金项目

中央高校基本科研业务费资助项目(2015xkms076);国家级大学生创新创业训练计划资助项目(201710290089)

作者简介

张玉婷(1995—), 女, 山东邹城人, 硕士研究生, 主要研究方向为复杂网络同步. E-mail:zhangyutingzjl@163.com

通讯作者

侍红军(1979—), 男, 江苏淮安人, 讲师, 博士研究生, 主要研究方向为复杂网络同步与控制, 图论染色. E-mail:hjshi79@gmail.com

文章历史

收稿日期:2016-04-13
网络出版时间:2017-03-29 16:32:36
不连续耦合的时滞复杂动态网络的同步
张玉婷1,3, 李望1,2, 王晨光1, 刘友权1, 侍红军1     
1. 中国矿业大学数学学院, 江苏 徐州 221116;
2. 江苏徐州医药高等学校基础部, 江苏 徐州 221116;
3. 哈尔滨工业大学人文社科与法学学院, 黑龙江 哈尔滨 150001
摘要:基于李雅普诺夫稳定性理论, 对不连续耦合的时滞复杂动态网络进行分析, 得到网络同步的充分条件, 并且给出网络实现同步时滞的上界估计。研究表明:即使网络之间的耦合是不连续的, 只要时滞满足一定条件, 网络也可以实现同步, 且网络容许的时滞上界与耦合强度、网络代数连通性以及耦合的开关率相关。数值模拟中利用Ikeda系统作为节点动力学, 采用误差函数作为网络同步性指标, 给出网络同步误差演化轨迹和各状态的演化轨迹, 并进一步分析控制参数对同步速度的影响, 模拟结果验证了理论结果的正确性。
关键词复杂网络    同步    不连续耦合    时滞    李雅普诺夫稳定性    
Synchronization of time-delayed complex dynamical networks with discontinuous coupling
ZHANG Yuting1,3, LI Wang1,2, WANG Chenguang1, LIU Youquan1, SHI Hongjun1     
1. School of Mathematics, China University of Mining and Technology, Xuzhou 221116, Jiangsu, China;
2. Basic Department, Jiangsu Xuzhou Medical College, Xuzhou 221116, Jiangsu, China;
3. School of Humanities and Social Science and Law, Harbin Institute of Technology, Harbin 150001, Heilongjiang, China
Abstract: The synchronization problem of complex dynamical networks with time delay and discontinuous coupling was investigated based on Lyapunov stability theory. The sufficient conditions for the networks synchronization was established and the upper bound estimation of the time delay was obtained. The acquired analytical results showed that network with discontinuous coupling could achieve synchronization if time delay met some conditions. The upper bound of the delay for synchronization depended on the coupling strength, the algebraic connectivity of network and on-off rate. The application of numerical simulation results proved that evolution trajectory of network synchronization error and different conditions, in which Ikeda system was used as node dynamics and error function as the network synchronization index. Furthermore, the effect of control parameters on the synchronization speed was analyzed. Numerical examples were provided to verify the effectiveness of the theoretical results.
Key words: complex networks    synchronization    discontinuous coupling    time delay    Lyapunov stability theory    
0 引言

近年来, 复杂网络已迅速发展成为一个新的研究领域, 引起包括工程、金融经济、神经科学、生物、数学等众多领域学者的关注。从因特网到万维网, 从物流网络到物联网, 从生态网络到交通网络, 从生物体中由各种神经网络构成的大脑到各种新陈代谢网络, 从政治经济网络到社会关系网络, 复杂网络广泛存在于我们生活中[1-3]。同步是指网络个体的状态对时间的一致性, 是复杂网络的典型动力学行为之一, 在商业、神经科学、社会科学、通讯通信等领域起着重要作用。近年来, 关于复杂网络的同步问题受到了广泛关注[4-9]。文献[10]给出了网络同步时网络耦合矩阵的谱分布范围。文献[11-12]研究了小世界网络与无标度网络的同步问题。文献[13]研究了时变动态网络的同步问题。文献[14]研究了复杂网络的外部同步问题。

在现实世界中, 时滞普遍地存在, 通常是由有限的信号传输和记忆效应引起的[15-17]。在很多生物和物理系统中, 时滞广泛存在, 例如基因调控网络、新陈代谢网络和神经网络。时滞对复杂动态网络的行为有重要影响。因此时滞复杂网络的同步问题引起了很多人的研究兴趣。文献[18]引入了随机耦合项研究时滞网络的同步问题。文献[19]研究了具有时变延迟的随机神经网络的同步问题。文献[20]研究了具有耦合时滞以及与噪声扰动相关的网络同步问题, 给出了同步时网络能承受的时滞上界。

然而以上研究成果都是假设网络耦合是连续的。但是在现实世界中, 网络之间的耦合连接关系有时会断开[21-23], 网络间的耦合可能是周期开关[24]。例如, 由于季节性, 一年中生态系统中食物链之间的耦合连接关系可能会被激活或者关闭, 这在一定程度上可以描述为周期开关耦合。因此将研究范围由连续耦合的复杂网络扩展到不连续耦合的复杂网络是必要的。

本研究探讨了具有不连续耦合的时滞复杂动态网络的网络同步问题。利用微分方程稳定性理论, 得到不连续耦合的时滞复杂动态网络实现同步的充分条件。分析结果表明:即使是不连续的耦合, 只要时间延迟满足一定条件仍然可以实现同步。数值模拟验证了理论结果的正确性, 并分析了控制参数对网络同步时间的影响。

1 网络模型和预备知识

文献[21-22]中由N个节点构成的具有不连续耦合的网络模型为

${{\mathit{\boldsymbol{\dot x}}}_i}\left( t \right) = f\left( {{\mathit{\boldsymbol{x}}_i}\left( t \right)} \right) - k\left( t \right)\sum\limits_{j = 1}^N {{g_{ij}}h\left( {{x_j}} \right)} ,i = 1,{\rm{ }}2, \cdots ,{\rm{ }}N,$ (1)

式中: xi(t)=(xi1, …, xin)TRn为第i个节点的状态变量; f: RnRn为连续可微的向量函数; h: RnRn是内部耦合函数; G=(gij)N×N是耦合矩阵, 表示网络节点间的耦合强度和网络的拓扑结构, 若结点i和节点j之间存在连接, 则gij=gji=-1, 否则gij=0(ij), G的对角线元素${g_{ii}} = - \sum\limits_{j = 1,j \ne i}^N {{g_{ij}}} ,i = 1,2, \cdots ,N;k\left( t \right)$为耦合强度, 为周期性函数, 当mTt<(m+θ)T, (m=0, 1, 2, …)时, k(t)=k, 当(m+θ)Tt<(m+1)T时, k(t)=0, 其中T为开关周期, T>0;θ为开关率, 0<θ≤1。易见当θ=1时网络(1) 为通常的连续耦合网络。

由于现实世界中时滞普遍存在, 特别是节点的动力学可能存在时滞。本研究考虑如下时滞复杂动态网络同步问题

${{\mathit{\boldsymbol{\dot x}}}_i}\left( t \right) = f\left( {{\mathit{\boldsymbol{x}}_i}\left( t \right),{\mathit{\boldsymbol{x}}_i}\left( {t - \tau } \right)} \right) - k\left( t \right)\sum\limits_{j = 1}^N {{g_{ij}}h\left( {{x_j}} \right)} ,i = 1,2, \cdots ,N,$ (2)

式中: f: Rn×RnRn是连续可微函数; τ为节点的动力学延迟, τ>0。

下面给出本研究中要用到的引理与假设。

引理1[25]    若矩阵JRn×n为对称阵, 则存在正交阵ΦRn×n使JTΦ=ΦΛ, 其中Λ=diag(λ1, …, λn), λiJ的特征值。

引理2[26]    考虑如下的切换时滞系统:

$\mathit{\boldsymbol{\dot x}}\left( t \right) = {\mathit{\boldsymbol{A}}_{\sigma (x(t))}}\mathit{\boldsymbol{x}}\left( t \right) + {\mathit{\boldsymbol{B}}_{\sigma (x(t))}}\mathit{\boldsymbol{x}}\left( {t - \tau } \right),\mathit{\boldsymbol{x}}({t_0}) = {x_0},$ (3)
$\mathit{\boldsymbol{x}}\left( t \right) = \mathit{\boldsymbol{ \boldsymbol{\varPsi} }}\left( t \right),{\rm{ }}t \in \left[ { - \tau ,0} \right],$ (4)

若存在正常数αi (1≤in), 且满足$\sum\limits_{i = 1}^N {{\alpha _i} = 1} $, 使得如下切换时滞系统的线性组合

$\mathit{\boldsymbol{\dot x}}\left( t \right) = \sum\limits_{i = 1}^n {{\alpha _i}\left[ {{\mathit{\boldsymbol{A}}_i}\left( t \right)\mathit{\boldsymbol{x}}\left( t \right) + {\mathit{\boldsymbol{B}}_i}\mathit{\boldsymbol{x}}\left( {x - \tau } \right)} \right]} ,$

是渐近稳定的, 则切换时滞系统(3) (4) 为渐近稳定的。

引理3[25]    设v(t)>0, ∀tR, τ≥0。若AB>0, 且

$\mathit{\boldsymbol{\dot v}}\left( t \right) \le - \mathit{\boldsymbol{Av}}\left( t \right) + B\left( {\mathop {{\rm{sup}}}\limits_{t - 2\tau \le s \le t} \mathit{\boldsymbol{v}}\left( s \right)} \right),{\rm{ }}\forall t > {t_0},$

则存在两个正常数Cγ, 使得

$\mathit{\boldsymbol{v}}\left( t \right) \le C{e^{ - \gamma (t - {t_0})}},\forall t > {t_0}$

假设1    假设函数f(x(t), x(t-τ))满足如下条件:存在α, β, μl 4个常数, 对任意t有:

$\begin{array}{l} {\lambda _{{\rm{max}}}}\left( {\mathit{\boldsymbol{J}}\left( t \right) + {\mathit{\boldsymbol{J}}^{\rm{T}}}\left( t \right)} \right) \le 2\alpha ,{\rm{ }}{\lambda _{{\rm{max}}}}\left( {{\mathit{\boldsymbol{J}}_\tau }\left( {t - \tau } \right)\mathit{\boldsymbol{J}}_\tau ^{\rm{T}}\left( {t - \tau } \right)} \right) \le 2\beta ,\\ {\lambda _{{\rm{max}}}}\left( {\mathit{\boldsymbol{J}}\left( t \right){\mathit{\boldsymbol{J}}^{\rm{T}}}\left( t \right)} \right) \le 2\mu ,{\rm{ }}{\lambda _{{\rm{max}}}}\left( {{\mathit{\boldsymbol{J}}_\tau }\left( {t - \tau } \right) + \mathit{\boldsymbol{J}}_\tau ^{\rm{T}}\left( {t - \tau } \right)} \right) \le 2l, \end{array}$

式中J(t), J(t-τ)分别为向量函数ft以及t-τ时刻的雅可比矩阵。

定义1    称网络(2) 实现同步, 如果

${\mathit{\boldsymbol{x}}_1}\left( t \right) = {\mathit{\boldsymbol{x}}_2}\left( t \right) = \cdots = {\mathit{\boldsymbol{x}}_N}\left( t \right) = \mathit{\boldsymbol{s}}\left( t \right),t \to \infty ,$

式中s(t)是一个孤立节点的解, 满足$\mathit{\boldsymbol{\dot s}}\left( t \right) = f\left( {\mathit{\boldsymbol{s}}\left( t \right),\mathit{\boldsymbol{s}}\left( {t - \tau } \right)} \right)$

2 主要结论

定理1    若假设1成立, 且不等式

$\begin{array}{l} \quad \quad \quad k\theta > \frac{{\alpha + l}}{{{\lambda _2}}},\\ \tau < {\tau ^*} = \frac{{\theta {\lambda _2}k - \left( {\alpha + l} \right)}}{{\mu + 3\beta }}, \end{array}$

成立, 则网络(2) 可以实现同步。

证明    令ei(t)=xi(t)-s(t), 由G的定义有

$\sum\limits_{j = 1}^N {{g_{ij}} = 0} ,$

注意到:

$\sum\limits_{j = 1}^n {{g_{ij}}{\mathit{\boldsymbol{x}}_j}\left( t \right)} = \sum\limits_{j = 1}^n {{g_{ij}}\left( {{\mathit{\boldsymbol{x}}_j}\left( t \right) - \mathit{\boldsymbol{s}}\left( t \right)} \right)} = \sum\limits_{j = 1}^n {{g_{ij}}{\mathit{\boldsymbol{e}}_j}\left( t \right)} ,$

从而由网络(2), 有如下误差系统

${{\mathit{\boldsymbol{\dot e}}}_i}\left( t \right) = f\left( {{\mathit{\boldsymbol{x}}_i}\left( t \right),{\mathit{\boldsymbol{x}}_i}\left( {t - \tau } \right)} \right) - f\left( {\mathit{\boldsymbol{s}}\left( t \right),{\rm{ }}\mathit{\boldsymbol{s}}\left( {t - \tau } \right)} \right) - k\left( t \right)\sum\limits_{j = 1}^n {{g_{ij}}{\mathit{\boldsymbol{e}}_j}\left( t \right)} 。$

易见, 如果线性时滞系统

${{\mathit{\boldsymbol{\dot e}}}_i}\left( t \right) = \mathit{\boldsymbol{J}}\left( t \right){\mathit{\boldsymbol{e}}_i}\left( t \right) + {\mathit{\boldsymbol{J}}_\tau }\left( {t - \tau } \right){\mathit{\boldsymbol{e}}_i}\left( {t - \tau } \right) - k\left( t \right)\sum\limits_{j = 1}^n {{g_{ij}}{\mathit{\boldsymbol{e}}_j}\left( t \right)} $ (5)

的零解为渐近稳定的, 则误差系统的解局部渐近稳定。

e(t)=(e1(t), e2(t), …, eN(t)), 则式(5) 可写成

$\mathit{\boldsymbol{\dot e}}\left( t \right) = \mathit{\boldsymbol{J}}\left( t \right)\mathit{\boldsymbol{e}}\left( t \right) + {\mathit{\boldsymbol{J}}_\tau }\left( {t - \tau } \right)\mathit{\boldsymbol{e}}\left( {t - \tau } \right) - k\left( t \right)\mathit{\boldsymbol{e}}\left( t \right){\mathit{\boldsymbol{G}}^{\rm{T}}},$ (6)

由引理1, 存在正交矩阵ΦRn×n, 使得GTΦ=ΦΛ, 其中Λ=diag(λ1, …, λN)。令w(t)=e(t)Φ, 其中w(t)=(w1 (t), w2 (t), …, wn (t))。因此由式(6) 得:

$\mathit{\boldsymbol{\dot w}}\left( t \right) = \mathit{\boldsymbol{J}}\left( t \right)\mathit{\boldsymbol{w}}\left( t \right) + {\mathit{\boldsymbol{J}}_\tau }\left( {t - \tau } \right)\mathit{\boldsymbol{w}}\left( {t - \tau } \right) - k\left( t \right)\mathit{\boldsymbol{w}}\left( t \right)\mathit{\boldsymbol{ \boldsymbol{\varLambda} }},$

或者

${{\mathit{\boldsymbol{\dot w}}}_i}\left( t \right) = \left( {\mathit{\boldsymbol{J}}\left( t \right) - {\lambda _i}k\left( t \right)\mathit{\boldsymbol{I}}} \right){\mathit{\boldsymbol{w}}_i}\left( t \right) + {\mathit{\boldsymbol{J}}_\tau }\left( {t - \tau } \right){\mathit{\boldsymbol{w}}_i}\left( {t - \tau } \right),{\rm{ }}i = 1,{\rm{ }}2, \cdots ,{\rm{ }}N。$

注意到λ1=0对应着同步流形, 因此误差系统(5) 的零解为渐近稳定的。如果如下N-1个系统为渐近稳定的:

${{\mathit{\boldsymbol{\dot w}}}_i}\left( t \right) = \left( {\mathit{\boldsymbol{J}}\left( t \right) - {\lambda _i}k\left( t \right)\mathit{\boldsymbol{I}}} \right){\mathit{\boldsymbol{w}}_i}\left( t \right) + {\mathit{\boldsymbol{J}}_\tau }\left( {t - \tau } \right){\mathit{\boldsymbol{w}}_i}\left( {t - \tau } \right),{\rm{ }}i = 2, \cdots ,{\rm{ }}N,$ (7)

由耦合强度k(t)的定义, 系统(7) 可以看作由下列两个独立系统构成的时滞切换系统:

$\begin{array}{l} \quad \quad {{\mathit{\boldsymbol{\dot w}}}_i}\left( t \right) = \mathit{\boldsymbol{J}}\left( t \right){\mathit{\boldsymbol{w}}_i}\left( t \right) + {\mathit{\boldsymbol{J}}_\tau }\left( {t - \tau } \right){\mathit{\boldsymbol{w}}_i}\left( {t - \tau } \right) \buildrel \Delta \over = {S_1},i = 1,2, \cdots ,N,\\ {{\mathit{\boldsymbol{\dot w}}}_i}\left( t \right) = \left( {\mathit{\boldsymbol{J}}\left( t \right) - {\lambda _i}k\mathit{\boldsymbol{I}}} \right){\mathit{\boldsymbol{w}}_i}\left( t \right) + {\mathit{\boldsymbol{J}}_\tau }\left( {t - \tau } \right){\mathit{\boldsymbol{w}}_i}\left( {t - \tau } \right) \buildrel \Delta \over = {S_2},i = 1,2, \cdots ,N。\end{array}$

由引理2, 系统(7) 为渐近稳定。如果下列N-1个n维时滞系统

${{\mathit{\boldsymbol{\dot w}}}_i}\left( t \right) = \theta {S_1} + \left( {1 - \theta } \right){S_2} = \left( {\mathit{\boldsymbol{J}}\left( t \right) - \theta {\lambda _i}k\mathit{\boldsymbol{I}}} \right){\mathit{\boldsymbol{w}}_i}\left( t \right) + {\mathit{\boldsymbol{J}}_\tau }\left( {t - \tau } \right){\mathit{\boldsymbol{w}}_i}\left( {t - \tau } \right),i = 2, \cdots ,N,$

为渐近稳定的, 为此只需要分析如下n维系统

$\mathit{\boldsymbol{\dot \delta }}\left( t \right)\mathit{\boldsymbol{ = }}\left( {\mathit{\boldsymbol{J}}\left( t \right) - \theta {\lambda _i}k\mathit{\boldsymbol{I}}} \right)\mathit{\boldsymbol{\delta }}\left( t \right) + {\mathit{\boldsymbol{J}}_\tau }\left( {t - \tau } \right)\mathit{\boldsymbol{\delta }}\left( {t - \tau } \right),$

的稳定性。易见$\mathit{\boldsymbol{\delta }}\left( t \right) - \mathit{\boldsymbol{\delta }}\left( {t - \tau } \right) = \int_{t - \tau }^t {\mathit{\boldsymbol{\delta }}\left( s \right){\rm{d}}s} $, 因此有

$\mathit{\boldsymbol{\dot \delta }}\left( t \right) = \left( {\mathit{\boldsymbol{J}}\left( t \right) - \theta {\lambda _i}k\mathit{\boldsymbol{I}}} \right)\mathit{\boldsymbol{\delta }}\left( t \right) + {\mathit{\boldsymbol{J}}_\tau }\left( {t - \tau } \right)\mathit{\boldsymbol{\delta }}\left( t \right) - {\mathit{\boldsymbol{J}}_\tau }\left( {t - \tau } \right)\int_{t - \tau }^t {\mathit{\boldsymbol{\dot \delta }}(s){\rm{d}}s} 。$ (8)

下面证明过程中构造如下李雅普诺夫函数

$\mathit{\boldsymbol{V}}\left( t \right) = {\mathit{\boldsymbol{\delta }}^{\rm{T}}}(t)\mathit{\boldsymbol{\delta }}\left( t \right),$

则由式(8), V(t)的导数为

$\begin{array}{l} \frac{{{\rm{d}}\mathit{\boldsymbol{V}}}}{{{\rm{d}}t}} = {{\mathit{\boldsymbol{\dot \delta }}}^{\rm{T}}}(t){\mathit{\boldsymbol{\delta }}^{\rm{T}}}\left( t \right) + \mathit{\boldsymbol{\dot \delta }}\left( t \right) = 2{\mathit{\boldsymbol{\delta }}^{\rm{T}}}\left( t \right)\mathit{\boldsymbol{J}}\left( t \right)\mathit{\boldsymbol{\delta }}\left( t \right) - 2{\mathit{\boldsymbol{\delta }}^{\rm{T}}}(t)\mathit{\boldsymbol{\delta }}\left( t \right)\theta {\lambda _i}k\mathit{\boldsymbol{I}} + 2{\mathit{\boldsymbol{\delta }}^{\rm{T}}}\left( t \right){\mathit{\boldsymbol{J}}_\tau }\left( {t - \tau } \right)\mathit{\boldsymbol{\delta }}\left( t \right) - \\ \quad \quad \quad 2{\mathit{\boldsymbol{\delta }}^{\rm{T}}}\left( t \right){\mathit{\boldsymbol{J}}_\tau }\left( {t - \tau } \right)\int_{t - \tau }^s {\left( {\mathit{\boldsymbol{J}}\left( s \right) - \theta {\lambda _i}k\mathit{\boldsymbol{I}}} \right)\mathit{\boldsymbol{\delta }}\left( s \right)} + {\mathit{\boldsymbol{J}}_\tau }\left( {s - \tau } \right)\mathit{\boldsymbol{\delta }}\left( {s - \tau } \right){\rm{d}}s, \end{array}$

根据假设1, 有:

$\begin{array}{l} \frac{{{\rm{d}}\mathit{\boldsymbol{V}}}}{{{\rm{d}}t}} \le 2{\mathit{\boldsymbol{\delta }}^{\rm{T}}}\left( t \right)\mathit{\boldsymbol{J}}\left( t \right)\mathit{\boldsymbol{\delta }}\left( t \right) - 2{\mathit{\boldsymbol{\delta }}^{\rm{T}}}\left( t \right)\mathit{\boldsymbol{\delta }}\left( t \right)\theta {\lambda _i}k\mathit{\boldsymbol{I}} + 2{\mathit{\boldsymbol{\delta }}^{\rm{T}}}\left( t \right){\mathit{\boldsymbol{J}}_\tau }\left( {t - \tau } \right)\mathit{\boldsymbol{\delta }}\left( t \right) + \\ \quad \quad 2{\mathit{\boldsymbol{\delta }}^{\rm{T}}}\left( t \right){\mathit{\boldsymbol{J}}_\tau }\left( {t - \tau } \right)\int_{t - \tau }^t {\left( {\left( {\mathit{\boldsymbol{J}}\left( s \right) - \theta {\lambda _i}k\mathit{\boldsymbol{I}}} \right)\mathit{\boldsymbol{\delta }}\left( s \right) + {\mathit{\boldsymbol{J}}_\tau }\left( {s - \tau } \right)\mathit{\boldsymbol{\delta }}\left( {s - \tau } \right)} \right){\rm{d}}s} \le \\ \quad \quad 2{\mathit{\boldsymbol{\delta }}^{\rm{T}}}(t)\mathit{\boldsymbol{J}}\left( t \right)\mathit{\boldsymbol{\delta }}\left( t \right) - 2{\mathit{\boldsymbol{\delta }}^{\rm{T}}}\left( t \right)\mathit{\boldsymbol{\delta }}\left( t \right)\theta {\lambda _i}k\mathit{\boldsymbol{I}} + 2{\mathit{\boldsymbol{\delta }}^{\rm{T}}}\left( t \right){\mathit{\boldsymbol{J}}_\tau }\left( {t - \tau } \right)\mathit{\boldsymbol{\delta }}\left( t \right) + \\ \quad \quad 2{\mathit{\boldsymbol{\delta }}^{\rm{T}}}\left( t \right){\mathit{\boldsymbol{J}}_\tau }\left( {t - \tau } \right)\int_{t - \tau }^t {\left( {\mathit{\boldsymbol{J}}\left( s \right)\mathit{\boldsymbol{\delta }}\left( s \right) + {\mathit{\boldsymbol{J}}_\tau }\left( {s - \tau } \right)\mathit{\boldsymbol{\delta }}\left( {s - \tau } \right)} \right){\rm{d}}s} , \end{array}$

式中: 2δT(t)J(t)δ(t)≤2αδT(t)δ(t), 2δT(t)Jτ(t-τ)δ(t)≤2lδT(t)δ(t)。

对式(9) 最后一项, 有:

$\begin{array}{l} \quad \quad 2{\mathit{\boldsymbol{\delta }}^{\rm{T}}}\left( t \right){\mathit{\boldsymbol{J}}_\tau }\left( {t - \tau } \right)\int_{t - \tau }^t {\left( {\mathit{\boldsymbol{J}}\left( s \right)\mathit{\boldsymbol{\delta }}\left( s \right) + {\mathit{\boldsymbol{J}}_\tau }\left( {s - \tau } \right)\mathit{\boldsymbol{\delta }}\left( {s - \tau } \right)} \right){\rm{d}}s} = \\ \int_{t - \tau }^t {\left( {2{\mathit{\boldsymbol{\delta }}^{\rm{T}}}\left( t \right){\mathit{\boldsymbol{J}}_\tau }\left( {t - \tau } \right)\mathit{\boldsymbol{J}}\left( s \right)\mathit{\boldsymbol{\delta }}\left( s \right) + 2{\mathit{\boldsymbol{\delta }}^{\rm{T}}}\left( t \right){\mathit{\boldsymbol{J}}_\tau }\left( {t - \tau } \right){\mathit{\boldsymbol{J}}_\tau }\left( {s - \tau } \right)\mathit{\boldsymbol{\delta }}\left( {s - \tau } \right)} \right)} {\rm{d}}s, \end{array}$

式中:

$\begin{array}{l} 2{\mathit{\boldsymbol{\delta }}^{\rm{T}}}\left( t \right){\mathit{\boldsymbol{J}}_\tau }\left( {t - \tau } \right)\mathit{\boldsymbol{J}}\left( s \right)\mathit{\boldsymbol{\delta }}\left( s \right) \le {\mathit{\boldsymbol{\delta }}^{\rm{T}}}\left( t \right){\mathit{\boldsymbol{J}}_\tau }\left( {t - \tau } \right)\mathit{\boldsymbol{J}}_\tau ^{\rm{T}}\left( {t - \tau } \right)\mathit{\boldsymbol{\delta }}\left( t \right) + {\mathit{\boldsymbol{\delta }}^{\rm{T}}}\left( s \right){\mathit{\boldsymbol{J}}^{\rm{T}}}\left( s \right)\mathit{\boldsymbol{J}}\left( s \right)\mathit{\boldsymbol{\delta }}\left( s \right) \le \\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad 2\beta {\mathit{\boldsymbol{\delta }}^{\rm{T}}}\left( t \right)\mathit{\boldsymbol{\delta }}\left( t \right) + 2\mu {\mathit{\boldsymbol{\delta }}^{\rm{T}}}\left( s \right)\mathit{\boldsymbol{\delta }}\left( s \right),\\ 2{\mathit{\boldsymbol{\delta }}^{\rm{T}}}\left( t \right){\mathit{\boldsymbol{J}}_\tau }\left( {t - \tau } \right){J_\tau }\left( {s - \tau } \right)\mathit{\boldsymbol{\delta }}\left( {s - \tau } \right) \le \\ \quad \quad {\mathit{\boldsymbol{\delta }}^{\rm{T}}}\left( t \right){\mathit{\boldsymbol{J}}_\tau }\left( {t - \tau } \right)\mathit{\boldsymbol{J}}_\tau ^{\rm{T}}\left( {t - \tau } \right)\mathit{\boldsymbol{\delta }}\left( t \right) + {\mathit{\boldsymbol{\delta }}^{\rm{T}}}\left( {s - \tau } \right)\mathit{\boldsymbol{J}}_\tau ^{\rm{T}}\left( {s - \tau } \right){\mathit{\boldsymbol{J}}_\tau }\left( {s - \tau } \right)\mathit{\boldsymbol{\delta }}\left( {s - \tau } \right) \le \\ \quad \quad 2\beta {\mathit{\boldsymbol{\delta }}^{\rm{T}}}\left( t \right)\mathit{\boldsymbol{\delta }}\left( t \right) + 2\beta {\mathit{\boldsymbol{\delta }}^{\rm{T}}}\left( {s - \tau } \right)\mathit{\boldsymbol{\delta }}\left( {s - \tau } \right), \end{array}$

因此, 有:

$\begin{array}{l} \frac{{{\rm{d}}\mathit{\boldsymbol{V}}}}{{{\rm{d}}t}} \le (2\alpha - 2\theta {\lambda _i}k + 2l){\mathit{\boldsymbol{\delta }}^{\rm{T}}}\left( t \right)\mathit{\boldsymbol{\delta }}\left( t \right) + \int_{t - \tau }^t {(4\beta {\mathit{\boldsymbol{\delta }}^{\rm{T}}}\left( t \right)\mathit{\boldsymbol{\delta }}\left( t \right) + 2\mu {\mathit{\boldsymbol{\delta }}^{\rm{T}}}\left( s \right)\mathit{\boldsymbol{\delta }}\left( s \right) + 2\beta {\mathit{\boldsymbol{\delta }}^{\rm{T}}}\left( {s - \tau } \right)} \\ \quad \quad \mathit{\boldsymbol{\delta }}(s - \tau )){\rm{d}}s \le 2(\alpha - \theta {\lambda _i}k + l){\mathit{\boldsymbol{\delta }}^{\rm{T}}}\left( t \right)\mathit{\boldsymbol{\delta }}\left( t \right) + 4\beta \tau {\mathit{\boldsymbol{\delta }}^{\rm{T}}}\left( t \right)\mathit{\boldsymbol{\delta }}\left( t \right) + \left( {2\mu + 2\beta } \right)\tau \mathop {{\rm{sup}}}\limits_{t - 2\tau \le s \le t} {\mathit{\boldsymbol{\delta }}^{\rm{T}}}\left( s \right)\\ \quad \quad \mathit{\boldsymbol{\delta }}\left( s \right) = - 2\left( {\theta {\lambda _i}k - \left( {\alpha + l} \right)} \right){\mathit{\boldsymbol{\delta }}^{\rm{T}}}\left( t \right)\mathit{\boldsymbol{\delta }}\left( t \right) + 4\beta \tau {\mathit{\boldsymbol{\delta }}^{\rm{T}}}\left( t \right)\mathit{\boldsymbol{\delta }}\left( t \right) + \left( {2\mu + 2\beta } \right)\tau \mathop {sup}\limits_{t - 2\tau \le s \le t} \mathit{\boldsymbol{V}}\left( s \right), \end{array}$

此外, 如果θλi kα+l, 且$\tau < {\tau ^*} = \frac{{\theta {\lambda _i}k - \left( {\alpha + l} \right)}}{{\mu + 3\beta }}$, 由引理3知存在2个正常数Cγ使得

$V\left( t \right) \le C{e^{ - \gamma t}},\forall t > 0,$

等价的有, $\left\| {\mathit{\boldsymbol{\delta }}\left( t \right)} \right\| \le \sqrt C {e^{ - \frac{\gamma }{2}t}},\forall t > 0$

由于0=λ1λ2λ3≤…≤λN, 且λ1=0对应着同步流形。所以当$k\theta > \frac{{\alpha + l}}{{{\lambda _2}}},\tau < {\tau ^*} = \frac{{\theta {\lambda _2}k - \left( {\alpha + l} \right)}}{{\mu + 3\beta }}$时, 网络(2) 可以实现同步。

注1    特征值λ2通常称为网络图的代数连通性, 文献[10-11]关于非时滞网络的研究中表明网络的同步能力依赖于网络的代数连通性。具体而言, 网络的同步能力与λ2正相关。本研究关于具有不连续耦合的时滞网络的研究表明:网络同步能力以及网络所能容忍的时滞上界均与λ2正相关。另外, 定理1条件表明:对于给定的网络, 可以增加耦合强度以及开关率θ来提高网络的同步能力。

3 数值模拟

下面通过数值模拟来验证上述理论结果的正确性。考虑图 1中由8个节点构成的网络。在数值模拟中, 网络的初始条件在区间[-1, 1]内随机选取, 用误差函数${\Delta _i}\left( t \right) \buildrel \Delta \over = {\mathit{\boldsymbol{x}}_i}\left( t \right) - \frac{1}{N}\sum\limits_{i = 1}^N {{\mathit{\boldsymbol{x}}_i}\left( t \right)} $作为同步指标, 易见Δi (t)→0时网络实现同步。

图 1 具有8个节点的网络拓扑结构图 Figure 1 Network topology with eight nodes

数值模拟时选取Ikeda系统作为网络的节点动力学。Ikeda系统为典型的时滞混沌系统, 可用如下微分方程描述:

$\mathit{\boldsymbol{\dot x}} = f(x,{\rm{ }}{x_\tau }) = - p\mathit{\boldsymbol{x}} + q\;{\rm{sin}}\;{\mathit{\boldsymbol{x}}_\tau },$

经简单计算知:Ikeda系统满足假设1且α=-pβ=q2/2、μ=p2/2、l=q。数值模拟时参数取值如下:p=15、q=10、τ=0.1、k=60、θ=0.7、T=0.01。图 1所示网络耦合矩阵的代数连通数λ2 (G)=0.5858, 经简单计算得τ*=0.12, 从而定理1中条件成立。图 2给出了网络同步误差函数Δi (t)的演化轨迹和网络各状态xi (t)的演化轨迹, 可见网络实现了同步。

图 2 网络同步误差及网络状态的演化轨迹 Figure 2 Trajactories of network synchronization error and network states

取时滞τ分别为0.1、0.2、0.3、0.5, 进行数值模拟得到对网络同步的影响。结果发现, 只有满足定理1的时滞τ=0.1时能达到同步, 其他均不能。数值模拟验证了结论的正确性。图 3给出了不同时滞对应的网络同步误差Δ(t)的演化轨迹, 其中p=15、q=10、k=60、θ=0.7、T=0.01, 分别取τ=0.1、0.2、0.3、0.5。另外由定理1可知:时滞的上界与开关率、耦合强度等系数有关, 不同的开关率和耦合强度对网络同步的影响进行仿真对比。图 4为不同开关率下对应的网络同步误差的演化轨迹, 参数取值为p=15、q=10、τ=0.1、k=60、T=0.01, 分别取θ=0.1、0.3、0.5、0.7。图 5为不同耦合强度下对应的网络同步误差的演化轨迹, 参数取值为p=15、q=10、τ=0.1、θ=0.7、T=0.01, 分别取k=30、40、70、90。

图 3 不同时滞对应的网络同步误差Δ(t)的演化轨迹 Figure 3 Evolution of the network synchronization error Δ(t) with different values of time delays
图 4 不同开关率的网络误差Δ(t)的演化轨迹 Figure 4 Evolution of the network synchronization error Δ(t) with different values of on-off rate
图 5 不同耦合强度的网络误差Δ(t)的演化轨迹 Figure 5 Evolution of the network synchronization error Δ(t) with different values of coupling strength
4 结语

本研究研究探讨了具有不连续耦合的时滞复杂网络的同步问题。根据微分方程的稳定性定理得到了网络同步的充分条件。与以往文献不同, 本研究的同步条件依赖于时滞。研究发现:网络同步时所能容忍的时滞的上界依赖于网络的耦合强度、网络拓扑的代数连通性以及周期耦合的开关率, 在满足定理1的条件下网络即可达到同步。另外研究结果表明:对于给定的网络, 可以增加耦合强度以及开关率来提高网络的同步能力。数值模拟验证了理论结果的正确性。

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