复杂网络协同，同步，中性稳定性和 Pinning 控制

陈天平

复旦大学

ConsensusSynchronizationNeutral stability of manifoldStabilityPinning control

定义同步 , 协同 , 正确定义被接受

错误的定义被淘汰

接受

接受

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非线性系统的线性耦合

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线性系统的线性耦合

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协同

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三个模型

模型 1 。非线性系统的同步

模型 2 。线性系统的同步

模型 3 。协同

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显然 , 模型 2 , 3 是模型 1 的特例非线性同步 :

全局同步 : Lyapunov 函数 局部同步 : Jordan 分解

线性同步 : 全局同步等价于局部同步

有关文章W.L. Lu and T.P. Chen, Synchronization of Coupled Connected Neural Networks With Delays, IEEE Trans. Circ. Sys. I , vol. 51, 2004, pp.2491-2503 被引频次 : 86W. Lu and T. Chen, ``Synchronization Analysis of Linearly Coupled Networks of Discrete Time Systems'', Physica D 198(2004) 148-168 被引频次 : 61R. Olfati-Saber and R.M. Murray, Consensus problems in networks of agents with switching topology and time delays, IEEE Trans. Automat. Control, vol. 49, 2004, pp. 1520-1533 被引频次 : 超过800 次

在文章 W.L. Lu and T.P. Chen, Synchronization

of Coupled Connected Neural Networks With Delays, IEEE Trans. Circ. Sys. I , vol. 51, 2004, pp.2491-2503

中 , 讨论了耦合非线性系统的全局和局部同步。后面的的讨论证完全来自上述文章关于局部同步的讨论。

因此，文章R. Olfati-Saber and R.M. Murray, Consensus problems in networks of agents with switching topology and time delays, IEEE Trans. Automat. Control, vol. 49, 2004, pp. 1520-1533

的结果可以从我们的工作中得到。

耦合矩阵的 Jordan 分解

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线性系统同步的简单推导记

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非线性耦合协同和同步模型下列文章给出非线性耦合协同的很好结果Xiwei, Liu, Tianping Chen, Wenlian Lu, Consensus problem in directed networks of multi-agents via nonlinear protocols, Physics Letters A 373 (2009) 3122–3127

非线性耦合协同Theorem 1. Assume the nonlinear functions satisfy the conditions

1. are continuous mappings and are local Lipschitzian

2. ⇔x = y;3. The consensus model

can realize consensus, if and only if the directed graph has a spanning tree.

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非线性耦合协同 ( 续 ) Theorem 2. Assume the function h(·) is continuous and strictly increasing, the digraph G has its graph Laplacian L. If there exists a positive constant l > 0, which may be related to the initial states, such that (h(w)−h(v))/(w− v) , for any

, then consensus

can be realized exponentially by the above

consensus model Furthermore, the final consensus state is a linear combination

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非线性耦合同步Theorem . Assume the function h(·) is continuous and strictly increasing, the digraph G has its graph Laplacian L. If there exists a positive constant l > 0, which may be related to the initial states, such that (h(w)−h(v))/(w− v) >0 , then synchronization through the model

can be realized exponentially . Moreover,

If f(.) satisfies “QUAD” condition and the coupling is symmetric

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非线性耦合同步问题

Tianping Chen, Zhimiao Zhu, “Exponential Synchronization of Nonlinear Coupled Dynamical Networks”, International Journal of Bifurcation and Chaos, 17(3), (2007), 999-1005

对于同步问题 , 联结矩阵必须为对称的 . 而对于协同问题 , 连接矩阵可以为非对称的 .

问题

有限时间收敛不同的内联矩阵（控制论中最关心）不同的非线性耦合各种时变系统切换系统和随机切换系统时滞系统协同系统中有噪声分群同步和协同不连续系统

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中性稳定性 (Neutral stability)

协同流形 上的每个状态是系统

的一个平衡点沿流形 S 切方向的导数为 0. 每个状态都是不稳定的 流形 S 是吸引的 . ( 横切方向吸引 )最简单系统

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主成份分析 (PCA)

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有关文章 (10 年前 )

T. Chen Y. Hua and W. Yan, "Global Convergence of Oja's Subspace Algorithm for Principal Component Extraction“ IEEE Transactions on Neural Networks, Vol.8, (1998) pp.57-68Tianping Chen Shun-Ichi Amari and Qing Lin, "A Unified Algorithm For Principal and Minor Component Extraction", Neural Networks 11(3), (1998), pp.365-369Tianping Chen and Shun-ichi Amari, "New Theorems on Global Convergence of Some Dynamical Systems," Neural Networks, V. 14:(3), (2001), p.p. 251-255.

PCA 算法的中性稳定性由个最大的” p” 个特征值对应的特征向量构成的 Stiefel 流形 S(n,p) 上的每个点都是系统

的一个平衡点 . 沿 Stiefel 流形 S(n,p) 切空间方向导数为 0 。亦即 S 上每个点都不稳定 流形 S 是吸引的。即 X(t) 收敛到 S

)()()()()( tCXtXtXtCXtX T

各种主（微）成份分析算法主成份分析算法 （流形稳定，平衡点中性稳定） :

微成份分析算法 （不稳定）：

主（微）成份分析算法 （流形中性稳定） :

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Pinning 控制

控制耦合系统

的轨迹收敛到确定的轨道 Tianping Chen, Xiwei Liu, and Wenlian Lu, "Pinning Complex Networks by a Single Controller", IEEE Transactions on Circuits and Systems—I: Regular Papers, 54(6), 2007, 1317-1326, 被引频次 : 52

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Pinning 控制模型

Pinning 控制模型

用自适应算法，耦合强度并不需要很大。2000 节点，自适应算法

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随机耦合 2000 Lorenz 系统一个 Pinning 控制自适应算法

同步转化为稳定性

记 则，

同步问题转化为稳定性问题耦合项 加了

n(N-1) 约束，保证同步 . 再加上负反馈 ，保证收敛

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特征值分析数学上 , 矩阵 是一个奇异 M- 矩 阵。 的特征值

加了负反馈后， 变成非奇异 M- 矩阵。 的所有特征值

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时变和切换耦合网络时变耦合系统

Wei Wu and Tianping Chen, Global Synchronization Criteria of Linearly Coupled Neural Network Systems With Time-Varying Coupling, IEEE TRANSACTIONS ON NEURAL NETWORKS, 19(2), 2008, 319-332

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时变和切换耦合网络 ( 有关文章 )Wenlian Lu ，“ Comment on ‘Adaptive-feedback control algorithm”’ Physical Review E, 75, 018201, 2007.Wenlian Lu, Fatihcan M. Atay, Jurgen Jost ， “ Synchronization of discrete-time dynamical networks with time-varying couplings” SIAM Journal on Mathematical Analysis, 39:4, 1231-1259, 2007.Wenlian Lu, Fatihcan M. Atay, Jurgen Jost “Chaos synchronization in networks of coupled maps with time-varying topologies” European Physical Journal B, 63, 399-406, 2008.Bo Liu and Tianping Chen ， Consensus in Networks of Multiagents With Cooperation and Competition Via Stochastically Switching Topologies ， IEEE Transactions on Neural Networks, 19(11), 1967-1973

.

相同振子耦合网络的分群同步Wei Wu, Wenjuan Zhou, and Tianping Chen, Cluster Synchronization of Linearly Coupled Complex Networks Under Pinning Control, IEEE Transactions on Circuits and Systems—I: Regular Papers, 56(4), 2009 829-839Wei Wu, Tianping Chen, Partial synchronization in linearly and symmetrically coupled ordinary differential systems, Physica D 238 (2009) 355-364

不同振子耦合网络的分群同步

Wenlian Lu, Bo Liu and Tianping Chen, Cluster synchronization in networks of coupled non-identical dynamical systems , Chaos (to appear)Wenlian Lu, Bo Liu and Tianping Chen, Cluster synchronization in networks of coupled nonidentical maps, EUJB (to appear)

无界时滞系统及同步 Tianping Chen and Lili Wang, Global $\mu$- Stability of Delayed Neural Networks With Unbounded Time-Varying Delays, IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 18, NO. 6, 2007, 1836-1840Tianping Chen, Wei Wu, and Wenjuan Zhou Global -Synchronization of Linearly Coupled Unbounded Time-Varying Delayed Neural Networks With Unbounded Delayed Coupling, IEEE TRANSACTIONS ON NEURAL NETWORKS, 2008, Nov. 1809-1915

有关文章Xiwei Liu, Tianping Chen, Synchronization analysis for nonlinearly coupled complex networks with an asymmetric coupling matrix, Physica A 387, (2008), 4429-4439Xiwei Liu, Tianping Chen, Robust $\mu$-stability for uncertain stochastic neural networks with unbounded time-varying delays, Physica A 387, (2008) 2952-2962Xiwei Liu, Tianping Chen, Boundedness and synchronization of y-coupled Lorenz systems with or without controllers, Physica D 237, (2008) 630-639

Model

Where , f and g satisy Lipschiz condition.This model is inappropriate. Because the coupling term g does not provide any help for synchronization.

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New Approach to Synchronization Analysis of Linearly Coupled Map Lattices 摘要： In this paper, we present a new approach to analyze synchronization of Linearly Coupled Map Lattices (LCMLs). A reference vector is introduced as projection of the trajectory of the coupled system on the synchronization manifold. The stability analysis of the synchronization manifold can be regarded as investigating difference between the trajectory and the projection. By this method, criteria are given for both local and global synchronization. These criteria indicate that the left and right eigenvectors corresponding to eigenvalue "0" of the coupling matrix play key roles in the stability of synchronization manifold for coupled system. Moreover, we reveal that stability of synchronization manifold for the coupled system is different from the stability for dynamical system in usual sense.

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作者声称提出了一种新的研究耦合振子同步的方法 . 但遗憾的是 , 审稿人并不认为这是一种全新的方法 , 只不过是以前扰动分析分析方法的一种改进工作 . 不学无术但又霸道。特别的 , 审稿人有下面一些评论 : 1) 题目显然需要修改 , 不能说是一种新方法 ; 首次引入2) 第 2 页第一段第 8 行 是错误的 , 的极限是什么 , 函数还是具体的一个数值 ? 不懂就好好学，不要胡说八道。3) 文中的参考向量 从本质上说就是系统扰动后的解 , 扰动解可以表示为系统的解加上误差 . 所以自然的有系统的解属于稳定流形 . 作者只不过限定这种扰动解为系统各个节点解的线性组合 , 当然在这种条件下误差向量也比较特殊 . 又在胡说八道。真是以其（己）昏昏，使人昭昭4) 第 4 页的定义 1-2 只不过是一个记号 , 没有必要用定义表示 ;定义 6-8 可以合并成一个简单的定义 ; 荒唐5) 第 10-12 的例子和讨论并不能说明任何问题 . 实际上 , 前几年就有人证明了误差系统的横截 Lyapunov 指数都小于零并不能保证耦合系统同步 . 而且 , 扰动分析的结论大多是充分条件 , 不是必要条件 , 所以也没有什么可以比较的 . 连文章内容都没有看懂。

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学术水平低下。不懂装懂，还要教训人。真所谓：以其（己）昏昏，使人昭昭；学术道德卑劣；玩弄肮脏的学术手段（ academic politics) ；教训人也要有资本；

Blind Source Separation