摘要
复杂网络是研究复杂系统的一门新兴学科,近年来受到国内外研究者的广泛关注。从人体细胞内分子运动到世界各国的相互联系,我们都是网络的一部分。这种特殊的现象使得科学家们正在将网络分析推广到各个学科领域。今天,对网络系统的认识已经成为一系列传统学科的共同目标：细胞生物学家使用网络来研究信号传导和代谢过程；计算机科学家正在绘制互联网和万维网的结构；流行病学家追踪病毒传播的传输网络；脑研究者正在致力于研究连接体,一种神经水平的大脑连接图。
噪声通常被视为随机的和持续的干扰,会遮蔽或减弱信号的清晰度。在网络系统中噪声的作用是两方面的。一方面,噪声的存在往往容易导致网络的不稳定性。另一方面,噪声的存在也会产生好的作用,人们可以通过它来改善系统的动力学特性,将不稳定的状态通过加入随机噪声而变成稳定的状态,或者使本来稳定的状态变得更加稳定。由于噪声普遍存在于自然系统与人造系统中,因此考虑噪声对系统动力学性质的影响就显得非常必要。
由于多智能体网络的协调一致性问题的研究在无人飞行器控制,队列控制,卫星群姿态的一致性控制以及通信网络的拥塞控制等众多领域有着广泛的应用,近年来已经成为复杂网络研究中的热点,受到国内外研究者的广泛关注。本论文在前人工作基础上对多智能体网络的一致性问题做了一些深入的探索和研究,主要研究了随机噪声、通信延迟及网络拓扑结构的切换等因素对智能体网络一致性行为的影响。
随着混沌同步研究的不断深入,人们发现它在保密通信、纳米振荡器等领域有着重要的应用。由于噪声普遍存在于现实系统与人造系统中,混沌同步不可避免会受到噪声的影响。近年来,带噪声的混沌系统的同步问题受到了众多研究者的关注。本论文主要研究了具有噪声干扰的延迟系统以及两个不同混沌系统的同步问题。
另外,本文还研究了掌声同步问题,以及小世界网络上传染病模型的稳定性、分岔与混沌。这些研究无论在理论上还是在实际应用中都具有重要意义。本文的主要内容可概述如下：
第一章介绍复杂网络与混沌同步的研究背景与进展,给出本文所要用到的预备知识,并给出全文的结构。
第二章研究了噪声环境中多智能体网络的一致性问题。研究了噪声干扰的平均一致性问题、具有通信延迟的平均一致性问题、具有虚拟领导者以及通信延迟的一致性问题,以及噪声耦合的多智能体网络的平均一致性问题。利用随机微分方程的稳定性理论,给出了多智能体系统达到一致性的充分条件。理论与数值模拟结果表明：噪声的干扰可以减慢多智能体系统趋于一致性的速度,甚至可以破坏一致性的出现。
第三章研究了具有带头鼓掌者的掌声同步问题,在掌声传播的时滞对观众有或无明显影响的两种情形下给出了掌声同步的充分条件。理论结果不仅说明了人们在日常生活中观察到的现象：在上座率高的剧场内更容易听到有节奏的掌声,而且也得到了一个有趣的结论：即使在很大的剧场内一位带头鼓掌者就足以带领所有观众以相同的节奏鼓掌。
第四章建立了小世界网络上具有延迟的非线性传染病模型,研究了模型平衡点的局部稳定性与hopf分岔的存在性,并研究了相应离散模型的动力学性质。研究表明：防治措施的强度不仅可以决定平衡点的稳定性,而且可以用来将周期震荡甚至混沌的传播行为控制到稳定状态。
第五章研究了噪声干扰下的混沌系统的同步问题。首先,研究了含参数误差的延迟系统的完全同步与反同步,给出了具有常时滞与变时滞的混沌系统完全同步与反同步的充分条件。然后,研究了Lorenz系统与噪声干扰的陈系统与吕系统的同步,利用随机微分方程的稳定性理论给出了耦合系统同步的充分条件。
第六章对论文工作进行了总结,并对今后的研究工作进行了展望。
Complex networks have been considered as an important approach for describing and understanding complex systems by many researchers recently. From the movement of molecules within our cells to communication across an entire planet, we are part of networks. This special section shows how scientists are pushing network analysis to its limits across disciplinary fields. Today the understanding of networks is a common goal of an unprecedented array of traditional disciplines:Cell biologists use networks to make sense of signal transduction cascades and metabolism; computer scientists are mapping the Internet and the WWW; epidemiologists follow transmission networks through which viruses spread; and brain researchers are after the connectome, a neural-level connectivity map of the brain.
Noise is commonly regarded as a random and persistent disturbance obscuring or reducing the clarity of a signal. The role of noise in network systems is quite versatile. In certain cases noise is detrimental to the stability of the networks. In certain cases it's presences is necessary, noise can be used to stabilize a given unstable system or to make a system even more stable when it is already stable. Since noise is ubiquitous in both nature and manmade systems, it is necessary to consider the effect of noise on the dynamics of network systems.
Recently, consensus problems of multi-agent networks have attracted a great deal of attention in many fields. This is partly due to broad applications of multi-agent sys-tems in many areas including cooperative control of unmanned air vehicles, formation control, attitude alignment of clusters of satellites, and congestion control in communi-cation networks. Based on the work of many researchers, this dissertation extend the previous results of consensus in networks of multi-agents. The effect of noise, commu-nication time-delay and the switching topology are considered.
Chaos synchronization is of fundamental importance in secure communication, and nano-oscillators, etc. Since noise is ubiquitous in both nature and manmade sys-tems, synchronization of concrete models is unavoidably subject to internal and exter-nal noise. Recently, synchronization in noise-perturbed chaotic systems was studied by many researchers. In this dissertation, the synchronization of the noise-perturbed time-delayed systems and two different systems are investigated.
In addition, the synchronization of hand clapping, and the stability, bifurcation and chaos of the epidemic model on small-world networks are also considered. These results are significant not only in mathematical theory but also in many applied fields. The main work in this dissertation is listed as follows:
In chapter 1, the research background and progress on complex networks and chaos synchronization are introduced. Moreover, some preliminaries and the structure of this dissertation are given.
In chapter 2, the consensus problems on networks of multi-agent systems in noisy environment are presented. Firstly the average consensus problem is considered. Then the average consensus problem with time delay and the leader-following consensus problem with time delay are considered. Finally the average consensus problem with noise coupling is investigated. Based on the stability theory of stochastic differential equations, the sufficient conditions ensuring the successful consensus are given. Both the theoretical and numerical results show that the speed of convergence in the environ-ments with large noise intensity is lower than that in the environments with relatively small noise intensity, the strong noise even can prevent the consensus from occurring.
In chapter 3, synchronization of the applause with leaders is investigated. Two cases of the effect of the applause transmission time delays on the spectators are con-sidered. Sufficient conditions for the applause synchronization are presented for both cases. The established theoretical results not only support the observation that it is easy to hear the rhythmic applause in a theater with high attendance rate, but also show a sur-prising conclusion that one leader is enough to make all spectators clap with the same frequency in a very large theater.
In chapter 4, a nonlinear epidemic model in small-world networks with time-delay is presented to investigate the impact of the treatment measures on the epidemic dynam-ics. Local stability and Hopf bifurcation are considered. It is shown that the strength of treatment measures not only determine the stability of the local equilibrium, but also can be applied to stabilize a periodic or chaotic spreading behavior onto a stable equi-librium.
In chapter 5, the synchronization of noise-perturbed chaos systems is investigated. Firstly complete synchronization and anti-synchronization of a class of coupled time-delayed systems with parameter mismatch and noise perturbation are investigated. Suf-ficient conditions guaranteeing complete synchronization and anti-synchronization with constant and varying time delay are developed. Then chaos synchronization between Lorenz system and one of the noise-perturbed Chen and Lu systems is investigated. Based on the Lyapunov theory in stochastic differential equations, sufficient conditions for the stability of the error dynamics are derived.
At the end of this dissertation, the conclusions and some topics for future work are given.
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