复杂网络建模及其动力学性质的若干研究
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摘要
复杂网络近年来受到来自科学与工程各个领域研究者越来越多的关注,成为近年来研究的一个热点。本论文将统计方法、非线性系统理论、控制理论以及矩阵理论等理论和方法应用到复杂网络的研究中,对复杂网络的动力学性质和加权复杂网络的建模两个方面进行了研究。这两方面的研究无论在理论上还是在实际应用中都具有重要意义。通过对复杂网络动力学性质的研究,一方面可以使我们更好地了解和解释现实世界中复杂网络所呈现出来的各种动力学现象,如稳定、同步、振荡等;另一方面我们可以将对复杂网络动力学性质研究的理论成果应用到具体问题当中去,如可以设计出具有更好特性的实际网络或使网络处于对我们有利的状态,使得网络理论可以为我们所用。另外,现实世界中很多网络都是各个连接间具有不同权值的加权网络,过去比较多研究的无权网络模型只是对复杂网络的一种近似简化描述,加权网络模型则能够对实际复杂网络的动力学演化特性提供更加真实的细致和全面的描述。因此,对加权网络建模研究的重要意义是显而易见的。
本文的主要主要内容和创新之处可概述如下:
1.关于时延小世界网络的局部稳定性及分岔的研究。
由于信号传输速度有限,以及节点间竞争和通道拥塞等因素,在复杂网络中通常存在时延。我们研究了一个带有时延的小世界网络模型的局部稳定性和Hopf分岔。并用中心流形定理和正规型理论确定了分岔周期解的稳定性和分岔方向。对小世界网络的分岔现象研究的意义在于:一方面,由于Hopf分岔和振荡现象密切相关,对小世界网络Hopf分岔的研究可以使我们更好地解释现实世界中的许多小世界网络,如Internet、电网、生物神经网络中发生的对参数敏感的现象;另一方面,若能深入地了解小世界网络中的分岔现象及其规律,则通过利用比较成熟的分岔控制理论和方法,我们就可将现实世界中的小世界网络控制到所期望的有利的状态中去。
Recently, complex networks attract more and more attentions from various fields of science and engineering. In this dissertation, we apply statistical method, nonlinear system theory, control theory and matrix theory to the research of complex networks, and we study the dynamic properties of complex networks, as well as the modeling of weighted complex networks. These studies are very important both in theory and in practical applications. By studying the dynamic properties of complex networks, on the one hand, we can understand and explain the dynamic properties presented in real-world networks, such as stability, synchronization and oscillation phenimena; and on the other hand, we can apply these theoretical results to some practical applications, for example, we can apply these results to the design of real networks to achieve good performance or to the control of real networks to achieve some desirable network behaviors that benefit the networks. In addition, many real-world networks are weighted networks with different weights in different connections. The unweighted network models studied in many existing literatures are simplified modeling of real networks, while weighted network models can provide more realistic and comprehensive descriptions of real networks. So, the importance of the modelling of weighted networks is clearly self-evident.The main contents and originalities in this paper can be summarized as follows: 1. Local stability and Hopf bifurcation in a delayed small-world network modelDue to limited transmitting speed, competition and congestions, usually there are time delays in complex networks. We study the local stability and Hopf bifurcation of a delayed small-world network model. We determine the stability of the bifurcating periodic solutions and directions of Hopf bifurcation by applying the center manifold theorem and the normal form theory. The studies of Hopf bifurcation in small-world networks are quite important. On the one hand, the bifurcations, which involve emergence of oscillatory behaviors, may provide an explanation for the parameter sensitivity observed in practice in many realistic small-world networks such as the
Internet, the electrical power grids, and the biological neural networks; and on the other hand, if we understand more about the bifurcation behaviors of small-world networks, we can apply the existing effective bifurcation control method to achieve some desirable system behaviors that benefit the networks.2. Synchronization of complex networksThe synchronization has attracted increasing attentions due to its importance both in theory and in practical applications. In this dissertation we study the synchronization problems of delayed small-world networks of phase oscillators and coupled maps. And we study the synchronization of a general delayed complex network model by using Lyapunov-krasovskii functions and linear matrix inequalities (LMIs). We derive some easy-verified synchronization criteria. Further, we study the chaotic phase synchronization in small-world networks.3. On-off intermittency in small-world network of chaotic mapsWe study how the small-world topology would affect on-off intermittency of small-world networks of chaotic maps. When the globally coupled chaotic maps are synchronous, we fix the coupling coefficient. We find that by decreasing the connection probability gradually, when the probability slightly less than a critical value, the synchronous chaos is no longer stable and on-off intermittency appears. By further decreasing the probability, the intermittent dynamics will eventually be replaced by fully developed asynchronous chaos.4. Study on a neural network model with weighted small-world connections There are many biological neural networks that present small-world connections.But in most existing literatures, the authors studied neural network models with regular topologies. Note also that in most of the existing small-world network models no special weights in the connections were taken into account. However, this kind of simplified network models cannot well characterize biological neural networks, in which there are different weights in different synaptic connections. Here, we present a neural network model with small-world topology and with random weight values in different connections, and further investigate the stability of this model by using the Lyapunov function method and statistical method. We derive explicit relationship between the stability of small-world neural networks and the values of the network connection probability p and the number of neurons N.
本文的主要主要内容和创新之处可概述如下:
1.关于时延小世界网络的局部稳定性及分岔的研究。
由于信号传输速度有限,以及节点间竞争和通道拥塞等因素,在复杂网络中通常存在时延。我们研究了一个带有时延的小世界网络模型的局部稳定性和Hopf分岔。并用中心流形定理和正规型理论确定了分岔周期解的稳定性和分岔方向。对小世界网络的分岔现象研究的意义在于:一方面,由于Hopf分岔和振荡现象密切相关,对小世界网络Hopf分岔的研究可以使我们更好地解释现实世界中的许多小世界网络,如Internet、电网、生物神经网络中发生的对参数敏感的现象;另一方面,若能深入地了解小世界网络中的分岔现象及其规律,则通过利用比较成熟的分岔控制理论和方法,我们就可将现实世界中的小世界网络控制到所期望的有利的状态中去。
Recently, complex networks attract more and more attentions from various fields of science and engineering. In this dissertation, we apply statistical method, nonlinear system theory, control theory and matrix theory to the research of complex networks, and we study the dynamic properties of complex networks, as well as the modeling of weighted complex networks. These studies are very important both in theory and in practical applications. By studying the dynamic properties of complex networks, on the one hand, we can understand and explain the dynamic properties presented in real-world networks, such as stability, synchronization and oscillation phenimena; and on the other hand, we can apply these theoretical results to some practical applications, for example, we can apply these results to the design of real networks to achieve good performance or to the control of real networks to achieve some desirable network behaviors that benefit the networks. In addition, many real-world networks are weighted networks with different weights in different connections. The unweighted network models studied in many existing literatures are simplified modeling of real networks, while weighted network models can provide more realistic and comprehensive descriptions of real networks. So, the importance of the modelling of weighted networks is clearly self-evident.The main contents and originalities in this paper can be summarized as follows: 1. Local stability and Hopf bifurcation in a delayed small-world network modelDue to limited transmitting speed, competition and congestions, usually there are time delays in complex networks. We study the local stability and Hopf bifurcation of a delayed small-world network model. We determine the stability of the bifurcating periodic solutions and directions of Hopf bifurcation by applying the center manifold theorem and the normal form theory. The studies of Hopf bifurcation in small-world networks are quite important. On the one hand, the bifurcations, which involve emergence of oscillatory behaviors, may provide an explanation for the parameter sensitivity observed in practice in many realistic small-world networks such as the
Internet, the electrical power grids, and the biological neural networks; and on the other hand, if we understand more about the bifurcation behaviors of small-world networks, we can apply the existing effective bifurcation control method to achieve some desirable system behaviors that benefit the networks.2. Synchronization of complex networksThe synchronization has attracted increasing attentions due to its importance both in theory and in practical applications. In this dissertation we study the synchronization problems of delayed small-world networks of phase oscillators and coupled maps. And we study the synchronization of a general delayed complex network model by using Lyapunov-krasovskii functions and linear matrix inequalities (LMIs). We derive some easy-verified synchronization criteria. Further, we study the chaotic phase synchronization in small-world networks.3. On-off intermittency in small-world network of chaotic mapsWe study how the small-world topology would affect on-off intermittency of small-world networks of chaotic maps. When the globally coupled chaotic maps are synchronous, we fix the coupling coefficient. We find that by decreasing the connection probability gradually, when the probability slightly less than a critical value, the synchronous chaos is no longer stable and on-off intermittency appears. By further decreasing the probability, the intermittent dynamics will eventually be replaced by fully developed asynchronous chaos.4. Study on a neural network model with weighted small-world connections There are many biological neural networks that present small-world connections.But in most existing literatures, the authors studied neural network models with regular topologies. Note also that in most of the existing small-world network models no special weights in the connections were taken into account. However, this kind of simplified network models cannot well characterize biological neural networks, in which there are different weights in different synaptic connections. Here, we present a neural network model with small-world topology and with random weight values in different connections, and further investigate the stability of this model by using the Lyapunov function method and statistical method. We derive explicit relationship between the stability of small-world neural networks and the values of the network connection probability p and the number of neurons N.
引文
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