复杂网络上的动力学模型分析及随机影响
|
摘要
复杂网络是研究复杂系统的一门新兴学科,近几年受到国内外研究学者的广泛关注。任何复杂系统都可以抽象成为由相互作用的个体组成的网络,因而网络无处不在,遍及自然界、生物系统和人类社会。这些网络中比较为人所熟知的如神经网络、互联网、疾病传播网、蛋白质网、合作网等。为了理解这些复杂网络的结构和性质,人们首先就需要研究这些网络的动力学行为。通过对复杂网络动力学行为的研究,不仅可以更好地理解现实世界中的网络所呈现出来的各种动力学现象,而且还可以根据不同的网络特性采取不同的方法为我们的工作和生活服务,更重要的是在某些方面我们还可以设计出具有更好特性的网络为人类造福。
传染病在人群中的流行、病毒在计算机网络上的蔓延、谣言在社会中的扩散等,都可以看成服从某种规律的传播行为。传统的传播动力学模型假定所研究范围的个体是充分混合的,即每个个体接触其他个体的数量基本相等,但近年来关于复杂网络的研究显示绝大部分的个体不是充分混合的,而是具有某些特殊结构。这些特殊结构导致了复杂网络上的传播行为与传统动力学模型有较大差异,这说明在考察传播行为时,必须结合特定的网络结构才能建立更加符合实际的动力学模型。本论文在前人工作基础上对复杂网络上的传播行为做了一些深入的探索和研究,主要考虑了非线性传播率的传播动力学,两种群的传播动力学。另外我们知道,随机因素会对复杂网络的动力学行为产生一定的影响,因此我们还研究了随机噪音对网络传播的影响,随机离散的神经网络模型,以及共同噪音引起的离散系统的同步问题。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(包括神经网络)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。
第二章研究复杂网络上带有非线性传播率的模型。研究了带有非线性传播率的均匀网络模型,给出了模型的平衡点的存在条件、稳定性以及Hopf分岔分析等,得出了在网络传播中出现周期性现象的原因;研究了带有非线性传播率的非均匀网络模型,给出了无病平衡点的稳定性分析。
第三章研究复杂网络上两种群的传播动力学模型。主要介绍了均匀网络上的两种群的传播模型,研究了非均匀网络上两种群的传播模型,给出了疾病传播的阈值和平衡点的稳定性分析。另外还给出了两种群传播模型三种免疫策略,得出了均匀网络和非均匀网络上的有效免疫策略。
第四章研究复杂网络上的随机传播模型。研究了均匀网络上的随机传播模型和非均匀网络上的随机传播模型,得出了随机稳定性和随机分岔的条件。研究表明随机噪音对不同拓扑结构的网络传播模型会起到不同的作用。
第五章研究随机噪音对离散神经网络的影响。考虑了两类随机噪音影响的离散神经网络模型,利用随机动力系统中的鞅理论,得出了随机离散神经网络的平凡解是几乎必然渐近稳定的条件。
第六章研究离散状态下共同噪音引起的同步问题。先讨论了一般的离散系统发生同步的条件,然后讨论了带有时滞的离散系统发生同步的条件。
第七章对论文工作进行了总结,并对今后要研究的工作提出了展望。
Complex networks have been considered as an important approach for describing and understanding complex systems by many researchers recently. Any complex system is composed of interacted individuals, which can be naturally represented by graphs with individuals denoted by nodes and interactions by links. From this point of view, complex networks are ubiquitous, ranging from nature and biological system to society. The well-known networks include neural networks, Internet, epidemic spreading networks, protein-protein interaction networks, collaboration networks, etc. To understand the structures and properties of networks, the study of dynamical behaviors on these complex networks are needed firstly. By studying the dynamic properties of complex networks, one can not only understand the dynamic properties presented in real-world networks but also take different measures for networks with different structures. Furthermore, these results can been applied to design real networks to achieve some desirable that benefit all over the world.
The spread of epidemics, computer virus and rumors can be seen as same behaviors with some laws. In the traditional spreading models, the individuals are considered to be mixed adequately, that means each individual contacts other individuals with the same rate. However the recent study of complex networks shows that the distribution of individuals is not homogeneous, but has some special structures. These special structures make the spreading behavior on complex networks different from the traditional models. In order to make the model more practical, one must consider the special structures in studying the spreading behavior. Based on the work of many researchers this dissertation extend the previous results on complex networks, which are the spreading model with non-linear incidence rate and with two interacting species. Furthermore, stochastic factor will affect the dynamic behaviors of complex networks, so the effect of the stochastic noise on complex network, stochastic discrete-time neural networks, and the synchronization of discrete-time systems with common noise are also investigated. 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, including neural networks, are introduced. Moreover, some preliminaries and the structure of dissertation are given.
In chapter 2, the spreading models on complex networks with a generalized nonlinear incidence rate are presented. Firstly the model on homogeneous networks with nonlinear incidence rate is considered, and the existence, the stability of equilibria and the Hopf bifurcation of the model are given. Then the model on a heterogenous scale-free network are considered, and the stability of the disease-free equilibrium is obtained. It is shown that the basic reproductive number is independent of the functional form of the nonlinear incidence rate, while the number of the equilibria and the behaviors are indeed different from the corresponding model with linear incidence rate.
In chapter 3, models for the spread of two interacting species on complex networks are presented. The dynamic behaviors of the models on the homogeneous network and heterogenous scale-free network are considered, and the stability of the disease-free equilibrium is obtained. Three immunization strategies are applied to models. The analytical and simulated results are given to show that the proportional immunization strategy is effective on heterogenous scale-free networks.
In chapter 4, the general stochastic nonlinear models of spreading, describing the effect of random fluctuations on complex networks are proposed. It has been found that fluctuation noise would trigger a state of networks from instability to stability. The probability density function for the proportion of infected individuals are found explicitly, and the stochastic bifurcation is analyzed by probability density function. It is a better explanation of occurring the different phenomena with sensitive parameters in many real-world complex networks.
In chapter 5, the effect of stochastic noise about the discrete-time stochastic neural networks are investigated. By using the martingale convergence theorem, the almost surely stability condition of two sub-classed of stochastic neural networks are analyzed. Furthermore, numerical examples are provided to illustrate some possible applications of the theoretical results.
In chapter 6, the synchronization of discrete-time system and the discrete-time system with delay are discussed. These systems are all induced by common external noise. A set of sufficient conditions for the synchronization is found. These conditions show that the synchronization occurs in a wide class of discrete-time system and the discrete-time system with delay.
At the end of this dissertation, the conclusions and some topics for future work which include complex networks and stochastic dynamical systems are given.
传染病在人群中的流行、病毒在计算机网络上的蔓延、谣言在社会中的扩散等,都可以看成服从某种规律的传播行为。传统的传播动力学模型假定所研究范围的个体是充分混合的,即每个个体接触其他个体的数量基本相等,但近年来关于复杂网络的研究显示绝大部分的个体不是充分混合的,而是具有某些特殊结构。这些特殊结构导致了复杂网络上的传播行为与传统动力学模型有较大差异,这说明在考察传播行为时,必须结合特定的网络结构才能建立更加符合实际的动力学模型。本论文在前人工作基础上对复杂网络上的传播行为做了一些深入的探索和研究,主要考虑了非线性传播率的传播动力学,两种群的传播动力学。另外我们知道,随机因素会对复杂网络的动力学行为产生一定的影响,因此我们还研究了随机噪音对网络传播的影响,随机离散的神经网络模型,以及共同噪音引起的离散系统的同步问题。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(包括神经网络)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。
第二章研究复杂网络上带有非线性传播率的模型。研究了带有非线性传播率的均匀网络模型,给出了模型的平衡点的存在条件、稳定性以及Hopf分岔分析等,得出了在网络传播中出现周期性现象的原因;研究了带有非线性传播率的非均匀网络模型,给出了无病平衡点的稳定性分析。
第三章研究复杂网络上两种群的传播动力学模型。主要介绍了均匀网络上的两种群的传播模型,研究了非均匀网络上两种群的传播模型,给出了疾病传播的阈值和平衡点的稳定性分析。另外还给出了两种群传播模型三种免疫策略,得出了均匀网络和非均匀网络上的有效免疫策略。
第四章研究复杂网络上的随机传播模型。研究了均匀网络上的随机传播模型和非均匀网络上的随机传播模型,得出了随机稳定性和随机分岔的条件。研究表明随机噪音对不同拓扑结构的网络传播模型会起到不同的作用。
第五章研究随机噪音对离散神经网络的影响。考虑了两类随机噪音影响的离散神经网络模型,利用随机动力系统中的鞅理论,得出了随机离散神经网络的平凡解是几乎必然渐近稳定的条件。
第六章研究离散状态下共同噪音引起的同步问题。先讨论了一般的离散系统发生同步的条件,然后讨论了带有时滞的离散系统发生同步的条件。
第七章对论文工作进行了总结,并对今后要研究的工作提出了展望。
Complex networks have been considered as an important approach for describing and understanding complex systems by many researchers recently. Any complex system is composed of interacted individuals, which can be naturally represented by graphs with individuals denoted by nodes and interactions by links. From this point of view, complex networks are ubiquitous, ranging from nature and biological system to society. The well-known networks include neural networks, Internet, epidemic spreading networks, protein-protein interaction networks, collaboration networks, etc. To understand the structures and properties of networks, the study of dynamical behaviors on these complex networks are needed firstly. By studying the dynamic properties of complex networks, one can not only understand the dynamic properties presented in real-world networks but also take different measures for networks with different structures. Furthermore, these results can been applied to design real networks to achieve some desirable that benefit all over the world.
The spread of epidemics, computer virus and rumors can be seen as same behaviors with some laws. In the traditional spreading models, the individuals are considered to be mixed adequately, that means each individual contacts other individuals with the same rate. However the recent study of complex networks shows that the distribution of individuals is not homogeneous, but has some special structures. These special structures make the spreading behavior on complex networks different from the traditional models. In order to make the model more practical, one must consider the special structures in studying the spreading behavior. Based on the work of many researchers this dissertation extend the previous results on complex networks, which are the spreading model with non-linear incidence rate and with two interacting species. Furthermore, stochastic factor will affect the dynamic behaviors of complex networks, so the effect of the stochastic noise on complex network, stochastic discrete-time neural networks, and the synchronization of discrete-time systems with common noise are also investigated. 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, including neural networks, are introduced. Moreover, some preliminaries and the structure of dissertation are given.
In chapter 2, the spreading models on complex networks with a generalized nonlinear incidence rate are presented. Firstly the model on homogeneous networks with nonlinear incidence rate is considered, and the existence, the stability of equilibria and the Hopf bifurcation of the model are given. Then the model on a heterogenous scale-free network are considered, and the stability of the disease-free equilibrium is obtained. It is shown that the basic reproductive number is independent of the functional form of the nonlinear incidence rate, while the number of the equilibria and the behaviors are indeed different from the corresponding model with linear incidence rate.
In chapter 3, models for the spread of two interacting species on complex networks are presented. The dynamic behaviors of the models on the homogeneous network and heterogenous scale-free network are considered, and the stability of the disease-free equilibrium is obtained. Three immunization strategies are applied to models. The analytical and simulated results are given to show that the proportional immunization strategy is effective on heterogenous scale-free networks.
In chapter 4, the general stochastic nonlinear models of spreading, describing the effect of random fluctuations on complex networks are proposed. It has been found that fluctuation noise would trigger a state of networks from instability to stability. The probability density function for the proportion of infected individuals are found explicitly, and the stochastic bifurcation is analyzed by probability density function. It is a better explanation of occurring the different phenomena with sensitive parameters in many real-world complex networks.
In chapter 5, the effect of stochastic noise about the discrete-time stochastic neural networks are investigated. By using the martingale convergence theorem, the almost surely stability condition of two sub-classed of stochastic neural networks are analyzed. Furthermore, numerical examples are provided to illustrate some possible applications of the theoretical results.
In chapter 6, the synchronization of discrete-time system and the discrete-time system with delay are discussed. These systems are all induced by common external noise. A set of sufficient conditions for the synchronization is found. These conditions show that the synchronization occurs in a wide class of discrete-time system and the discrete-time system with delay.
At the end of this dissertation, the conclusions and some topics for future work which include complex networks and stochastic dynamical systems are given.
引文
[1]汪小帆,李翔,陈关荣,复杂网络理论及其应用,北京:清华大学出版社,2006.
[2]郭雷,许晓鸣,复杂网络,上海:上海科技教育出版社,2006.
[3]M.E.J.Newman,The structure and function of complex networks,SIAM Rev.,2003,45:167-256.
[4]阮炯,顾凡及,蔡志杰,神经网络动力学模型方法和应用,北京:科学出版社,2002.
[5]焦李成,神经网络系统理论,西安:西安电子科技大学出版社,1990.
[6]张立明,人工神经网络的模型及应用,上海:复旦大学出版社,1993.
[7]L.O.Chua,CNN:A paradigm for complexity,Singapore:World Scientific,1998.
[8]A.-L.Barabasi,Linked:The new science of networks,Massachusetts:Persus Publishing,2002.
[9]D.J.Watts,The 'new' science of networks,Annual Review of Sociology,2004,30:243-270.
[10]S.H.Strogatz,Exploring complex networks,Nature,2001,410:268-276..
[11]S.Wiggins,Introduction to applied nonlinear dynamical systems and chaos,New York:Springer,1990.
[12]P.Glendinning,Stability,instability and chaos,New York:Cambridge University,1994.
[13]廖晓昕,动力系统的稳定性理论和应用,北京:国防工业出版社,2000.
[14]马知恩,周义仓,常微分方程定性与稳定性方法,北京:科学出版社,2001.
[15]钟守铭,刘碧森,王晓梅,范小明,神经网络稳定性理论,北京:科学出版社,2008.
[16]张芷芬,李承治,郑志明,李伟固,向量场的分岔理论基础,北京:高等教育出版社,1993.
[17]韩茂安,朱德明,微分方程分支理论,北京:煤炭工业出版社,1994.
[18]张琪昌,王洪礼,褚致文,沈菲,任爱娣,刘海英,分岔与混沌理论及应用,天津:天津大学出版社,2005.
[19]Y.A.Kuznetsov,Elements of applied bifurcation theory,New York:Springer,2004.
[20]X.S.Yang,Chaos in small-world networks,Phys.Rev.E,2001,63:046206.
[21]X.S.Yang,Fractals in small-world networks with time-delay,Chaos,Solitons and Fractals,2002,13:215-219.
[22]X.Li,G.Chen,Local stability and Hopf bifurcation m small-world networks,Chaos,Solitons and Fractals,2004,20:353-361.
[23]X.Li,G.Chen,C.Li,Stability and bifurcation of disease spreading in cimplex networks,International Journal of Systems Science,2004,35:527-536.
[24]H.C.Tuckwell,L.Tot.biana,J.E Vibert,Enhancement of epidemic spread by noise and stochastic resonance in spatial network models with viral dynamics,Phys.Rev.E,2000,6 i:5611-5619.
[25]X.Mao,G.Marion,E.Renshaw,Environmental Brownian noise suppresses explosions in population dynamics,Stochastic Processes and their Applications,2002,97:95-110.
[26]T.Y.Li,J.A.Yorke,Period three implies chaos,Amer Math Monthly,1975,82:985-992.
[27]R.M.May,Simple mathematical models with very complicated dynamics,Nature,1976,261:456-467.
[28]M.J.Feigenbaum,Quantitative universality for a class of nonlinear transformations,J.State Phys.,1978,19:25-52.
[29]R Erd(o|¨)s,A.Renyi,On changes of signs in infinite series.Pub.of the Math.Ins.of the Hungarian Acad.of Sci.,1960,5:17-60.
[30]S.Milgram,The small-world problem,Psychology Today,1967,2:60-67.
[31]D.J.Watts,S.H.Strogatz,Collective dynamics of 'small-world' networks,Nature,1998,393(6684):440-442.
[32]A.-L.Barabasi,R.Albert,Emergence of scaling in random networks,Science,1999,286(5439):509-512.
[33]M.E.J.Newman,D.J.Watts,Scaling and percolation in the small-world network model,Phys.Rev.E,1999,60:7332-7342.
[34]彭文伟,传染病学.北京:人民卫生出版社,2001.
[35]W.O.Kermazk,A.G.McKendrick,A contribution to the mathematical theory of epidemic,Proc.R.Soc.London,1927,All5:700-721.
[36]H.W.Hethcote,The mathematics of infectious diseases,SIAM Rev.,2000,42:599-653.
[37]马知恩,王稳地,周义仓,靳祯,传染病动力学的数学建模与研究,北京:科学出版社,2004.
[38]R.M.Anderson and R.M.May,Infectious Diseases of Humans,Oxford:Oxford University Press,1992.
[39]F.Brauer,P.van den Driessche,J.Wu,Mathematical Epidemiology,New York:Springer,2008.
[40]R.M.May,A.L.Lloyd,Infection dynamics on scale-free networks,Phys.Rev.E,2001,64:066112.
[41]R.Pastor-Satorras,A.Vespignani,Epidemic dynamics and endemic states in complex networks,Phys.Rev.E,2001,63:066117.
[42]R.Pastor-Satorras,A.Vespignani,Epidemic Spreading in Scale-Free Networks,Phys.Rev.Lett.,2001,86:3200.
[43]R.Pastor-Satorras,A.Vespignani,Imunization of complex networks,Phys.Rev.E,2002,65:036104.
[44]M.E.J.Newman,Spread of epidemic disease on networks,Phys,Rev.E,2002,66:016128.
[45]M.Boguna,R.Pastor-Satorras,Epidemic spreading in correlated complex networks.Phys.Rev.E,2002,66:047104.
[46]R.Cohen,S.Havlin,D.Ben-Avraham,Efficient immunization strategies for computer networks and populations,Phys.Rev.Lett.,2003,91:247901.
[47]J.Z.Liu,J.S.Wu,Z.R.Yang,The spread of infectious disease on complex networks with household-structure,Physica A,2004,341:273-280.
[48]J.Z.Liu,Y.Tang,Z.R.Yang,The spread of disease with birth and death networks.J.Star.Mech.,2004,P08008.
[49]Y.Hayashi,M.Minoura,J.Matsukubo,Oscillatory epidemic prevalence in growing scale-free networks,Phys.Rev.E,2004,69:016112.
[50]T.Zhou,J.G.Liu,W.J.Bai,G.R.Chen,B.H.Wang,Behaviors of susceptible-infected epidemics on scale-free networks with identical infectivity,Phys.Rev.E,2006,74:056109.
[51]R.Yang,B.H.Wang,J.Ren,W.J.Bai,Z.W.Shi,W.Xu,T.Zhou,Epidemic spreading on heterogeneous networks with identical infectivity,Phys.Lett.A,2007,364:189-193.
[52]H.Shi,Z.Duan,G.Chen,An SIS model with infective medium on complex networks Physica A,2008,387:2133-2144.
[53]唐三,肖燕妮,单种群生物动力系统,北京:科学出版社,2008.
[54]E.Allen,Modeling with Ito Stochastic Differential Equations,New York:Springer,2007.
[55]L.J.S.Allen,A.M.Burgin,Comparison of deterministic and stochastic SIS and SIR models in discrete time,Math.Biosci.,2000,163(1):1-33.
[56]H.C.Tuckwell,R.J.Williams,Some properties of a simple stochastic epidemic model of SIR type,Math.Biosci.,2007,208(1):76-97.
[57]W.S.McCulloch,W.Pitts,A logical calculus of the ideas immanent in nervous activity,Bulletin of Mathematical Biophysics,1943,5:115-133.
[58]D.O.Hebb,The Organization of Behaviour,New York:Wiley,1949.
[59]F.Rosenblatt,The perceptron:a probabilistic model for information storage and organization in the brain,Rsychological Review,1958,65:386-408.
[60]J.J.Hopfield,Neural networks as physical systems with emergent collective conputational abilities,Proc.National Academy of Science of USA,1982,72:2554-2558.
[61]J.J.Hopfield,Neurons with graded response have collective computational properties like those of two-state neurons,Proc.National Academy of Science of USA,1984,81:3088-3092.
[62]T.Wang,Z.H.Sheng,Asymptotic stationarity of discrete-time stochastic neural networks,Neural Network,1996,9(6):957-963.
[63]E.Kaslik,St.Balint,Bifurcation analysis for a two dimensinal delayed discrete-time Hopfield neural network,Chaos,Solitons & Fractals,34(2007)1245-1253.
[64]Z.Yuan,D.Hu,L.Huang,Stability and bifurcation analysis on a discrete-time neural network,J.Comput.Appl.Math.,2005,177:89-100.
[65]胡适耕,黄乘明,吴付科,随机微分方程,北京:科学出版社,2008.
[66]L.Arnold,Random Dynamical Systems,Heidelberg:Springer,1998.
[67]G.Chen,X.Dong,From Chaos to Order:Methodologies;Perspectives and Applications,Singapore:World Scientific,1998.
[68]X.Wang,G.Chen,Chaotification via arbitrarily small feedback controls:Thory,method,and applications,Int.J.Bifur.Chaos,2000,10:549-570.
[69]Z.Chen,P.Yu,Controlling and anti-controlling Hopf bifurcations in discrete maps using polynomial functions,Chaos,Solitons and Fractals,2005,26:1231-1248.
[70]E.Ott,C.Grebogi,J.A.Yorke,Controlling chao,Phys.Rev.Lett.,1990,64:1196-1199.
[71]A.Maritan,J.R.Banavar,Chaos,noise,and synchronization,Phys.Rev.Lett.,1994,72:1451-1454.
[72]A.S.Pikovsky.Comment on 'Chaos,Noise,and Synchronization',Phys.Rev.Lett.,1994,73:2931.
[73]G.Malescio.Noise and synchronization in chaotic systems.Phys.Rev.E,1996,53:6551-6553.
[74]L.Longa,E.M.K Curado,and F.A.Oliveira.Roundoff-induced coalescence of chaotic trajectories.Phys.Rev.E,1996,54:2201-2204.
[75]L.Gammaitoni,P.Hanggi,P.Jung,and F.Marchesoni.Stochastic resonance.Reviews of Modern Physics,1998,70:223-287.
[76]M.S.Wang,Z.H.Hou,and H.W.Xin.Internal noise-enhanced phase synchronization of coupled chemical chaotic oscillators.J.Phys.A:Math.Gen.,2005,38:145-152.
[77]W.Lin,Y.B.He.Complete synchronization of the noise-perturbed Chua's circuits,Chaos,2005,15:023705.
[78]W.Lin,G.Chen,Using white noise to enhance synchronization of coupled chaotic systems,Chaos,2006,16:013134.
[79]Z.F.Mainen,T.J.Sejnowski,Reliability of Spike Timing in Neocortical Neurons,Science,1995,268:1503-1506.
[80]T.Royama,Analytical Population Dynamics,Chapman and Hall:London,1992.
[81]丁玖,周爱辉,确定性系统的统计性质,北京:清华大学出版社,2006.
[82]J.A.Yorke,W.R London,Recurrent outbreaks of measles,chickenpox and mumps Ⅱ,Am.J.Epidemiol.,1973,98:469-482.
[83]V.Capasso,G.Serio,A generalization of the Kermack-Mckendrick deterministic epidemic model,Math.Biosci.,1978,42:43-61.
[84]W.M.Liu,H.W.Hethcote,S.A.Levin,Dynamical behavior of epidemiological models with nonlinear incidence rates,J.Math.Biol.,1987,25:359-380.
[85]W.M.Liu,S.A.Levin,Y.Iwasa,Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models,J.Math.Biol.,1986,23:187-204.
[86]M.E.Alexander,S.M.Moghadas,Periodicity in an epidemic model with a generalized non-linear incidence,Math.Biosci.,2004,189:75-96.
[87]S.M.Moghadas,M.E.Alexander,Bifurcation of an epidemic model with non-linear incidence and infection-dependent removal rate,Math.Med.and Bio.,2006,23:231-254.
[88]W.R.Derrick,P.van den Driessche,A disease transmission model in a nonconstant population,J.Math.Biol.,1993,31:495-512.
[89]S.Ruan,W.Wang,Dynamical behavior of an epidemic model with a nonlinear incidence rate,J.Differ.Equations,2003,188:135-163
[90]P.van den Driessche,J.Watmough,A simple SIS epidemic model with a backward bifurcation,J.Math.Biol.,2000,40:525-540.
[91]D.Xiao,S.Ruan,Global analysis of an epidemic model with nonmonotone incidence rate,Math.Biosic.,2007,208:419-429.
[92]D.Xiao,Y.Zhou,Qualitative analysis of an epidemic mosel,Canadian Applied Mathmatics Quarterly,2006,14:469-492.
[93]X.C.Fu,M.Small,D.M.Walker,H.F.Zhang,Epidemic dynamics on scale-free networks with piecewise linear infectivity and immunization,Phys.Rev.E,2008,77:036113.
[94]R.M.Anderson,R.M.May,The invasion,persistence and spread of infectious diseases within animal and plant communitcs,Phil.Trans.R.Soc.London,1986,(B314):533-570.
[95]Y.N.Xiao,L.S.Chen,Modeling and analysis of a predator-prey model with disease in the prey,Math.Biosci.,2001,171:58-82.
[96]L.T.Han,Z.E.Ma,S.Tan,An SIRS epidemic model of two competitive species,Mathl.Comput.Modelling,2003,37:87-108.
[97]M.E.J.Newman,Threshold effects for two pathogens spreading on a network,Phys.Rev.Lett.,2005,95:108701.
[98]Y.Y.Ahn,H.Jeong,N.Masuda,J.D.Nob,Epidemic dynamics of two species of interacting particles on scale-free networks,Phys.Rev.E,2006,74:066113.
[99]N.Masuda,N.Konno,Multi-state epidemic processes on complex networks,Journal of Theoretical Biology,2006,243:64-75.
[100]C.Castillo-Chavez,W.Huang,J.Li,Competitive exclusion and coexistence of multiple strains in an SIS STD model,SIAM J.Appl.Math.,1999,59:1790-1811.
[101]F.Liljeros,C.R.Edling,L.A.N.Amaral,H.E.Stanley,Y.Aberg,The web of human sexual contaxts,Nature,2001,411:907-908.
[102]F.Liljeros,C.R.Edling,L.A.N.Amaral,Sexual networks:implications for the transmission of sexually transmitted infections,Microbes and Infection,2003,5:189-196.
[103]H.J.Jones,M.S.Handcock,Sexual contacts and epidemic thresholds,Nature 2003,423:605-606.
[104]W.P.Guo,X.Li,X.F.Wang,Epidemics and immunization on Euclidean distance preferred smallworld networks,2007,Physica A,380:684-690.
[105]X.Li,X.F.Wang,Controlling the spreading in small-world evolving networks:Stability,oscilla tion,and topology,IEEE Trans.Automat.Control,2006,5(3):534-540.
[106]X.Li,X.F.Wang,On the stability of epidemic spreading in small-world networks:how prompt the recovery should be? Int.J.Syst.Sci.,2007,38:401-411.
[107]H.Risken,T.Frank,The Fokker-Planck Equation:Methods of Solutions and Applications,New York:Springer,1996.
[108]V.L.Oseledec,A multiplicativee rgodict heorem Lyapunovc haracteristic numbers for dynamical systems,Transaction of the Moscow Mathematical Society,1968,19:197-231.
[109]L.Arnold,G.Papanicolsou,V.Wihstutz,Asymptotic analysis of the Lyapunov exponent and rotation number of the random oscillator and applications,SIAM Journal of Applied Mechanics,1986,46(3):427-450.
[110]N.S.Namachiwaya,V.Roessel,S.Talwar,Maximal Lyapunov exponent and almost-sure stability for coupled two degree of fieedom stochastic systems,ASME Journal of Applied Mechanics,1994,61:446-452.
[111]S.T.Ariaratnam,W.C.Xie,Lyapunov exponents and stochastic stability of coupled linear systems under real noise excitation,ASME Journal of Applied Mechanics,1992,59:664-673.
[112]刘先斌,陈虬,陈大鹏,非线性随机动力系统的稳定性和分岔研究,力学进展,1996,26(4):437-452.
[113]刘先斌,一类随机分叉系统概率1分叉研究,固体力学学报,2001,22(3):297-302.
[114]戌海武,徐伟,方同,二自由度棍合非线性随机系统的最大Lyapunov指数和稳定性,应用力学学报,1998,15(1):22-29.
[115]戎海武,孟光,徐伟,方同,二自由度祸合线性随机系统的最大Lyapunov指数和稳定性,应用力学学报,2000,17(3):46-53.
[116]C.M.Marcus,R.M.Westervelt,Dynamics of iterated-map neural networks,Physical Review A,1989,40:501-504.
[117]林元烈,应用随机过程,北京:清华大学出版社,2002.
[118]A.Shiryaev,Probability,Berlin:Springer,1996.
[119]J.A.D.Appleby,X.Mao,A.Rodkina,On stochastic stabilization of difference equations,Discrete and continuous dynamical systems,2006,15(3):843-857.
[120]S.Haykin,Neural Networks,New Jersey:Prentice-Hall,1994.
[121]A.Uchida,R.McAllister,R.Roy,Consistency of Nonlinear System Response to Complex Drive Signals,Phys.Rev.Lett.,2004,93:244102.
[122]R.Toral,C.R.Mirasso,E.Hemandez-Garcia,O.Piro,Analytical and numerical studies of noiseinduced synchronization of chaotic systems,Chaos,2001,11:665-673.
[123]C.Zhou,J.Kurths,Noise-Induced Phase Synchronization and Synchronization Transitions in Chaotic Oscillators,Phys.Rev.Lett.,2002,88:230602.
[124]K.Yoshimura,I.Valiusaityte,P.Davis,Synchronization induced by common colored noise in limit cycle and chaotic systems,Phys.Rev.E,2007,75:026208.
[125]K.Yoshimura,J.Muramatsu,P.Davis,Conditions for common-noise-induced synchronization in time-delay systems,Physica D,2008,237:3146-3152.
[2]郭雷,许晓鸣,复杂网络,上海:上海科技教育出版社,2006.
[3]M.E.J.Newman,The structure and function of complex networks,SIAM Rev.,2003,45:167-256.
[4]阮炯,顾凡及,蔡志杰,神经网络动力学模型方法和应用,北京:科学出版社,2002.
[5]焦李成,神经网络系统理论,西安:西安电子科技大学出版社,1990.
[6]张立明,人工神经网络的模型及应用,上海:复旦大学出版社,1993.
[7]L.O.Chua,CNN:A paradigm for complexity,Singapore:World Scientific,1998.
[8]A.-L.Barabasi,Linked:The new science of networks,Massachusetts:Persus Publishing,2002.
[9]D.J.Watts,The 'new' science of networks,Annual Review of Sociology,2004,30:243-270.
[10]S.H.Strogatz,Exploring complex networks,Nature,2001,410:268-276..
[11]S.Wiggins,Introduction to applied nonlinear dynamical systems and chaos,New York:Springer,1990.
[12]P.Glendinning,Stability,instability and chaos,New York:Cambridge University,1994.
[13]廖晓昕,动力系统的稳定性理论和应用,北京:国防工业出版社,2000.
[14]马知恩,周义仓,常微分方程定性与稳定性方法,北京:科学出版社,2001.
[15]钟守铭,刘碧森,王晓梅,范小明,神经网络稳定性理论,北京:科学出版社,2008.
[16]张芷芬,李承治,郑志明,李伟固,向量场的分岔理论基础,北京:高等教育出版社,1993.
[17]韩茂安,朱德明,微分方程分支理论,北京:煤炭工业出版社,1994.
[18]张琪昌,王洪礼,褚致文,沈菲,任爱娣,刘海英,分岔与混沌理论及应用,天津:天津大学出版社,2005.
[19]Y.A.Kuznetsov,Elements of applied bifurcation theory,New York:Springer,2004.
[20]X.S.Yang,Chaos in small-world networks,Phys.Rev.E,2001,63:046206.
[21]X.S.Yang,Fractals in small-world networks with time-delay,Chaos,Solitons and Fractals,2002,13:215-219.
[22]X.Li,G.Chen,Local stability and Hopf bifurcation m small-world networks,Chaos,Solitons and Fractals,2004,20:353-361.
[23]X.Li,G.Chen,C.Li,Stability and bifurcation of disease spreading in cimplex networks,International Journal of Systems Science,2004,35:527-536.
[24]H.C.Tuckwell,L.Tot.biana,J.E Vibert,Enhancement of epidemic spread by noise and stochastic resonance in spatial network models with viral dynamics,Phys.Rev.E,2000,6 i:5611-5619.
[25]X.Mao,G.Marion,E.Renshaw,Environmental Brownian noise suppresses explosions in population dynamics,Stochastic Processes and their Applications,2002,97:95-110.
[26]T.Y.Li,J.A.Yorke,Period three implies chaos,Amer Math Monthly,1975,82:985-992.
[27]R.M.May,Simple mathematical models with very complicated dynamics,Nature,1976,261:456-467.
[28]M.J.Feigenbaum,Quantitative universality for a class of nonlinear transformations,J.State Phys.,1978,19:25-52.
[29]R Erd(o|¨)s,A.Renyi,On changes of signs in infinite series.Pub.of the Math.Ins.of the Hungarian Acad.of Sci.,1960,5:17-60.
[30]S.Milgram,The small-world problem,Psychology Today,1967,2:60-67.
[31]D.J.Watts,S.H.Strogatz,Collective dynamics of 'small-world' networks,Nature,1998,393(6684):440-442.
[32]A.-L.Barabasi,R.Albert,Emergence of scaling in random networks,Science,1999,286(5439):509-512.
[33]M.E.J.Newman,D.J.Watts,Scaling and percolation in the small-world network model,Phys.Rev.E,1999,60:7332-7342.
[34]彭文伟,传染病学.北京:人民卫生出版社,2001.
[35]W.O.Kermazk,A.G.McKendrick,A contribution to the mathematical theory of epidemic,Proc.R.Soc.London,1927,All5:700-721.
[36]H.W.Hethcote,The mathematics of infectious diseases,SIAM Rev.,2000,42:599-653.
[37]马知恩,王稳地,周义仓,靳祯,传染病动力学的数学建模与研究,北京:科学出版社,2004.
[38]R.M.Anderson and R.M.May,Infectious Diseases of Humans,Oxford:Oxford University Press,1992.
[39]F.Brauer,P.van den Driessche,J.Wu,Mathematical Epidemiology,New York:Springer,2008.
[40]R.M.May,A.L.Lloyd,Infection dynamics on scale-free networks,Phys.Rev.E,2001,64:066112.
[41]R.Pastor-Satorras,A.Vespignani,Epidemic dynamics and endemic states in complex networks,Phys.Rev.E,2001,63:066117.
[42]R.Pastor-Satorras,A.Vespignani,Epidemic Spreading in Scale-Free Networks,Phys.Rev.Lett.,2001,86:3200.
[43]R.Pastor-Satorras,A.Vespignani,Imunization of complex networks,Phys.Rev.E,2002,65:036104.
[44]M.E.J.Newman,Spread of epidemic disease on networks,Phys,Rev.E,2002,66:016128.
[45]M.Boguna,R.Pastor-Satorras,Epidemic spreading in correlated complex networks.Phys.Rev.E,2002,66:047104.
[46]R.Cohen,S.Havlin,D.Ben-Avraham,Efficient immunization strategies for computer networks and populations,Phys.Rev.Lett.,2003,91:247901.
[47]J.Z.Liu,J.S.Wu,Z.R.Yang,The spread of infectious disease on complex networks with household-structure,Physica A,2004,341:273-280.
[48]J.Z.Liu,Y.Tang,Z.R.Yang,The spread of disease with birth and death networks.J.Star.Mech.,2004,P08008.
[49]Y.Hayashi,M.Minoura,J.Matsukubo,Oscillatory epidemic prevalence in growing scale-free networks,Phys.Rev.E,2004,69:016112.
[50]T.Zhou,J.G.Liu,W.J.Bai,G.R.Chen,B.H.Wang,Behaviors of susceptible-infected epidemics on scale-free networks with identical infectivity,Phys.Rev.E,2006,74:056109.
[51]R.Yang,B.H.Wang,J.Ren,W.J.Bai,Z.W.Shi,W.Xu,T.Zhou,Epidemic spreading on heterogeneous networks with identical infectivity,Phys.Lett.A,2007,364:189-193.
[52]H.Shi,Z.Duan,G.Chen,An SIS model with infective medium on complex networks Physica A,2008,387:2133-2144.
[53]唐三,肖燕妮,单种群生物动力系统,北京:科学出版社,2008.
[54]E.Allen,Modeling with Ito Stochastic Differential Equations,New York:Springer,2007.
[55]L.J.S.Allen,A.M.Burgin,Comparison of deterministic and stochastic SIS and SIR models in discrete time,Math.Biosci.,2000,163(1):1-33.
[56]H.C.Tuckwell,R.J.Williams,Some properties of a simple stochastic epidemic model of SIR type,Math.Biosci.,2007,208(1):76-97.
[57]W.S.McCulloch,W.Pitts,A logical calculus of the ideas immanent in nervous activity,Bulletin of Mathematical Biophysics,1943,5:115-133.
[58]D.O.Hebb,The Organization of Behaviour,New York:Wiley,1949.
[59]F.Rosenblatt,The perceptron:a probabilistic model for information storage and organization in the brain,Rsychological Review,1958,65:386-408.
[60]J.J.Hopfield,Neural networks as physical systems with emergent collective conputational abilities,Proc.National Academy of Science of USA,1982,72:2554-2558.
[61]J.J.Hopfield,Neurons with graded response have collective computational properties like those of two-state neurons,Proc.National Academy of Science of USA,1984,81:3088-3092.
[62]T.Wang,Z.H.Sheng,Asymptotic stationarity of discrete-time stochastic neural networks,Neural Network,1996,9(6):957-963.
[63]E.Kaslik,St.Balint,Bifurcation analysis for a two dimensinal delayed discrete-time Hopfield neural network,Chaos,Solitons & Fractals,34(2007)1245-1253.
[64]Z.Yuan,D.Hu,L.Huang,Stability and bifurcation analysis on a discrete-time neural network,J.Comput.Appl.Math.,2005,177:89-100.
[65]胡适耕,黄乘明,吴付科,随机微分方程,北京:科学出版社,2008.
[66]L.Arnold,Random Dynamical Systems,Heidelberg:Springer,1998.
[67]G.Chen,X.Dong,From Chaos to Order:Methodologies;Perspectives and Applications,Singapore:World Scientific,1998.
[68]X.Wang,G.Chen,Chaotification via arbitrarily small feedback controls:Thory,method,and applications,Int.J.Bifur.Chaos,2000,10:549-570.
[69]Z.Chen,P.Yu,Controlling and anti-controlling Hopf bifurcations in discrete maps using polynomial functions,Chaos,Solitons and Fractals,2005,26:1231-1248.
[70]E.Ott,C.Grebogi,J.A.Yorke,Controlling chao,Phys.Rev.Lett.,1990,64:1196-1199.
[71]A.Maritan,J.R.Banavar,Chaos,noise,and synchronization,Phys.Rev.Lett.,1994,72:1451-1454.
[72]A.S.Pikovsky.Comment on 'Chaos,Noise,and Synchronization',Phys.Rev.Lett.,1994,73:2931.
[73]G.Malescio.Noise and synchronization in chaotic systems.Phys.Rev.E,1996,53:6551-6553.
[74]L.Longa,E.M.K Curado,and F.A.Oliveira.Roundoff-induced coalescence of chaotic trajectories.Phys.Rev.E,1996,54:2201-2204.
[75]L.Gammaitoni,P.Hanggi,P.Jung,and F.Marchesoni.Stochastic resonance.Reviews of Modern Physics,1998,70:223-287.
[76]M.S.Wang,Z.H.Hou,and H.W.Xin.Internal noise-enhanced phase synchronization of coupled chemical chaotic oscillators.J.Phys.A:Math.Gen.,2005,38:145-152.
[77]W.Lin,Y.B.He.Complete synchronization of the noise-perturbed Chua's circuits,Chaos,2005,15:023705.
[78]W.Lin,G.Chen,Using white noise to enhance synchronization of coupled chaotic systems,Chaos,2006,16:013134.
[79]Z.F.Mainen,T.J.Sejnowski,Reliability of Spike Timing in Neocortical Neurons,Science,1995,268:1503-1506.
[80]T.Royama,Analytical Population Dynamics,Chapman and Hall:London,1992.
[81]丁玖,周爱辉,确定性系统的统计性质,北京:清华大学出版社,2006.
[82]J.A.Yorke,W.R London,Recurrent outbreaks of measles,chickenpox and mumps Ⅱ,Am.J.Epidemiol.,1973,98:469-482.
[83]V.Capasso,G.Serio,A generalization of the Kermack-Mckendrick deterministic epidemic model,Math.Biosci.,1978,42:43-61.
[84]W.M.Liu,H.W.Hethcote,S.A.Levin,Dynamical behavior of epidemiological models with nonlinear incidence rates,J.Math.Biol.,1987,25:359-380.
[85]W.M.Liu,S.A.Levin,Y.Iwasa,Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models,J.Math.Biol.,1986,23:187-204.
[86]M.E.Alexander,S.M.Moghadas,Periodicity in an epidemic model with a generalized non-linear incidence,Math.Biosci.,2004,189:75-96.
[87]S.M.Moghadas,M.E.Alexander,Bifurcation of an epidemic model with non-linear incidence and infection-dependent removal rate,Math.Med.and Bio.,2006,23:231-254.
[88]W.R.Derrick,P.van den Driessche,A disease transmission model in a nonconstant population,J.Math.Biol.,1993,31:495-512.
[89]S.Ruan,W.Wang,Dynamical behavior of an epidemic model with a nonlinear incidence rate,J.Differ.Equations,2003,188:135-163
[90]P.van den Driessche,J.Watmough,A simple SIS epidemic model with a backward bifurcation,J.Math.Biol.,2000,40:525-540.
[91]D.Xiao,S.Ruan,Global analysis of an epidemic model with nonmonotone incidence rate,Math.Biosic.,2007,208:419-429.
[92]D.Xiao,Y.Zhou,Qualitative analysis of an epidemic mosel,Canadian Applied Mathmatics Quarterly,2006,14:469-492.
[93]X.C.Fu,M.Small,D.M.Walker,H.F.Zhang,Epidemic dynamics on scale-free networks with piecewise linear infectivity and immunization,Phys.Rev.E,2008,77:036113.
[94]R.M.Anderson,R.M.May,The invasion,persistence and spread of infectious diseases within animal and plant communitcs,Phil.Trans.R.Soc.London,1986,(B314):533-570.
[95]Y.N.Xiao,L.S.Chen,Modeling and analysis of a predator-prey model with disease in the prey,Math.Biosci.,2001,171:58-82.
[96]L.T.Han,Z.E.Ma,S.Tan,An SIRS epidemic model of two competitive species,Mathl.Comput.Modelling,2003,37:87-108.
[97]M.E.J.Newman,Threshold effects for two pathogens spreading on a network,Phys.Rev.Lett.,2005,95:108701.
[98]Y.Y.Ahn,H.Jeong,N.Masuda,J.D.Nob,Epidemic dynamics of two species of interacting particles on scale-free networks,Phys.Rev.E,2006,74:066113.
[99]N.Masuda,N.Konno,Multi-state epidemic processes on complex networks,Journal of Theoretical Biology,2006,243:64-75.
[100]C.Castillo-Chavez,W.Huang,J.Li,Competitive exclusion and coexistence of multiple strains in an SIS STD model,SIAM J.Appl.Math.,1999,59:1790-1811.
[101]F.Liljeros,C.R.Edling,L.A.N.Amaral,H.E.Stanley,Y.Aberg,The web of human sexual contaxts,Nature,2001,411:907-908.
[102]F.Liljeros,C.R.Edling,L.A.N.Amaral,Sexual networks:implications for the transmission of sexually transmitted infections,Microbes and Infection,2003,5:189-196.
[103]H.J.Jones,M.S.Handcock,Sexual contacts and epidemic thresholds,Nature 2003,423:605-606.
[104]W.P.Guo,X.Li,X.F.Wang,Epidemics and immunization on Euclidean distance preferred smallworld networks,2007,Physica A,380:684-690.
[105]X.Li,X.F.Wang,Controlling the spreading in small-world evolving networks:Stability,oscilla tion,and topology,IEEE Trans.Automat.Control,2006,5(3):534-540.
[106]X.Li,X.F.Wang,On the stability of epidemic spreading in small-world networks:how prompt the recovery should be? Int.J.Syst.Sci.,2007,38:401-411.
[107]H.Risken,T.Frank,The Fokker-Planck Equation:Methods of Solutions and Applications,New York:Springer,1996.
[108]V.L.Oseledec,A multiplicativee rgodict heorem Lyapunovc haracteristic numbers for dynamical systems,Transaction of the Moscow Mathematical Society,1968,19:197-231.
[109]L.Arnold,G.Papanicolsou,V.Wihstutz,Asymptotic analysis of the Lyapunov exponent and rotation number of the random oscillator and applications,SIAM Journal of Applied Mechanics,1986,46(3):427-450.
[110]N.S.Namachiwaya,V.Roessel,S.Talwar,Maximal Lyapunov exponent and almost-sure stability for coupled two degree of fieedom stochastic systems,ASME Journal of Applied Mechanics,1994,61:446-452.
[111]S.T.Ariaratnam,W.C.Xie,Lyapunov exponents and stochastic stability of coupled linear systems under real noise excitation,ASME Journal of Applied Mechanics,1992,59:664-673.
[112]刘先斌,陈虬,陈大鹏,非线性随机动力系统的稳定性和分岔研究,力学进展,1996,26(4):437-452.
[113]刘先斌,一类随机分叉系统概率1分叉研究,固体力学学报,2001,22(3):297-302.
[114]戌海武,徐伟,方同,二自由度棍合非线性随机系统的最大Lyapunov指数和稳定性,应用力学学报,1998,15(1):22-29.
[115]戎海武,孟光,徐伟,方同,二自由度祸合线性随机系统的最大Lyapunov指数和稳定性,应用力学学报,2000,17(3):46-53.
[116]C.M.Marcus,R.M.Westervelt,Dynamics of iterated-map neural networks,Physical Review A,1989,40:501-504.
[117]林元烈,应用随机过程,北京:清华大学出版社,2002.
[118]A.Shiryaev,Probability,Berlin:Springer,1996.
[119]J.A.D.Appleby,X.Mao,A.Rodkina,On stochastic stabilization of difference equations,Discrete and continuous dynamical systems,2006,15(3):843-857.
[120]S.Haykin,Neural Networks,New Jersey:Prentice-Hall,1994.
[121]A.Uchida,R.McAllister,R.Roy,Consistency of Nonlinear System Response to Complex Drive Signals,Phys.Rev.Lett.,2004,93:244102.
[122]R.Toral,C.R.Mirasso,E.Hemandez-Garcia,O.Piro,Analytical and numerical studies of noiseinduced synchronization of chaotic systems,Chaos,2001,11:665-673.
[123]C.Zhou,J.Kurths,Noise-Induced Phase Synchronization and Synchronization Transitions in Chaotic Oscillators,Phys.Rev.Lett.,2002,88:230602.
[124]K.Yoshimura,I.Valiusaityte,P.Davis,Synchronization induced by common colored noise in limit cycle and chaotic systems,Phys.Rev.E,2007,75:026208.
[125]K.Yoshimura,J.Muramatsu,P.Davis,Conditions for common-noise-induced synchronization in time-delay systems,Physica D,2008,237:3146-3152.
© 2004-2018 中国地质图书馆版权所有 京ICP备05064691号 京公网安备11010802017129号 地址：北京市海淀区学院路29号 邮编：100083 电话：办公室：(+86 10)66554848；文献借阅、咨询服务、科技查新：66554700 |