传染病动力系统分析与种群混沌系统控制问题研究
|
摘要
传染病一直以来威胁人类的健康与生命,历史上曾给人类带来巨大的灾难。时至今日,虽然一些传染病由于人类采取的有力措施而被消灭或得到控制,但有再次抬头且不断蔓延的趋势,各种新型的传染病也层出不穷,给人们的生命财产带来了巨大的损失。为了研究传染病在种群内外的传播规律及与之有关的社会因素,建立能反映传染病流行规律的数学模型并研究其发生、发展与传播的规律,了解疾病的发展过程,揭示发展规律并预测其变化趋势,以便为相关部门制定防治策略提供科学依据,使人们能更好的抵御疾病,这一工作至关重要。
生物动力系统是典型的非线性复杂系统。系统除自身的复杂性与结构的庞大性外,还有种群内部、种群之间的相互作用及外界环境的干扰等诸多因素,导致系统存在分岔乃至混沌等复杂的动力学行为。目前有许多专家对生物动力系统存在的复杂现象问题进行了研究,并得到了丰富的结果。对生物系统控制以达到服务人类、保持生态平衡的目的,是值得研究的问题。然而应用工程上的控制理论或方法对系统所存在的复杂动态特性的控制问题的研究刚刚开始,研究结果尚不多见,还有大量问题未得到解决。
本文针对一类具有垂直传染方式的传染病模型,利用动力系统稳定性理论,对系统采取连续接种与脉冲接种的方式,研究了平衡点与周期解的存在性与稳定性问题。针对一些种群混沌模型,利用OGY (Ott, Grebogi, Yorke)方法、非线性反馈控制、反馈线性化控制、直线稳定化、混沌跟踪控制等控制理论和方法,研究了混沌现象及相应的混沌控制问题。主要研究内容如下：
(1)讨论了具有垂直传染和连续预防接种的双线性SIRS传染病模型稳定性问题。根据不同类型的新生儿采取不同的连续接种方式,利用Lyapunov函数方法和LaSalle不变原理,研究了无病平衡点和地方病平衡点(正平衡点)的存在与全局稳定性问题。分别得到了无病平衡点存在与稳定的条件,地方病平衡点存在与稳定的条件。
(2)针对具有垂直传染的双线性SIRS传染病模型,对易感者新生儿分别采取连续预防接种和脉冲预防接种两种方式,分别给出了这两种接种方式下SIRS传染病模型的基本再生数。利用Lyapunov函数方法和LaSalle不变原理,得到了连续预防接种下无病平衡点和地方病平衡点的全局稳定性条件；利用脉冲微分方程的Floquet乘子理论、比较定理和非线性分析方法,得到了无病周期解的存在性和全局稳定性条件。
(3)针对一类具有物种流动特定区域内的种群增长差分模型,利用Lyapunov指数方法,简单直观地验证了混沌现象的存在。为了消除影响种群正常增长及生态平衡的混沌现象,利用OGY方法,设计了调节平衡点和保证周期轨道稳定的控制器。针对一类具有收获系数的单种群模型,利用Lyapunov指数方法,验证了混沌现象的存在。通过分析系统分岔图,展示了种群收获系数在不同范围下的系统动力学行为。分别利用OGY方法、非线性反馈控制方法、轨迹跟踪控制方法,消除种群系统中存在的有害混沌,并使混沌轨道稳定在期望的周期轨上。仿真表明跟踪控制的目标比较灵活,根据实际问题的需要,可以是相空间内的任意一点,也可以是任意给定周期轨道。研究结果对维持具有混沌现象的种群稳定生存具有一定的意义。
(4)针对一类离散捕食-食饵模型,利用Lyapunov指数方法,验证了混沌现象的存在。为了消除种群中的混沌现象,利用反馈线性化方法设计控制器,将种群稳定到目标周期轨道,并制定出合理的开发策略；应用混沌跟踪控制设计控制器,将种群稳定到任意周期状态。针对具有共生作用的离散耦合Logistic模型,首先采用Lyapunov指数方法验证了混沌现象的存在,通过详细分析系统状态随参数变化的分岔图得知耦合系统比Logistic模型存在更复杂的动力学行为；然后利用混沌跟踪控制方法,设计控制器,使种群稳定到任意周期状态。针对一类离散功能性反应模型,采用混沌跟踪控制方法,设计控制器,使种群稳定到任意合理周期状态,实现了混沌控制。
(5)针对一类受延迟影响的种群生态模型,利用Lyapunov指数方法,验证了混沌现象的存在。应用改进的直线稳定化方法设计控制器,使种群稳定到不动点轨道,消除了种群中存在的混沌现象。
Infection has always threatened the health and life of people, and it had brought tremendous mischance to pelope in history. Today, some infection have been eliminated or controlled because of powerful measures that people adopt, but there is a kind of trend for infection to occur again and spread, and various new types of infection emerge endlessly which has made huge loss of life and wealth of people. For studying laws of spread of the interior and interspecific population and correlative social factors, building the mathmaticl models which show the course of disease development is very important. The aims of studying the models are to understand and discover the laws of development and forecast the trend of development, in order to offer the evidence to the correlative department for making policy and resisting the disease better. This work is very important.
Biology dynamical systems are a representative kind of nonlinear complex systems. In addition to complexity of system itself and hugeness of structure, there are many factors such as internal population interaction, the interaction between the population and disturbance from outside environment These factors will lead systems to the existence of complex dynamic behavior such as quasi-periodic solutions and bifurcationand chaos. There have been many researches to control biology systems in order to achieve aim of people, but the problem of control complex dynamic characters of biology systems that exist by applying control theory and methods of engineering has begun just now, there are many unsolved problem.
In this disertation, to a class of epidemical model, by using the stability theory of dynamical system, the existence and stability problems of equilibrium points and period solutions of are studied when continuous vaccination and pulse vaccination are considered. To some population chaotic models, the problem of chaotic phenomena and chaotic control of population systems are investigated by the control theory and methods such as OGY (Ott, Grebogi, Yorke) method, nonlinear feedback control, feedback linearization control, straight-line stabilization method, and chaotic tracking control. Main results of this distertation are as follows.
(1) To two kinds of bilinear incidence SIRS (Susceptible-Infected-Removed-Susceptible) epidemic model with vertical infection and continuous vaccination, different contiounous vaccination metohods are adopted according to different types of new-born, the existence and global stability of infection-free equilibrium and endemic equilibrium (positive equilibrium) is studied by using Lyapunov function method and LaSalle invariable principle respectively. The conditions of the existence and stability of infection-free equilibrium are obtained, as well as the conditions of the existence and stability of endemic equilibrium are obtained.
(2) To a class of bilinear incidence epidemic models with continuous vaccination and pulse vaccination and vertical infection, the basic reproduce numbers of SIRS epidemic model are given respectively. Global stability of infection-free equilibrium and endemic equilibrium is proven by using Lyapunov function method and LaSalle invariable principle; the existence and stability of the infection-free period solution is proved by using Floquet multipler theory、comparison theorem and nonlinear analysis method. The conclusion shows that adopting the strategy of pulse vaccination is better than adopting the strategy of continuous vaccination for the same SIRS model with bilinear incidence.
(3) It is shown that there exists the chaotic behavior for a class of discrete population difference equation with species flowing in special domain by Lyapunov exponents distinctly. A controller is designed to control the stabilities of equilibria.and periodic orbits by using the essential ideas of the OGY method and eliminate bad chaos that can affect population increase and ecology balance. To a class of single-species model with harvesting coefficient, the existence of chaotic behaviors is validated by Lyapunov exponents. System dynamic behaviors are showed under the different harvesting coefficient range by studying bifurcation diagram detailedly. Bad chaos that can affect population increase and ecology balance is eliminated by using the OGY method, nonlinear feedback control method and chaotic tracking control method, chaotic orbits are stabilized on the anticipant period orbits and chaos is eliminated. The aim of tracking control is more flexible, it is any point in phase space or any given period orbits and chaotic orbits according to needs of actual problem. The result provides a kind of method to maintain population permanent survival.
(4) To a class of discrete predator-prey model, it is shown that there exists the chaotic behavior by lyapunov exponents distinctly. For eliminating chaotic phenomena of population, a feedback controller is designed to stabilize the chaotic population to the expected target periodic orbits by using feedback linearization method and establish rational exploiture strategy; at the same time, a controller is designed to stabilize population to any period states by applying chaotic tracking control method. To a discrete coupled logistic model with the symbiotic interaction of two species, firstly existence of chaotic phenomena is validated by applying Lyapunov exponent, System dynamic behaviors of the coupled logistic model is more complex dynamic behaviors than logistic model by analyzing bifurcation diagram detailedly. A controller is designed to stabilize the population to any periodic orbits by using chaotic tracking control method. To a discrete functional response model, a controller is designed to stabilize the population to any reasonable periodic orbits by using chaotic tracking control method and thereby chaos control is realized.
(5) To a class of time-delay population model, it is shown that there exists the chaotic behavior by Lyapunov exponents distinctly. A controller is designed to stabilize the chaotic population to fixed point orbits by using the method of straight-line stabilization for system, thereby chaotic phenomena of population are eliminated.
生物动力系统是典型的非线性复杂系统。系统除自身的复杂性与结构的庞大性外,还有种群内部、种群之间的相互作用及外界环境的干扰等诸多因素,导致系统存在分岔乃至混沌等复杂的动力学行为。目前有许多专家对生物动力系统存在的复杂现象问题进行了研究,并得到了丰富的结果。对生物系统控制以达到服务人类、保持生态平衡的目的,是值得研究的问题。然而应用工程上的控制理论或方法对系统所存在的复杂动态特性的控制问题的研究刚刚开始,研究结果尚不多见,还有大量问题未得到解决。
本文针对一类具有垂直传染方式的传染病模型,利用动力系统稳定性理论,对系统采取连续接种与脉冲接种的方式,研究了平衡点与周期解的存在性与稳定性问题。针对一些种群混沌模型,利用OGY (Ott, Grebogi, Yorke)方法、非线性反馈控制、反馈线性化控制、直线稳定化、混沌跟踪控制等控制理论和方法,研究了混沌现象及相应的混沌控制问题。主要研究内容如下：
(1)讨论了具有垂直传染和连续预防接种的双线性SIRS传染病模型稳定性问题。根据不同类型的新生儿采取不同的连续接种方式,利用Lyapunov函数方法和LaSalle不变原理,研究了无病平衡点和地方病平衡点(正平衡点)的存在与全局稳定性问题。分别得到了无病平衡点存在与稳定的条件,地方病平衡点存在与稳定的条件。
(2)针对具有垂直传染的双线性SIRS传染病模型,对易感者新生儿分别采取连续预防接种和脉冲预防接种两种方式,分别给出了这两种接种方式下SIRS传染病模型的基本再生数。利用Lyapunov函数方法和LaSalle不变原理,得到了连续预防接种下无病平衡点和地方病平衡点的全局稳定性条件；利用脉冲微分方程的Floquet乘子理论、比较定理和非线性分析方法,得到了无病周期解的存在性和全局稳定性条件。
(3)针对一类具有物种流动特定区域内的种群增长差分模型,利用Lyapunov指数方法,简单直观地验证了混沌现象的存在。为了消除影响种群正常增长及生态平衡的混沌现象,利用OGY方法,设计了调节平衡点和保证周期轨道稳定的控制器。针对一类具有收获系数的单种群模型,利用Lyapunov指数方法,验证了混沌现象的存在。通过分析系统分岔图,展示了种群收获系数在不同范围下的系统动力学行为。分别利用OGY方法、非线性反馈控制方法、轨迹跟踪控制方法,消除种群系统中存在的有害混沌,并使混沌轨道稳定在期望的周期轨上。仿真表明跟踪控制的目标比较灵活,根据实际问题的需要,可以是相空间内的任意一点,也可以是任意给定周期轨道。研究结果对维持具有混沌现象的种群稳定生存具有一定的意义。
(4)针对一类离散捕食-食饵模型,利用Lyapunov指数方法,验证了混沌现象的存在。为了消除种群中的混沌现象,利用反馈线性化方法设计控制器,将种群稳定到目标周期轨道,并制定出合理的开发策略；应用混沌跟踪控制设计控制器,将种群稳定到任意周期状态。针对具有共生作用的离散耦合Logistic模型,首先采用Lyapunov指数方法验证了混沌现象的存在,通过详细分析系统状态随参数变化的分岔图得知耦合系统比Logistic模型存在更复杂的动力学行为；然后利用混沌跟踪控制方法,设计控制器,使种群稳定到任意周期状态。针对一类离散功能性反应模型,采用混沌跟踪控制方法,设计控制器,使种群稳定到任意合理周期状态,实现了混沌控制。
(5)针对一类受延迟影响的种群生态模型,利用Lyapunov指数方法,验证了混沌现象的存在。应用改进的直线稳定化方法设计控制器,使种群稳定到不动点轨道,消除了种群中存在的混沌现象。
Infection has always threatened the health and life of people, and it had brought tremendous mischance to pelope in history. Today, some infection have been eliminated or controlled because of powerful measures that people adopt, but there is a kind of trend for infection to occur again and spread, and various new types of infection emerge endlessly which has made huge loss of life and wealth of people. For studying laws of spread of the interior and interspecific population and correlative social factors, building the mathmaticl models which show the course of disease development is very important. The aims of studying the models are to understand and discover the laws of development and forecast the trend of development, in order to offer the evidence to the correlative department for making policy and resisting the disease better. This work is very important.
Biology dynamical systems are a representative kind of nonlinear complex systems. In addition to complexity of system itself and hugeness of structure, there are many factors such as internal population interaction, the interaction between the population and disturbance from outside environment These factors will lead systems to the existence of complex dynamic behavior such as quasi-periodic solutions and bifurcationand chaos. There have been many researches to control biology systems in order to achieve aim of people, but the problem of control complex dynamic characters of biology systems that exist by applying control theory and methods of engineering has begun just now, there are many unsolved problem.
In this disertation, to a class of epidemical model, by using the stability theory of dynamical system, the existence and stability problems of equilibrium points and period solutions of are studied when continuous vaccination and pulse vaccination are considered. To some population chaotic models, the problem of chaotic phenomena and chaotic control of population systems are investigated by the control theory and methods such as OGY (Ott, Grebogi, Yorke) method, nonlinear feedback control, feedback linearization control, straight-line stabilization method, and chaotic tracking control. Main results of this distertation are as follows.
(1) To two kinds of bilinear incidence SIRS (Susceptible-Infected-Removed-Susceptible) epidemic model with vertical infection and continuous vaccination, different contiounous vaccination metohods are adopted according to different types of new-born, the existence and global stability of infection-free equilibrium and endemic equilibrium (positive equilibrium) is studied by using Lyapunov function method and LaSalle invariable principle respectively. The conditions of the existence and stability of infection-free equilibrium are obtained, as well as the conditions of the existence and stability of endemic equilibrium are obtained.
(2) To a class of bilinear incidence epidemic models with continuous vaccination and pulse vaccination and vertical infection, the basic reproduce numbers of SIRS epidemic model are given respectively. Global stability of infection-free equilibrium and endemic equilibrium is proven by using Lyapunov function method and LaSalle invariable principle; the existence and stability of the infection-free period solution is proved by using Floquet multipler theory、comparison theorem and nonlinear analysis method. The conclusion shows that adopting the strategy of pulse vaccination is better than adopting the strategy of continuous vaccination for the same SIRS model with bilinear incidence.
(3) It is shown that there exists the chaotic behavior for a class of discrete population difference equation with species flowing in special domain by Lyapunov exponents distinctly. A controller is designed to control the stabilities of equilibria.and periodic orbits by using the essential ideas of the OGY method and eliminate bad chaos that can affect population increase and ecology balance. To a class of single-species model with harvesting coefficient, the existence of chaotic behaviors is validated by Lyapunov exponents. System dynamic behaviors are showed under the different harvesting coefficient range by studying bifurcation diagram detailedly. Bad chaos that can affect population increase and ecology balance is eliminated by using the OGY method, nonlinear feedback control method and chaotic tracking control method, chaotic orbits are stabilized on the anticipant period orbits and chaos is eliminated. The aim of tracking control is more flexible, it is any point in phase space or any given period orbits and chaotic orbits according to needs of actual problem. The result provides a kind of method to maintain population permanent survival.
(4) To a class of discrete predator-prey model, it is shown that there exists the chaotic behavior by lyapunov exponents distinctly. For eliminating chaotic phenomena of population, a feedback controller is designed to stabilize the chaotic population to the expected target periodic orbits by using feedback linearization method and establish rational exploiture strategy; at the same time, a controller is designed to stabilize population to any period states by applying chaotic tracking control method. To a discrete coupled logistic model with the symbiotic interaction of two species, firstly existence of chaotic phenomena is validated by applying Lyapunov exponent, System dynamic behaviors of the coupled logistic model is more complex dynamic behaviors than logistic model by analyzing bifurcation diagram detailedly. A controller is designed to stabilize the population to any periodic orbits by using chaotic tracking control method. To a discrete functional response model, a controller is designed to stabilize the population to any reasonable periodic orbits by using chaotic tracking control method and thereby chaos control is realized.
(5) To a class of time-delay population model, it is shown that there exists the chaotic behavior by Lyapunov exponents distinctly. A controller is designed to stabilize the chaotic population to fixed point orbits by using the method of straight-line stabilization for system, thereby chaotic phenomena of population are eliminated.
引文
1. Brauer F and Castillo-Chavez C. Mathematical Models in Population Biology and Epidemiology [M], Tests in Applied Mathematics 40, Springer,2001.
2.(美)皮特-布鲁克史密斯(译者：马永波).未来的灾难：瘟疫复活与人类生存之战[M],海口：海南出版社,1999.
3.艾滋病预防与控制中心.艾滋病疫情动态,2000年全国艾滋病疫情简介[M],北京：卫生部艾滋病预防与控制中心.
4.马知恩,周义仓,王稳地,靳祯.传染病动力学的数学建模与研究[M],北京：科学出版社,2004.
5. Bernouli D. Essai d'une nouvelle analyse de la mortalite causee par la petite verole et des avantages de l'inoculation pour al prevenir, in Memoires de Mathematiques et de physique[J], Paris:Academie Royale des Sciences,1760,1-45.
6. Hamer W H. Epidemic disease in England [J], Lancet,1,1906,733-739.
7. Ross R. The Prevention of Malaria,2nd ed[M], Murray, London,1911.
8. Kermack W O, McKendrick A G. Contributions to the mathematical theory of epidemics[C], Proc.Roy.Soc.,1927, A115:700-721.
9. Kermack W O, McKendrick A G. Contributions to the mathematical theory of epidemics[C], Proc.Roy.Soc.,1932, A138:55-83.
10. N T J Bailey. The Mathematical Theory of Infectious Diseases,2nd [M], Hafner, New York,1975.
11.陆征一,周义仓.数学生物学进展[M],北京：科学出版社,2005,5-7.
12.韩丽涛.两种群相互作用的传染病模型的研究[D],西安交通大学博士论文,2002.
13.靳祯.在脉冲作用下的生态和流行病模型的研究[D],西安交通大学博士论文,2001.
14.高淑京.脉冲效应下种群动力系统和传染病模型的研究[D],西安交通大学博士论文,2001.
15. Meng X Z, Chen L S. The dynamics of a new SIR epidemic model concerning pulse vaccination strategy [J], Applied Mathematics and Computation,2008,197(2):582-597.
16. D'Onofrio A. Pulse Vaccination Strategy in the SIR Epidemic Model:Global Asymptotic Stable Eradication in Presence of vaccine Failures [J], Mathematical and Computer Modelling,2002, (36): 473-489.
17. D'Onofrio A. Stability properties of pulse vaccination strategy in SEIR epidemic model [J], Math. Biol,2002,179(1):57-72.
18. D'Onofrio A. Vaccination policies and nonlinear force of infection:generalization of an observation by Alexander and Moghadas(2004) [J], Applied Mathematics and Computation,2005, (168): 613-622.
19. D'Onofrio A. On pulse vaccination strategy in the SIR epidemic model with vertical transmission [J], Applied Mathematics Letters,2005, (18):729-732.
20. Stone L, Shulgin B, Agur Z. Theoretical examination of the pulse vaccination policy in the SIR epidemic model [J], Math. Comput. Model,2000,31(4-5):207-215.
21. Lakshmikantham V, Bainov D D, Simeonov P S. Theory of Impulsive Differential Equations [M], World Scientific,1989.
22.陈兰荪.数学生态模型与研究方法[M],北京：科学出版社,1991,199-232.
23. Hethcote H W, Ma Z, Liao S. Effects of quarantine in six endemic models for infectious diseases [J], Math. Biosci.,2002,180:141-160.
24. Brauer F. Models for diseases with vertical transmission and nonlinear population dynamics [J], Math. Biosci.,1995,128:13-24.
25. Busenberg S N, HadelerK P. Demography and epidemics [J], Math. Biosci.,1990,101:63-74.
26. Wang W D, Ma Z E. Global dynamics of an epidemic model with time delay [J], Nonlinear Analysis: Real World Applications,2002,3(3):365-373.
27. Wang W D. Global behavior of an SEIRS. epidemic model with time delays [J], Applied Mathematics Letters,2002,15(4):423-428.
28. Yoshida N, Hara T. Global stability of a delayed SIR epidemic model with density dependent birth and death rates [J], Journal of Computational and Applied Mathematics,2007,201(2):339-347.
29. Gao S J, Chen L S, Teng Z D. Pulse vaccination of an SEIR epidemic model with time delay [J], Nonlinear Analysis:Real World Applications,2008,9(2):599-607.
30. Zhang T L, Teng Z D. Pulse vaccination delayed SEIRS epidemic model with saturation incidence[J], Applied Mathematical Modeling,2008,32(7):1403-1416.
31. Zhang G S, Wang Y F. Minimal wave speed in a diffusive epidemic model with temporal delay [J], Applied Mathematics and Computation,2007,188(1):275-280.
32. Shulgin B, Stone L, Agur Z. Pulse vaccination strategy in the SIR epidemic model [J], Bull. Math.Biol,1998,60:1123-1148.
33. Stone L, Shulgin B, Agur Z. Theoretical examination of the pulse vaccination policy in the SIR epidemic model [J], Math.Comput.Modelling,2000,31:207-215.
34. Hui J, Chen L. Impulsive vaccination of SIR epidemic models with nonlinear incidence rates [J], Discrete and Continuous Dynamical Systems, series B 2004,4:595-605.
35. D'Onofrio A. On pulse vaccination strategy in the SIR epidemic model with vertical transmission [J], Appl. Math.Letters,2005,18:729-732.
36. Roberts M G, Kao R R. The dynamics of an infectious disease in a population with birth pulses [J], Math.Biol,1998,149:23-36.
37. Lu Z, Chi X, Chen L. The effect of constant and pulse vaccination on SIR epidemic model with horizontal and vertical transmission [J], Math Comput. Model,2002,36(10):1039-1057.
38. Liu W M, Levin S A, Iwasa Y. Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models [J], J. Math. Biol.,1986,23:187-204.
39. Hethcote H W, van den D P. Some epidemiological models with nonlinear incidence [J], J. Math. Biol.,1991,29:271-287.
40. Mena-lorca J, Hethcote H W. Dynamic models of infections disease as regulations of population size [J], J. Math. Biol.,1992,30:693-716.
41.王军,王辉.具有种群动力学和非线性接触率的SIRS流行病模型[J],兰州大学学报(自然科学版),2002,38(1)；22-29.
42.杨建雅,张凤琴.一类具有垂直传染的SIR传染病模型[J],生物数学学报,2006,21(3)；341-344.
43.杨俊元,张桂丽.具有垂直传染和因病死亡的SIR传染病模型[J],运城学院学报,2007,25(5)：21-23.
44. Michael Y. L, John R. G, Wang L C, et al. Global dynamics of a SEIR model with varying total population size [J], Mathematical Biosciences,1999,160:191-213.
45. Fan M, Michael Y. L, Wang K. Global stability of an SEIS epidemic model with recruitment and a varying total population size [J], Mathematical Biosciences,2001,170:199-208.
46. Feng Z, Thieme H R. Endemic models with arbitrarily distributed periods of infection, Ⅰ:General theory [J], SIAM. J. Appl.,2000,61:803.
47. Feng Z, Thieme H R. Endemic models with arbitrarily distributed periods of infection, Ⅱ:Fast disease dynamics and permanent recovery [J], SIAM. J. Appl.,2000,61:983.
48. Li J Q, Ma Z E. Qualitative analysis of SIS epidemic model with vaccination and varying total population size [J], Math.Comp. Model,2002,35:1235-1243.
49. Lizana M, Rivero J. Multiparametric bifurcations for a model in epidemiology [J], J. Math. Biol., 1996,35:21-36.
50. Wu L, Feng Z. Homoclinic bifurcation in an SIQR model for childhood disease [J], J. Diff. Eqns., 2000,168:150-167.
51. Van den D P, Watmongh J. A simple SIS epidemic model with a backward bifurcation [J], J. Math. Biol.,2000,40:525-540.
52. Ruan S G, Wang W D. Dynamic behavior of an epidemic model with a nonlinear incidence rate [J], J. Diff. Eqns.,2003,18:135-163.
53. Beretta E, Hara T, Ma W B, et al. Global asymptotic stability of an SIR epidemic model with distributed time delay [J], Nonlinear Analysis,2001,47:4107-4115.
54.周义仓.具有年龄与病程结构的流行病模型及其渐近性态[D],西安交通大学,1999.
55.刘汉武,徐洁,刘启明.连续接种的具有年龄结构的SIS传染病模型[J],工程数学学报,2002,19：25-29.
56. Iannelli M, Kim M Y, Park E J. Asymptotic behavior for an SIS epidemic model and its approximation [J], Nonlinear Analysis,1999,35:797-814.
57. Miiller J. Optimal vaccination patterns in age-structured populations [J], SIAM J. Appl. Math.,1998, 59:222-241.
58. Manuel D, Monica M B, Antonio S. Analysis of an age-structured predator-prey model with disease in the prey[J], Nonlinear Analysis:Real World Applications,2006,7(4):853-871.
59. Allen L J S, Jones M A, Martin C E. A discrete-time model with vaccination for a measles epidemic [J], Math. Biosci.,1991,105:111-131.
60. Allen L J S, Thrasher D B. The effects of vaccination in an age-dependent model for varicella and Herpes zoster [J], IEEE Trans. On Automatic Control,1998,43:779-789.
61.周毅,娄梅枝,娄洁.一类分年龄阶段的流行病模型研究[J],工程数学学报,2003,3：135-138.
62. Lu Z H, Gao S J, Chen L S. Analysis of an SI epidemic model with nonlinear transmission and stage structure[J], Acta Math. Sci.,2003,4:440-446.
63. Xiao Y N, Chen L S. A SIS epidemic model with stage structure and a delay [J], Acta Math.Appl. Sinica, English Series,2002,18:607-618.
64. Bainov D.D, Simeonov P.S. Impulsive differential equations:periodic solutions and applications [M], New York:Longman Scientific and Technical,1993.
65. Roberts M G, Kao R R. The dynamics of an infectious disease in a population with birth pulse [J], Math. Biosci,1998,149 (1):23-36.
66. Shulgin B, Stone L, Agur Z. Pulse vaccination strategy in the SIR epidemic model [J], Bull. Math. Biol,1998,60(6):1123-1148.
67.靳祯,马知恩.具有连续和脉冲接种的流行病模型[J],华北工学院学报,2003,24：235-243.
68. Gao S J, Chen L S, Sun L H. Dynamic complexities in a seasonal prevention epidemic model with birth pulses [J], Chaos, Solitons and Fractals,2005,26(4):1171-1181.
69. Gao S J, Chen L S, Sun L H. Optimal pulse fishing policy in stage-structured models with birth pulses [J], Chaos, Solitons and Fractals,2005,25(5):1209-1219.
70. Gao S J, Chen L S. The effect of seasonal harvesting on a single-species discrete population model with stage structure and birth pulses [J], Chaos, Solitons and Fractals,2005,24(4):1013-1023.
71. Gakkhar S, Negi K. Pulse vaccination in SIRS epidemic model with non-monotonic incidence rate [J], Chaos, Solitons and Fractals,2008,35(3):626-638.
72. Pang G P, Chen L S. A delayed SIRS epidemic model with pulse vaccination [J], Chaos, Solitons and Fractals,2007,34(5):1629-1635.
73.徐光启.《农政全书》卷四：《玄扈先生井田考》[M].
74. Malthus T R. An essay on the principle of population as it affects the future improvement of society, with remarks on the speculations of M.Godwin, M.Condorcot, and other writers, J.Johnson, London [M],1798.
75.马知恩.种群生态学的数学建模与研究[M],合肥：安徽教育出版社,1996,1-40.
76.徐克.生物数学[M],北京：科学出版社,1999,1-11.
77. Munteanu A, Sole R V. Phenotypic diversity and chaos in a minimal cell model [J], Journal of Theoretical Biology,2006,240(3):434-442.
78. D'Onofrio A. On a family of models of cell division cycle [J], Chaos Solitons and Fractals,2006, 27(5):1205-1212.
79. Furusawa C, Kaneko K. Theory of robustness of irreversible differentiation in a stem cell system: chaos hypothesis [J], Journal of Theoretical Biology,2001,209(4):21,395-416.
80. Sun P, Yang X B. Dynamic behaviors of the Ricker population model under a set of randomized perturbations [J], Mathematical Biosciences,2000,164:147-159.
81. Beckmann M J. On spatial population dynamics [J], Chaos, Solitons and Fractals,2003,18(3): 439-444.
82.荆海英,赵立纯.微生物连续培养模型的溢流量控制[J],生物数学学报,1998,13(5)：599-603.
83.赵立纯,荆海英.微生物连续培养模型的最优溢流量控制[J],生物数学学报,1999,14(1)：32-35.
84. Banasiak J, Lachowicz M, Moszyski M. Topological chaos:When topology meets medicine [J], Applied Mathematics Letters,2003,16(3):303-308.
85. Angelis E D, Delitala M. Modelling complex systems in applied sciences; methods and tools of the mathematical kinetic theory for active particles[J], Mathematical and Computer Modelling,2006, 43(11-12):1310-1328.
86. Autiero M, Smet F D, Claes F,et al. Role of neural guidance signals in blood vessel navigation[J], Cardiovascular Research,2005,65(3):629-638.
87. May R M. Biological populations with non-overlapping generations:stable points, stable cycles and chaos [J], Science,1974,186:645-647.
88. May R M. Simple mathematical models with very complex dynamics [J], Nature,1976,261: 459-467.
89. May R M, Oster G F. Bifurcations and dynamic complexity in simple ecological models [J], American Naturalist,1976,110:573-599.
90. Xiao Y N, Chen L S. Effects of toxicants on a stage-structured population growth model [J], Applied Mathematics and Computations,2001,123(1):63-73.
91. Duan L X, Lu Q S, Yang Z Z, et al. Effects of diffusion on a stage-structured population in a polluted environment [J], Applied Mathematics and Computation,2004,154(2):347-359.
92. Wang W D, Mulone G, Salemi F, et al. Permanence and stability of a stage-structured predator-prey model [J], Journal of Mathematical Analysis and Applications,2001,262(2):499-528.
93. Gao S J, Chen L S. Dynamic complexities in a single-species discrete population model with stage structure and birth pulses [J], Chaos, Solitons and Fractals,2005,23(2):519-527.
94. Wang J, Wang K. The optimal harvesting problems of a stage-structured population [J], Applied Mathematics and Computation,2004,148(1):235-247.
95. Jin Z, Ma Z E, Han M A. Global stability of an SIRS epidemic model with delays [J], Acta Mathematica Scientia,2006,26B(2):291-306.
96. Xu R, Chen L S, Hao F L. Periodic solutions of an n-species Lotka-Volterra type food-chain model with time delays [J], Applied Mathematics and Computation,2005,171(1):511-530.
97.付军,李健全,陈任昭.年龄相关的种群空间扩散系统的广义解与收获控制[J],控制理论与应用,2005,22(4)：588-596.
98. Gakkhar S, Naji R K. Chaos in seasonally perturbed ratio-dependent prey-predator system [J], Chaos, Soliton and Fractals,2003,15:107-118.
99. Kaitala V, Ylikarjula J, Heino M. Dynamic complexities in host-parasitoid interaction[J], Journal of Theoretical Biology,1999,197:331-341.
100. Hsu S B, Hwang T W, Kuang Y. A ratio-dependent food chain model and its applications to biological control [J], Mathematical Biosciences,2003,181:55-83.
101. O'Brien S J, Troyer J L, Roelke M, etc. Plagues and adaptation:Lessons from the Felidae models for SARS and AIDS [J], Biological Conservation,2006,131(2):255-267.
102. Lou J, Ma Z E, Shao Y M, etc. Modelling the antigen and interaction of T-cells presenting cells HIV-1 in vivo [J], Computers and Mathematics with Applications,2004,48:9-33.
103. Pinho M R, Ferreira M M, Ledzewicz U, et al. A model for cancer chemotherapy with state-space constraints [J], Nonlinear Analysis,2005,63(5-7):e2591-e2602.
104. Ware N C, Wyatt M A, Tugenberg T. Adherence, stereotyping and unequal HIV treatment for active users of illegal drugs [J], Social Science & Medicine,2005,61(3):565-576.
105. Ghezzi L L, Piccardi C. PID control of a chaotic system:An application to an epidemiological model [J], Automatica,1997,33(2):181-191.
106. Wang W M, Wang X Q, Lin Y Z. Complicated dynamics of a predator-prey system with Watt-type functional response and impulsive control strategy[J], Chaos, Solitons and Fractals,2008,37(5): 1427-1441.
107. Wang X Q, Wang W M, Lin X M. Chaotic behavior of a Watt-type predator-prey system with impulsive control strategy [J], Chaos, Solitons and Fractals,2008,37(3):706-718.
108.荆海英.可再生性资源开发系统的调控与最优管理[M],沈阳：辽宁人民出版社,2002,166-180.
109. Zhao L C, Zhang Q L, Yang Q C. Optimal impulsive harvesting for fish populations [J], Journal of System Science and Complexity,2003,16(4):466-474.
110.赵立纯,张庆灵,杨启昌.三种群食物链系统的耗散性控制[J],生物数学学报,2003,18(1)：82-92.
111.赵立纯,张庆灵,杨启昌.具有阶段结构单种群系统的诱导控制[J],数学物理学报,2005,25A(5)：710-717.
112.王庭槐,耿艺介.混沌动力学非线性分析方法在生物反馈研究中的应用[J],自然杂志,2004,26(4)：223-226.
113.赵立纯.种群动力学系统控制问题的研究[D],沈阳：东北大学,2003.
114.张化光,王智良,黄玮.混沌系统的控制理论[M],沈阳：东北大学出版社：2003,29-37.
115.黄润生.混沌及其应用[M],武汉：武汉大学出版社,2000,1-10.
116. Lorenz E N. Deterministic nonperiodic flow [J], J. Atmos. Sci.,1963,20(2):130-141.
117. Li T Y, Yorke J A. Period three implies chaos [J], Am. Math. Monthy,1975,82:985-992.
118.傅汝廉,常胜江,温午麒等.人体心脏脉动的混沌动力学研究[J],中华物理医学与康复杂志,1999,21(2)：100-102.
119.裴文江,何振亚,杨绿溪等.发现心脏节律中存在确定性混沌[J],科学通报,2001,46(8)：695-670.
120.王兴元,顾树生.心电动态生理及病理信息的非线性动力学研究[J],中国生物医学工程学报,2000,19(4)：397-403.
121.修志龙,曾安平.微生物细胞连续培养过程中振荡和混沌行为的研究进展[J],生物工程进展,1999,19(6)：58-63.
122. Summers D, Cranford J G, Healey B P. Chaos in periodically forced discrete-time ecosystem models [J], Chaos, Solitons and Fractals,2000,11:2331-2342.
123. Blackmore D, Chen J, Perez J, et al. Dynamical properties of discrete Lotka-Volterra equations [J], Chaos, Solitons and Fractals,2001,12:2553-2568.
124.Braverman E, Kinzebulatov D. On linear perturbations of the Ricker model [J], Mathematical Biosciences,2006,202:323-339.
125. Ruxton G D, Rohani P. Population floors and the persistence of chaos in ecological models [J], Theoretical Population Biology,1998,53:175-183.
126. Casagrandi R, Bolzoni L, Levin S A, et al. The SIRC model and influenza A [J], Mathematical Biosciences,2006,200:152-169.
127.王洪礼,冯剑丰,沈菲,孙景.赤潮藻类非线性动力学模型的分岔及稳定性研究[J],应用数学和力学,2005,26(6)：671-676.
128. Jing Z J, Chang Y, Guo B L. Bifurcation and chaos in discrete FitzHugh-Nagumo system [J], Chaos, Solitons & Fractals,2004,21:701-720.
129.张真,李典谟,查光济.马尾松毛虫种群动态的时间序列分析及复杂性动态研究[J],生态学报,2002,22(7)：1061-1067.
130.朱伟勇,王琰,方自平.猩红热和流脑流行过程的混沌分维研究[J],数学的实践与认识,2001,31(3)： 331-335.
131.邓小炎.一类单种群差分模型的周期解和混沌现象[J],华中农业大学学报,2000,19(4)：403-407.
132.李医民,周燕.一类具有收获系数的单种群模型的混沌分析[J],江苏大学学报(自然科学版),2005,26(4)：324-327.
133.张真,李典谟,张培义等.自然种群中混沌的检测及其在种群动态研究中的意义[J],生态学报,2003,23(10)：1951-1962.
134. Vayenas D V, Pavlou S. Chaotic dynamics of a microbial system of coupled food chains [J], Ecological Modelling,2001,136:285-295.
135. Misra J C, Misra A. Instabilities in single-species and host-parasite systems:period-doubling bifurcations and chaos [J], Computers and Mathematics with Applications,2006,52:525-538.
136. Tang S Y, Chen L S. Chaos in functional response host-parasitoid ecosystem models [J], Chaos, Solitons and Fractals,2002,13:875-884.
137. Xu C L, Boyce M S. Dynamic complexities in a mutual interference host-parasitoid model [J], Chaos, Solitons and Fractals,2005,24:175-182.
138. Jang S R J, Diamond S L. A host-parasitoid interaction with Allee effects on the host [J], Computers and Mathematics with Applications,2007,53:89-103.
139. Jing Z J, Yang J P. Bifurcation and chaos in discrete-time predator-prey system [J], Chaos, Solitons and Fractals,2006,27:259-277.
140. Liu X L, Xiao D M. Complex dynamic behaviors of a discrete-time predator-prey system [J], Chaos, Solitons and Fractals,2007,32:80-94.
141. Letellier C, Aziz-Alaoui M A. Analysis of the dynamics of a realistic ecological model [J], Chaos, Solitons and Fractals,2002,13:95-107.
142. Gakkhar S, Naji R K. Order and chaos in predator to prey ratio-dependent food chain [J], Chaos, Solitons and Fractals,2003,18:229-239.
143. Wang F Y, Pang G P. Chaos and Hopf bifurcation of a hybrid ratio-dependent three species food chain [J], Chaos, Solitons and Fractals,2008,36(5):1366-1376.
144. Naji R K, Balasim A T. Dynamical behavior of a three species food chain model with Beddington-DeAngelis functional response [J], Chaos, Solitons and Fractals,2007,32:1853-1866.
145. Wang K F, Wang W D, Pang H Y, et al. Complex dynamic behavior in a viral model with delayed immune response [J], Physica D,2007,226:197-208.
146.关新平,范正平,陈彩莲等.混沌控制及其在保密通信中的应用[M],北京：国防工业出版社,2002.
147.张树文,陈兰荪.具有脉冲效应和综合害虫控制的捕食系统[J],系统科学与数学,2005,25(3)：264-275.
148. Hui J, Zhu D M. Dynamic complexities for prey-dependent cosumption integrated pest management models with impulsive effects[J], Chaos, Solitons and Fractals,2006,29:233-251.
149.张玉娟.脉冲微分方程在种群生态管理数学模型研究中的应用[D],大连理工大学,2004.
150.贡崇颖,李医民,孙曦浩.肌型血管生物数学模型的自适应Backstepping控制设计[J],生物数学学报,2007,22(3)：503-508.
151.张悦,张庆灵,赵立纯等.一类种群增长模型的反馈线性化控制[J],东北大学学报(自然科学版),2005,10：926-929.
152. Awad E G, Al-Ruzaiza A S. Chaos and adaptive control in two prey, one predator system with nonlinear feedback[J], Chaos, Solitons and Fractals,2007,34(2):443-453.
153. Gomes A A, Manica E, Varriale M C. Applications of chaos control techniques to a three-species food chain [J], Chaos, Solitons and Fractals,2008,36(4):1097-1107.
154.黄润生,黄浩.混沌及其应用[M],武汉：武汉大学出版社,2005,270-312.
155.王光瑞,于熙龄,陈式刚.混沌的控制、同步与利用[M],北京：国防工业出版社,2001,563-571.
156. Ott E, Grebogi C, Yorke J A. Controlling chaos[J], Physics Review Letters,1990,64:1196-1199.
157. Zhang X Y, Shuai Z S, Wang K. Optimal impulsive harvesting for single population [J], Nonlinear Analysis:Real World Applications,2003(4):639-651.
158. Tsui A P M, Jones A J. The control of higher dimensional chaos:comparative results for the chaotic satellite attitude control problem [J], Physica D,2000,135:41-62.
159.王利清,魏学业,温伟刚等.混沌态电流模式Buck-Boost开关变换器的OGY控制[J],信息与控制,2005,34(6)：742-745.
160.王琳,彭建华,陈立宏.改进的OGY法控制超混沌的电路实验研究[J],东北师大学报(自然科学版),2006,38(1)：44-49.
161.薛怀庆.双光子光学双稳系统的混沌控制[J],兰州大学学报(自然科学版),2002,38(3)：27-32.
162.姚洪兴,盛昭瀚.经济混沌模型中一种改进的反馈控制方法[J],系统工程学报,2002,17(6)：507-511.
163.焉德军,刘向东,朱伟勇.六辊UC轧机的OGY混沌控制对策[J],控制与决策,1999,14(2)：156-160.
164.陈国华,李煜,盛昭瀚.基于不同决策规则的产出系统的混沌与控制[J],系统工程理论与实践,2004,5：84-90.
165. Isidori A. Nonlinear Control System [M], Heidelberg:Springer-Verlag Berlin,1985.
166. Chen L Q, Liu Y Z. Chaotic attitude motion of a magnetic rigid spacecraft and its control[J], International Journal of Non-Linear Mechanics,2002,37:493-504.
167.韦笃取,罗晓曙,方锦清等.基于微分几何方法的永磁同步电动机的混沌运动的控制[J],物理学报,2006,55(1)：54-59.
168. Harb A M, Abdel-Jabbar N. Controlling Hopf bifurcation and chaos in a small power system[J], Chaos, Solitons and Fractals,2003,18:1055-1063.
169. Kakmenia F M M, Bowong S, Tchawoua C, et al. Resonance, bifurcation and chaos control in electrostatic transducers with two external periodic forces [J], Physica A,2004,333:87-105.
170.于书敏,赵立纯,张庆灵等.一个三种群食物链系统的非线性控制[J],数学的实践与认识,2005,35(11)：162-171.
171.谭文,王耀南,黄丹等.不确定混沌系统的动态神经网络跟踪控制[J],控制与决策,2004,19(4)：455-458.
172.伍维根,古天祥.混沌系统的非线性反馈跟踪控制[J],物理学报,2000,49(10)：1922-1925.
173.窦春霞,张淑清.基于观测器的模型不确定的耦合时空混沌H∞跟踪控制[J],物理学报,2004,53(12)：4120-4125.
174.刘亚,胡寿松.基于自适应神经网络的不确定非线性系统的模糊跟踪控制[J],控制理论与应用,2004,21(5)：770-775.
175.张家树,肖先赐.连续混沌系统的非线性自适应预测跟踪控制[J],物理学报,2001,50(11)：2092-2096.
176.卢辉斌,李丽香,彭海朋等.一类不确定时滞混沌系统的输出追踪控制[J],仪器仪表学报,2003,24(2)：193-194,210.
177.任海鹏,刘丁.六辊UC轧机轧制过程混沌的轨迹跟踪控制[J],控制理论与应用,2002,19(6)：897-900.
178.李洁,任海鹏.永磁同步电动机中混沌运动的部分解耦控制[J],控制理论与应用,2005,22(4)：637-640.
179.李建芬,李农,林辉.适合传输快变信息信号的混沌调制保密通信[J],物理学报,2004,53(6)：1694-1698.
180.刘扬正,李平.一维混沌系统的非线性反馈控制[J],扬州大学学报(自然科学版),2003,6(3)：27-29.
181.郝百林.从抛物线谈起——一维混沌动力学引论[M],上海：上海科学技术出版社,1993,28.
182. Xu H B, Wang G R, Chen S G. Controlling chaos by a modified straight-line stabilization method [J], The European Physical Journal B,2001,22:65-69.
183.李健全.具有预防接种流行病的模型[D],西安交通大学,2003.
184.徐洁,周义仓.具有比例和脉冲接种的乙肝流行病模型[J],生物数学学报,2004,19(2)：149-155.
185. Fan M, Wang K. Optimal harvesting policy for single population with periodic coefficients [J], Mathematical Biosciences,1998,1(52):165-177.
186.任海鹏,刘丁.BOOST变换器中混沌现象及其控制的仿真研究[J],系统仿真学报,2004,16(11)：2529-2532.
187.陈关荣,胡刚.混沌系统的非线性反馈控制的可控性条件[J],北京师范大学学报(自然科学版),1996,32(2)：208-212.
188.李国辉,周世平,徐得名等.间歇线性反馈控制混沌[J],物理学报,2000,49(11)：2123-2127.
189.刘扬正,李平.利用间歇非线性时滞反馈控制一维Logistic系统的混沌运动[J],南京师范大学学报(自然科学版),2002,25(3)：53-57.
190. Hassan K K. Nonlinear Systems (Second Edition) [M], Upper Saddle River, New Jersey:Prentice Hall,1996.
191.FitzHugh R. Impulses and physiological states in theoretical models of nerve membrane [J], Biophys J.,1961,1:445-466.
192. Rasband S N. Chaotic Dynamics of Nonlinear System [M], New York:Wiley/Interscience,1990, 187-195.
193. Ricardo L R. Complex behavior in a discrete coupled logistic model for the symbiotic interaction of two species [J], Mathematical Biosciences and Engineering,2004,1(2):307-324.
194.王兴元,石其江.二维Logistic映射中的一种新型激变、回滞和分形[J],应用力学学报,2005,24(4)：501-506.
195. Neubert M G, Kot M. The subcritical collapse of predator populations in discrete-time predator-prey models [J], Mathematical Biosciences,1992,110:45-66.
196. Kocak H. Differential and difference equations through computer experiments [J], New York: Springer,1989.
197. Zhang Y, Zhang Q L, Zhao L C, et al. Dynamical behaviors and chaos control in a discrete functional response model[J], Chaos, Solitons and Fractals,2007,34:1318-1327.
198. Celik C. The stability and Hopf bifurcation for a predator-prey system with time delay [J], Chaos, Solitons and Fractals,2008,37(1):87-99.
199. Wang L, Wolkowicz G S K. A delayed chemostat model with general nonmonotone response functions and differential removal rates [J], Journal of Mathematical Analysis and Applications, 2006,321(1):452-468.
200. Choudhury S R. On bifurcations and chaos in predator-prey models with delay [J], Chaos Solitons and Fractals,1992,2(4):393-409.
201. Liao X F, Wong K W, Leung C S, et al. Hopf bifurcation and chaos in a single delayed neuron equation with non-monotonic activation function[J], Chaos,Solitons and Fractals,2001,12(8): 1535-1547.
202. Peng M S. Rich dynamics of discrete delay ecological models [J], Chaos, Solitons and Fractals, 2005,24(5):1279-1285.
203. Sun H J, Cao H J. Bifurcations and chaos of a delayed ecological model [J], Chaos, Solitons and Fractals,2007,33(4):1383-1393.
204. Babloyantz A, Lourenco C, Sepulchre J A. Control of chaos in delay differential equations, in a network of oscillators and in model cortex [J], Physica D:Nonlinear Phenomena,1995,86(1-2): 274-283.
205. Tian Y C, Gao F R. Adaptive control of chaotic continuous-time systems with delay [J], Physica D: Nonlinear Phenomena,1998,117(1-4):1-12.
206. Vladimir E. Bondarenko. Control and'anticontrol' of chaos in an analog neural network with time delay [J], Chaos, Solitons and Fractals,2002,13(1):139-154.
2.(美)皮特-布鲁克史密斯(译者：马永波).未来的灾难：瘟疫复活与人类生存之战[M],海口：海南出版社,1999.
3.艾滋病预防与控制中心.艾滋病疫情动态,2000年全国艾滋病疫情简介[M],北京：卫生部艾滋病预防与控制中心.
4.马知恩,周义仓,王稳地,靳祯.传染病动力学的数学建模与研究[M],北京：科学出版社,2004.
5. Bernouli D. Essai d'une nouvelle analyse de la mortalite causee par la petite verole et des avantages de l'inoculation pour al prevenir, in Memoires de Mathematiques et de physique[J], Paris:Academie Royale des Sciences,1760,1-45.
6. Hamer W H. Epidemic disease in England [J], Lancet,1,1906,733-739.
7. Ross R. The Prevention of Malaria,2nd ed[M], Murray, London,1911.
8. Kermack W O, McKendrick A G. Contributions to the mathematical theory of epidemics[C], Proc.Roy.Soc.,1927, A115:700-721.
9. Kermack W O, McKendrick A G. Contributions to the mathematical theory of epidemics[C], Proc.Roy.Soc.,1932, A138:55-83.
10. N T J Bailey. The Mathematical Theory of Infectious Diseases,2nd [M], Hafner, New York,1975.
11.陆征一,周义仓.数学生物学进展[M],北京：科学出版社,2005,5-7.
12.韩丽涛.两种群相互作用的传染病模型的研究[D],西安交通大学博士论文,2002.
13.靳祯.在脉冲作用下的生态和流行病模型的研究[D],西安交通大学博士论文,2001.
14.高淑京.脉冲效应下种群动力系统和传染病模型的研究[D],西安交通大学博士论文,2001.
15. Meng X Z, Chen L S. The dynamics of a new SIR epidemic model concerning pulse vaccination strategy [J], Applied Mathematics and Computation,2008,197(2):582-597.
16. D'Onofrio A. Pulse Vaccination Strategy in the SIR Epidemic Model:Global Asymptotic Stable Eradication in Presence of vaccine Failures [J], Mathematical and Computer Modelling,2002, (36): 473-489.
17. D'Onofrio A. Stability properties of pulse vaccination strategy in SEIR epidemic model [J], Math. Biol,2002,179(1):57-72.
18. D'Onofrio A. Vaccination policies and nonlinear force of infection:generalization of an observation by Alexander and Moghadas(2004) [J], Applied Mathematics and Computation,2005, (168): 613-622.
19. D'Onofrio A. On pulse vaccination strategy in the SIR epidemic model with vertical transmission [J], Applied Mathematics Letters,2005, (18):729-732.
20. Stone L, Shulgin B, Agur Z. Theoretical examination of the pulse vaccination policy in the SIR epidemic model [J], Math. Comput. Model,2000,31(4-5):207-215.
21. Lakshmikantham V, Bainov D D, Simeonov P S. Theory of Impulsive Differential Equations [M], World Scientific,1989.
22.陈兰荪.数学生态模型与研究方法[M],北京：科学出版社,1991,199-232.
23. Hethcote H W, Ma Z, Liao S. Effects of quarantine in six endemic models for infectious diseases [J], Math. Biosci.,2002,180:141-160.
24. Brauer F. Models for diseases with vertical transmission and nonlinear population dynamics [J], Math. Biosci.,1995,128:13-24.
25. Busenberg S N, HadelerK P. Demography and epidemics [J], Math. Biosci.,1990,101:63-74.
26. Wang W D, Ma Z E. Global dynamics of an epidemic model with time delay [J], Nonlinear Analysis: Real World Applications,2002,3(3):365-373.
27. Wang W D. Global behavior of an SEIRS. epidemic model with time delays [J], Applied Mathematics Letters,2002,15(4):423-428.
28. Yoshida N, Hara T. Global stability of a delayed SIR epidemic model with density dependent birth and death rates [J], Journal of Computational and Applied Mathematics,2007,201(2):339-347.
29. Gao S J, Chen L S, Teng Z D. Pulse vaccination of an SEIR epidemic model with time delay [J], Nonlinear Analysis:Real World Applications,2008,9(2):599-607.
30. Zhang T L, Teng Z D. Pulse vaccination delayed SEIRS epidemic model with saturation incidence[J], Applied Mathematical Modeling,2008,32(7):1403-1416.
31. Zhang G S, Wang Y F. Minimal wave speed in a diffusive epidemic model with temporal delay [J], Applied Mathematics and Computation,2007,188(1):275-280.
32. Shulgin B, Stone L, Agur Z. Pulse vaccination strategy in the SIR epidemic model [J], Bull. Math.Biol,1998,60:1123-1148.
33. Stone L, Shulgin B, Agur Z. Theoretical examination of the pulse vaccination policy in the SIR epidemic model [J], Math.Comput.Modelling,2000,31:207-215.
34. Hui J, Chen L. Impulsive vaccination of SIR epidemic models with nonlinear incidence rates [J], Discrete and Continuous Dynamical Systems, series B 2004,4:595-605.
35. D'Onofrio A. On pulse vaccination strategy in the SIR epidemic model with vertical transmission [J], Appl. Math.Letters,2005,18:729-732.
36. Roberts M G, Kao R R. The dynamics of an infectious disease in a population with birth pulses [J], Math.Biol,1998,149:23-36.
37. Lu Z, Chi X, Chen L. The effect of constant and pulse vaccination on SIR epidemic model with horizontal and vertical transmission [J], Math Comput. Model,2002,36(10):1039-1057.
38. Liu W M, Levin S A, Iwasa Y. Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models [J], J. Math. Biol.,1986,23:187-204.
39. Hethcote H W, van den D P. Some epidemiological models with nonlinear incidence [J], J. Math. Biol.,1991,29:271-287.
40. Mena-lorca J, Hethcote H W. Dynamic models of infections disease as regulations of population size [J], J. Math. Biol.,1992,30:693-716.
41.王军,王辉.具有种群动力学和非线性接触率的SIRS流行病模型[J],兰州大学学报(自然科学版),2002,38(1)；22-29.
42.杨建雅,张凤琴.一类具有垂直传染的SIR传染病模型[J],生物数学学报,2006,21(3)；341-344.
43.杨俊元,张桂丽.具有垂直传染和因病死亡的SIR传染病模型[J],运城学院学报,2007,25(5)：21-23.
44. Michael Y. L, John R. G, Wang L C, et al. Global dynamics of a SEIR model with varying total population size [J], Mathematical Biosciences,1999,160:191-213.
45. Fan M, Michael Y. L, Wang K. Global stability of an SEIS epidemic model with recruitment and a varying total population size [J], Mathematical Biosciences,2001,170:199-208.
46. Feng Z, Thieme H R. Endemic models with arbitrarily distributed periods of infection, Ⅰ:General theory [J], SIAM. J. Appl.,2000,61:803.
47. Feng Z, Thieme H R. Endemic models with arbitrarily distributed periods of infection, Ⅱ:Fast disease dynamics and permanent recovery [J], SIAM. J. Appl.,2000,61:983.
48. Li J Q, Ma Z E. Qualitative analysis of SIS epidemic model with vaccination and varying total population size [J], Math.Comp. Model,2002,35:1235-1243.
49. Lizana M, Rivero J. Multiparametric bifurcations for a model in epidemiology [J], J. Math. Biol., 1996,35:21-36.
50. Wu L, Feng Z. Homoclinic bifurcation in an SIQR model for childhood disease [J], J. Diff. Eqns., 2000,168:150-167.
51. Van den D P, Watmongh J. A simple SIS epidemic model with a backward bifurcation [J], J. Math. Biol.,2000,40:525-540.
52. Ruan S G, Wang W D. Dynamic behavior of an epidemic model with a nonlinear incidence rate [J], J. Diff. Eqns.,2003,18:135-163.
53. Beretta E, Hara T, Ma W B, et al. Global asymptotic stability of an SIR epidemic model with distributed time delay [J], Nonlinear Analysis,2001,47:4107-4115.
54.周义仓.具有年龄与病程结构的流行病模型及其渐近性态[D],西安交通大学,1999.
55.刘汉武,徐洁,刘启明.连续接种的具有年龄结构的SIS传染病模型[J],工程数学学报,2002,19：25-29.
56. Iannelli M, Kim M Y, Park E J. Asymptotic behavior for an SIS epidemic model and its approximation [J], Nonlinear Analysis,1999,35:797-814.
57. Miiller J. Optimal vaccination patterns in age-structured populations [J], SIAM J. Appl. Math.,1998, 59:222-241.
58. Manuel D, Monica M B, Antonio S. Analysis of an age-structured predator-prey model with disease in the prey[J], Nonlinear Analysis:Real World Applications,2006,7(4):853-871.
59. Allen L J S, Jones M A, Martin C E. A discrete-time model with vaccination for a measles epidemic [J], Math. Biosci.,1991,105:111-131.
60. Allen L J S, Thrasher D B. The effects of vaccination in an age-dependent model for varicella and Herpes zoster [J], IEEE Trans. On Automatic Control,1998,43:779-789.
61.周毅,娄梅枝,娄洁.一类分年龄阶段的流行病模型研究[J],工程数学学报,2003,3：135-138.
62. Lu Z H, Gao S J, Chen L S. Analysis of an SI epidemic model with nonlinear transmission and stage structure[J], Acta Math. Sci.,2003,4:440-446.
63. Xiao Y N, Chen L S. A SIS epidemic model with stage structure and a delay [J], Acta Math.Appl. Sinica, English Series,2002,18:607-618.
64. Bainov D.D, Simeonov P.S. Impulsive differential equations:periodic solutions and applications [M], New York:Longman Scientific and Technical,1993.
65. Roberts M G, Kao R R. The dynamics of an infectious disease in a population with birth pulse [J], Math. Biosci,1998,149 (1):23-36.
66. Shulgin B, Stone L, Agur Z. Pulse vaccination strategy in the SIR epidemic model [J], Bull. Math. Biol,1998,60(6):1123-1148.
67.靳祯,马知恩.具有连续和脉冲接种的流行病模型[J],华北工学院学报,2003,24：235-243.
68. Gao S J, Chen L S, Sun L H. Dynamic complexities in a seasonal prevention epidemic model with birth pulses [J], Chaos, Solitons and Fractals,2005,26(4):1171-1181.
69. Gao S J, Chen L S, Sun L H. Optimal pulse fishing policy in stage-structured models with birth pulses [J], Chaos, Solitons and Fractals,2005,25(5):1209-1219.
70. Gao S J, Chen L S. The effect of seasonal harvesting on a single-species discrete population model with stage structure and birth pulses [J], Chaos, Solitons and Fractals,2005,24(4):1013-1023.
71. Gakkhar S, Negi K. Pulse vaccination in SIRS epidemic model with non-monotonic incidence rate [J], Chaos, Solitons and Fractals,2008,35(3):626-638.
72. Pang G P, Chen L S. A delayed SIRS epidemic model with pulse vaccination [J], Chaos, Solitons and Fractals,2007,34(5):1629-1635.
73.徐光启.《农政全书》卷四：《玄扈先生井田考》[M].
74. Malthus T R. An essay on the principle of population as it affects the future improvement of society, with remarks on the speculations of M.Godwin, M.Condorcot, and other writers, J.Johnson, London [M],1798.
75.马知恩.种群生态学的数学建模与研究[M],合肥：安徽教育出版社,1996,1-40.
76.徐克.生物数学[M],北京：科学出版社,1999,1-11.
77. Munteanu A, Sole R V. Phenotypic diversity and chaos in a minimal cell model [J], Journal of Theoretical Biology,2006,240(3):434-442.
78. D'Onofrio A. On a family of models of cell division cycle [J], Chaos Solitons and Fractals,2006, 27(5):1205-1212.
79. Furusawa C, Kaneko K. Theory of robustness of irreversible differentiation in a stem cell system: chaos hypothesis [J], Journal of Theoretical Biology,2001,209(4):21,395-416.
80. Sun P, Yang X B. Dynamic behaviors of the Ricker population model under a set of randomized perturbations [J], Mathematical Biosciences,2000,164:147-159.
81. Beckmann M J. On spatial population dynamics [J], Chaos, Solitons and Fractals,2003,18(3): 439-444.
82.荆海英,赵立纯.微生物连续培养模型的溢流量控制[J],生物数学学报,1998,13(5)：599-603.
83.赵立纯,荆海英.微生物连续培养模型的最优溢流量控制[J],生物数学学报,1999,14(1)：32-35.
84. Banasiak J, Lachowicz M, Moszyski M. Topological chaos:When topology meets medicine [J], Applied Mathematics Letters,2003,16(3):303-308.
85. Angelis E D, Delitala M. Modelling complex systems in applied sciences; methods and tools of the mathematical kinetic theory for active particles[J], Mathematical and Computer Modelling,2006, 43(11-12):1310-1328.
86. Autiero M, Smet F D, Claes F,et al. Role of neural guidance signals in blood vessel navigation[J], Cardiovascular Research,2005,65(3):629-638.
87. May R M. Biological populations with non-overlapping generations:stable points, stable cycles and chaos [J], Science,1974,186:645-647.
88. May R M. Simple mathematical models with very complex dynamics [J], Nature,1976,261: 459-467.
89. May R M, Oster G F. Bifurcations and dynamic complexity in simple ecological models [J], American Naturalist,1976,110:573-599.
90. Xiao Y N, Chen L S. Effects of toxicants on a stage-structured population growth model [J], Applied Mathematics and Computations,2001,123(1):63-73.
91. Duan L X, Lu Q S, Yang Z Z, et al. Effects of diffusion on a stage-structured population in a polluted environment [J], Applied Mathematics and Computation,2004,154(2):347-359.
92. Wang W D, Mulone G, Salemi F, et al. Permanence and stability of a stage-structured predator-prey model [J], Journal of Mathematical Analysis and Applications,2001,262(2):499-528.
93. Gao S J, Chen L S. Dynamic complexities in a single-species discrete population model with stage structure and birth pulses [J], Chaos, Solitons and Fractals,2005,23(2):519-527.
94. Wang J, Wang K. The optimal harvesting problems of a stage-structured population [J], Applied Mathematics and Computation,2004,148(1):235-247.
95. Jin Z, Ma Z E, Han M A. Global stability of an SIRS epidemic model with delays [J], Acta Mathematica Scientia,2006,26B(2):291-306.
96. Xu R, Chen L S, Hao F L. Periodic solutions of an n-species Lotka-Volterra type food-chain model with time delays [J], Applied Mathematics and Computation,2005,171(1):511-530.
97.付军,李健全,陈任昭.年龄相关的种群空间扩散系统的广义解与收获控制[J],控制理论与应用,2005,22(4)：588-596.
98. Gakkhar S, Naji R K. Chaos in seasonally perturbed ratio-dependent prey-predator system [J], Chaos, Soliton and Fractals,2003,15:107-118.
99. Kaitala V, Ylikarjula J, Heino M. Dynamic complexities in host-parasitoid interaction[J], Journal of Theoretical Biology,1999,197:331-341.
100. Hsu S B, Hwang T W, Kuang Y. A ratio-dependent food chain model and its applications to biological control [J], Mathematical Biosciences,2003,181:55-83.
101. O'Brien S J, Troyer J L, Roelke M, etc. Plagues and adaptation:Lessons from the Felidae models for SARS and AIDS [J], Biological Conservation,2006,131(2):255-267.
102. Lou J, Ma Z E, Shao Y M, etc. Modelling the antigen and interaction of T-cells presenting cells HIV-1 in vivo [J], Computers and Mathematics with Applications,2004,48:9-33.
103. Pinho M R, Ferreira M M, Ledzewicz U, et al. A model for cancer chemotherapy with state-space constraints [J], Nonlinear Analysis,2005,63(5-7):e2591-e2602.
104. Ware N C, Wyatt M A, Tugenberg T. Adherence, stereotyping and unequal HIV treatment for active users of illegal drugs [J], Social Science & Medicine,2005,61(3):565-576.
105. Ghezzi L L, Piccardi C. PID control of a chaotic system:An application to an epidemiological model [J], Automatica,1997,33(2):181-191.
106. Wang W M, Wang X Q, Lin Y Z. Complicated dynamics of a predator-prey system with Watt-type functional response and impulsive control strategy[J], Chaos, Solitons and Fractals,2008,37(5): 1427-1441.
107. Wang X Q, Wang W M, Lin X M. Chaotic behavior of a Watt-type predator-prey system with impulsive control strategy [J], Chaos, Solitons and Fractals,2008,37(3):706-718.
108.荆海英.可再生性资源开发系统的调控与最优管理[M],沈阳：辽宁人民出版社,2002,166-180.
109. Zhao L C, Zhang Q L, Yang Q C. Optimal impulsive harvesting for fish populations [J], Journal of System Science and Complexity,2003,16(4):466-474.
110.赵立纯,张庆灵,杨启昌.三种群食物链系统的耗散性控制[J],生物数学学报,2003,18(1)：82-92.
111.赵立纯,张庆灵,杨启昌.具有阶段结构单种群系统的诱导控制[J],数学物理学报,2005,25A(5)：710-717.
112.王庭槐,耿艺介.混沌动力学非线性分析方法在生物反馈研究中的应用[J],自然杂志,2004,26(4)：223-226.
113.赵立纯.种群动力学系统控制问题的研究[D],沈阳：东北大学,2003.
114.张化光,王智良,黄玮.混沌系统的控制理论[M],沈阳：东北大学出版社：2003,29-37.
115.黄润生.混沌及其应用[M],武汉：武汉大学出版社,2000,1-10.
116. Lorenz E N. Deterministic nonperiodic flow [J], J. Atmos. Sci.,1963,20(2):130-141.
117. Li T Y, Yorke J A. Period three implies chaos [J], Am. Math. Monthy,1975,82:985-992.
118.傅汝廉,常胜江,温午麒等.人体心脏脉动的混沌动力学研究[J],中华物理医学与康复杂志,1999,21(2)：100-102.
119.裴文江,何振亚,杨绿溪等.发现心脏节律中存在确定性混沌[J],科学通报,2001,46(8)：695-670.
120.王兴元,顾树生.心电动态生理及病理信息的非线性动力学研究[J],中国生物医学工程学报,2000,19(4)：397-403.
121.修志龙,曾安平.微生物细胞连续培养过程中振荡和混沌行为的研究进展[J],生物工程进展,1999,19(6)：58-63.
122. Summers D, Cranford J G, Healey B P. Chaos in periodically forced discrete-time ecosystem models [J], Chaos, Solitons and Fractals,2000,11:2331-2342.
123. Blackmore D, Chen J, Perez J, et al. Dynamical properties of discrete Lotka-Volterra equations [J], Chaos, Solitons and Fractals,2001,12:2553-2568.
124.Braverman E, Kinzebulatov D. On linear perturbations of the Ricker model [J], Mathematical Biosciences,2006,202:323-339.
125. Ruxton G D, Rohani P. Population floors and the persistence of chaos in ecological models [J], Theoretical Population Biology,1998,53:175-183.
126. Casagrandi R, Bolzoni L, Levin S A, et al. The SIRC model and influenza A [J], Mathematical Biosciences,2006,200:152-169.
127.王洪礼,冯剑丰,沈菲,孙景.赤潮藻类非线性动力学模型的分岔及稳定性研究[J],应用数学和力学,2005,26(6)：671-676.
128. Jing Z J, Chang Y, Guo B L. Bifurcation and chaos in discrete FitzHugh-Nagumo system [J], Chaos, Solitons & Fractals,2004,21:701-720.
129.张真,李典谟,查光济.马尾松毛虫种群动态的时间序列分析及复杂性动态研究[J],生态学报,2002,22(7)：1061-1067.
130.朱伟勇,王琰,方自平.猩红热和流脑流行过程的混沌分维研究[J],数学的实践与认识,2001,31(3)： 331-335.
131.邓小炎.一类单种群差分模型的周期解和混沌现象[J],华中农业大学学报,2000,19(4)：403-407.
132.李医民,周燕.一类具有收获系数的单种群模型的混沌分析[J],江苏大学学报(自然科学版),2005,26(4)：324-327.
133.张真,李典谟,张培义等.自然种群中混沌的检测及其在种群动态研究中的意义[J],生态学报,2003,23(10)：1951-1962.
134. Vayenas D V, Pavlou S. Chaotic dynamics of a microbial system of coupled food chains [J], Ecological Modelling,2001,136:285-295.
135. Misra J C, Misra A. Instabilities in single-species and host-parasite systems:period-doubling bifurcations and chaos [J], Computers and Mathematics with Applications,2006,52:525-538.
136. Tang S Y, Chen L S. Chaos in functional response host-parasitoid ecosystem models [J], Chaos, Solitons and Fractals,2002,13:875-884.
137. Xu C L, Boyce M S. Dynamic complexities in a mutual interference host-parasitoid model [J], Chaos, Solitons and Fractals,2005,24:175-182.
138. Jang S R J, Diamond S L. A host-parasitoid interaction with Allee effects on the host [J], Computers and Mathematics with Applications,2007,53:89-103.
139. Jing Z J, Yang J P. Bifurcation and chaos in discrete-time predator-prey system [J], Chaos, Solitons and Fractals,2006,27:259-277.
140. Liu X L, Xiao D M. Complex dynamic behaviors of a discrete-time predator-prey system [J], Chaos, Solitons and Fractals,2007,32:80-94.
141. Letellier C, Aziz-Alaoui M A. Analysis of the dynamics of a realistic ecological model [J], Chaos, Solitons and Fractals,2002,13:95-107.
142. Gakkhar S, Naji R K. Order and chaos in predator to prey ratio-dependent food chain [J], Chaos, Solitons and Fractals,2003,18:229-239.
143. Wang F Y, Pang G P. Chaos and Hopf bifurcation of a hybrid ratio-dependent three species food chain [J], Chaos, Solitons and Fractals,2008,36(5):1366-1376.
144. Naji R K, Balasim A T. Dynamical behavior of a three species food chain model with Beddington-DeAngelis functional response [J], Chaos, Solitons and Fractals,2007,32:1853-1866.
145. Wang K F, Wang W D, Pang H Y, et al. Complex dynamic behavior in a viral model with delayed immune response [J], Physica D,2007,226:197-208.
146.关新平,范正平,陈彩莲等.混沌控制及其在保密通信中的应用[M],北京：国防工业出版社,2002.
147.张树文,陈兰荪.具有脉冲效应和综合害虫控制的捕食系统[J],系统科学与数学,2005,25(3)：264-275.
148. Hui J, Zhu D M. Dynamic complexities for prey-dependent cosumption integrated pest management models with impulsive effects[J], Chaos, Solitons and Fractals,2006,29:233-251.
149.张玉娟.脉冲微分方程在种群生态管理数学模型研究中的应用[D],大连理工大学,2004.
150.贡崇颖,李医民,孙曦浩.肌型血管生物数学模型的自适应Backstepping控制设计[J],生物数学学报,2007,22(3)：503-508.
151.张悦,张庆灵,赵立纯等.一类种群增长模型的反馈线性化控制[J],东北大学学报(自然科学版),2005,10：926-929.
152. Awad E G, Al-Ruzaiza A S. Chaos and adaptive control in two prey, one predator system with nonlinear feedback[J], Chaos, Solitons and Fractals,2007,34(2):443-453.
153. Gomes A A, Manica E, Varriale M C. Applications of chaos control techniques to a three-species food chain [J], Chaos, Solitons and Fractals,2008,36(4):1097-1107.
154.黄润生,黄浩.混沌及其应用[M],武汉：武汉大学出版社,2005,270-312.
155.王光瑞,于熙龄,陈式刚.混沌的控制、同步与利用[M],北京：国防工业出版社,2001,563-571.
156. Ott E, Grebogi C, Yorke J A. Controlling chaos[J], Physics Review Letters,1990,64:1196-1199.
157. Zhang X Y, Shuai Z S, Wang K. Optimal impulsive harvesting for single population [J], Nonlinear Analysis:Real World Applications,2003(4):639-651.
158. Tsui A P M, Jones A J. The control of higher dimensional chaos:comparative results for the chaotic satellite attitude control problem [J], Physica D,2000,135:41-62.
159.王利清,魏学业,温伟刚等.混沌态电流模式Buck-Boost开关变换器的OGY控制[J],信息与控制,2005,34(6)：742-745.
160.王琳,彭建华,陈立宏.改进的OGY法控制超混沌的电路实验研究[J],东北师大学报(自然科学版),2006,38(1)：44-49.
161.薛怀庆.双光子光学双稳系统的混沌控制[J],兰州大学学报(自然科学版),2002,38(3)：27-32.
162.姚洪兴,盛昭瀚.经济混沌模型中一种改进的反馈控制方法[J],系统工程学报,2002,17(6)：507-511.
163.焉德军,刘向东,朱伟勇.六辊UC轧机的OGY混沌控制对策[J],控制与决策,1999,14(2)：156-160.
164.陈国华,李煜,盛昭瀚.基于不同决策规则的产出系统的混沌与控制[J],系统工程理论与实践,2004,5：84-90.
165. Isidori A. Nonlinear Control System [M], Heidelberg:Springer-Verlag Berlin,1985.
166. Chen L Q, Liu Y Z. Chaotic attitude motion of a magnetic rigid spacecraft and its control[J], International Journal of Non-Linear Mechanics,2002,37:493-504.
167.韦笃取,罗晓曙,方锦清等.基于微分几何方法的永磁同步电动机的混沌运动的控制[J],物理学报,2006,55(1)：54-59.
168. Harb A M, Abdel-Jabbar N. Controlling Hopf bifurcation and chaos in a small power system[J], Chaos, Solitons and Fractals,2003,18:1055-1063.
169. Kakmenia F M M, Bowong S, Tchawoua C, et al. Resonance, bifurcation and chaos control in electrostatic transducers with two external periodic forces [J], Physica A,2004,333:87-105.
170.于书敏,赵立纯,张庆灵等.一个三种群食物链系统的非线性控制[J],数学的实践与认识,2005,35(11)：162-171.
171.谭文,王耀南,黄丹等.不确定混沌系统的动态神经网络跟踪控制[J],控制与决策,2004,19(4)：455-458.
172.伍维根,古天祥.混沌系统的非线性反馈跟踪控制[J],物理学报,2000,49(10)：1922-1925.
173.窦春霞,张淑清.基于观测器的模型不确定的耦合时空混沌H∞跟踪控制[J],物理学报,2004,53(12)：4120-4125.
174.刘亚,胡寿松.基于自适应神经网络的不确定非线性系统的模糊跟踪控制[J],控制理论与应用,2004,21(5)：770-775.
175.张家树,肖先赐.连续混沌系统的非线性自适应预测跟踪控制[J],物理学报,2001,50(11)：2092-2096.
176.卢辉斌,李丽香,彭海朋等.一类不确定时滞混沌系统的输出追踪控制[J],仪器仪表学报,2003,24(2)：193-194,210.
177.任海鹏,刘丁.六辊UC轧机轧制过程混沌的轨迹跟踪控制[J],控制理论与应用,2002,19(6)：897-900.
178.李洁,任海鹏.永磁同步电动机中混沌运动的部分解耦控制[J],控制理论与应用,2005,22(4)：637-640.
179.李建芬,李农,林辉.适合传输快变信息信号的混沌调制保密通信[J],物理学报,2004,53(6)：1694-1698.
180.刘扬正,李平.一维混沌系统的非线性反馈控制[J],扬州大学学报(自然科学版),2003,6(3)：27-29.
181.郝百林.从抛物线谈起——一维混沌动力学引论[M],上海：上海科学技术出版社,1993,28.
182. Xu H B, Wang G R, Chen S G. Controlling chaos by a modified straight-line stabilization method [J], The European Physical Journal B,2001,22:65-69.
183.李健全.具有预防接种流行病的模型[D],西安交通大学,2003.
184.徐洁,周义仓.具有比例和脉冲接种的乙肝流行病模型[J],生物数学学报,2004,19(2)：149-155.
185. Fan M, Wang K. Optimal harvesting policy for single population with periodic coefficients [J], Mathematical Biosciences,1998,1(52):165-177.
186.任海鹏,刘丁.BOOST变换器中混沌现象及其控制的仿真研究[J],系统仿真学报,2004,16(11)：2529-2532.
187.陈关荣,胡刚.混沌系统的非线性反馈控制的可控性条件[J],北京师范大学学报(自然科学版),1996,32(2)：208-212.
188.李国辉,周世平,徐得名等.间歇线性反馈控制混沌[J],物理学报,2000,49(11)：2123-2127.
189.刘扬正,李平.利用间歇非线性时滞反馈控制一维Logistic系统的混沌运动[J],南京师范大学学报(自然科学版),2002,25(3)：53-57.
190. Hassan K K. Nonlinear Systems (Second Edition) [M], Upper Saddle River, New Jersey:Prentice Hall,1996.
191.FitzHugh R. Impulses and physiological states in theoretical models of nerve membrane [J], Biophys J.,1961,1:445-466.
192. Rasband S N. Chaotic Dynamics of Nonlinear System [M], New York:Wiley/Interscience,1990, 187-195.
193. Ricardo L R. Complex behavior in a discrete coupled logistic model for the symbiotic interaction of two species [J], Mathematical Biosciences and Engineering,2004,1(2):307-324.
194.王兴元,石其江.二维Logistic映射中的一种新型激变、回滞和分形[J],应用力学学报,2005,24(4)：501-506.
195. Neubert M G, Kot M. The subcritical collapse of predator populations in discrete-time predator-prey models [J], Mathematical Biosciences,1992,110:45-66.
196. Kocak H. Differential and difference equations through computer experiments [J], New York: Springer,1989.
197. Zhang Y, Zhang Q L, Zhao L C, et al. Dynamical behaviors and chaos control in a discrete functional response model[J], Chaos, Solitons and Fractals,2007,34:1318-1327.
198. Celik C. The stability and Hopf bifurcation for a predator-prey system with time delay [J], Chaos, Solitons and Fractals,2008,37(1):87-99.
199. Wang L, Wolkowicz G S K. A delayed chemostat model with general nonmonotone response functions and differential removal rates [J], Journal of Mathematical Analysis and Applications, 2006,321(1):452-468.
200. Choudhury S R. On bifurcations and chaos in predator-prey models with delay [J], Chaos Solitons and Fractals,1992,2(4):393-409.
201. Liao X F, Wong K W, Leung C S, et al. Hopf bifurcation and chaos in a single delayed neuron equation with non-monotonic activation function[J], Chaos,Solitons and Fractals,2001,12(8): 1535-1547.
202. Peng M S. Rich dynamics of discrete delay ecological models [J], Chaos, Solitons and Fractals, 2005,24(5):1279-1285.
203. Sun H J, Cao H J. Bifurcations and chaos of a delayed ecological model [J], Chaos, Solitons and Fractals,2007,33(4):1383-1393.
204. Babloyantz A, Lourenco C, Sepulchre J A. Control of chaos in delay differential equations, in a network of oscillators and in model cortex [J], Physica D:Nonlinear Phenomena,1995,86(1-2): 274-283.
205. Tian Y C, Gao F R. Adaptive control of chaotic continuous-time systems with delay [J], Physica D: Nonlinear Phenomena,1998,117(1-4):1-12.
206. Vladimir E. Bondarenko. Control and'anticontrol' of chaos in an analog neural network with time delay [J], Chaos, Solitons and Fractals,2002,13(1):139-154.