具有离散和分布时滞的几类生物模型的分支问题研究
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摘要
离散时滞与分布时滞的微分方程被广泛地应用在生物系统当中，用于解释相应系统中具有的稳定性、周期振荡以及混沌等动力学行为。
本文主要利用非线性动力学、泛函微分方程等相关理论和方法，研究离散时滞与分布时滞的非线性生物模型的动力学性质。分析了时滞项对非线性生物模型的一些动力学行为的影响，主要包括系统的稳定性、系统的持久性、局部Hopf分支、全局Hopf分支和两类余维二分支等方面的理论结果。并给出了从模型中得到的数学理论结果代表的生物学意义。
首先，利用中心流形定理、Hassard规范型理论以及Faria和Magalhaes规范型理论，研究了一个具有休眠特征的离散时滞的捕食-被捕食模型的Hopf及Hopf-fold分支。分析了时滞对系统稳定性、持久性、周期解以及混沌的影响。给出了系统产生Hopf分支和Hopf-fold分支的条件，并从中得到了离散时滞的引入可以使系统产生稳定性切换和多周期轨共存等现象。推证出了带有原系统参数在Hopf-fold临界点附近的规范型及分支性质。利用Lyapunov-Razumikhin有界定理等相关知识证明了时滞系统的持久性质。通过数值模拟展示了周期解的稳定性和大范围存在等性质。
其次，研究了具有休眠和脉冲扰动的捕食-被捕食模型的稳定性及分支问题。利用脉冲微分方程理论、小振幅扰动理论和比较原理得到了被捕食者灭绝的周期解的全局渐近稳定条件和系统处于持久状态的条件。利用数值模拟得到了随着脉冲扰动项的增加会导致周期解发生倍周期分支从而导致混沌的现象，分析描述了脉冲扰动项和孵化项对系统固有周期行为的影响。
然后，利用规范型理论和拓扑度等相关理论研究了两个具有分布时滞的向日葵模型的局部Hopf分支和全局Hopf分支性质。利用儒歇定理和辐角原理讨论了呈对数生长的向日葵模型的稳定性及产生局部Hopf分支条件，给出了模型的周期解的近似表达形式。将呈对数生长的向日葵模型的Hopf分支性质与已有的呈线性生长的向日葵模型的Hopf分支性质进行了比较，揭示了周期解的周期变化和近似表达形式之间的差别。分别从理论和数值模拟上，揭示了呈线性生长的向日葵模型的周期解可以大范围存在的现象。
最后，研究了一个带有离散及分布混合时滞的神经元系统的Hopf-pitchfork分支。通过对特征方程的分析，给出了时滞神经元系统发生余维二Hopf-pitchfork分支的条件。利用中心流形定理与Faria和Magalhaes规范型理论得到了系统在Hopf-pitchfork临界点附近的规范型和带有原系统参数的分支集，细化了分支图。从理论和数值模拟两个方面揭示了当时滞项和系数项在临界点附近扰动时，神经元系统会出现两个稳定平衡点共存以及两个稳定平衡点与一个稳定周期轨共存的现象。
The diferential equations with discrete and distributed time delays are widely ap-plied in biological systems for describing biology dynamical behaviors, such as stability,periodic oscillation and chaos of systems.
In this thesis, we study the dynamical behaviors of nonlinear biology models withdiscrete and distributed time-delays by using nonlinear dynamics systems, functional dif-ferential equations and other related theories and methods. We analyze the efects of timedelay on the dynamics behavior of nonlinear biology models, including the stability ofthe system, the permanence of the system, local Hopf bifurcation, global Hopf bifurca-tion, and two types of codimension two bifurcation and other related theoretical results.The corresponding biological results, which is represented by the obtained mathematicaltheoretical results from studying the biological system, are revealed.
First, we investigate Hopf bifurcation and Hopf-fold bifurcation of predator-preymodel with dormancy of predators and discrete delay by using the center manifold theo-rem, the Hassard normal form theory and the Faria and Magalhaes normal form theory.The efects of time delay on the stability, persistence, periodic solution and chaos ofsystem are analyzed. The occurrence conditions of Hopf bifurcation and Hopf-fold bi-furcation are obtained. It shows that time delay in the prey-species growth can lead tothe phenomena of stability switches and the coexistence of multiple periodic orbits. Wededuce the normal form with original parameters in the model and bifurcation propertiesnear the bifurcation point. By using the Lyapunov-Razumikhin theorem and other relatedknowledge, the uniform persistence of predator-prey model with time delay is proved.Numerical simulation shows the stability as well as a large-scale existence of periodicsolution.
Next, we investigate the stability and bifurcation of predator-prey model with dor-mancy of predators and impulsive perturbations. By using the theories of impulsive e-quations, small amplitude perturbation skills and the comparison technique, we get theconditions which guarantee the global asymptotical stability of the prey-eradication peri-odic solution and the permanence of the system. Further, influences of the impulsive per-turbation on the inherent oscillation are studied numerically, which shows rich dynamics,such as period-doubling bifurcation, chaos, and period-halving bifurcation. Moreover, the efects of the impulsive perturbation and hatching rate on the chaos of the system are ob-tained by numerical simulation. In addition, using the theories of the normal form and thetopological degree, we study the local and global Hopf bifurcation properties of two sun-flower equations with distributed delay. Using the Rouche theorem and the principle ofargument, we discuss the occurrence conditions of local Hopf bifurcation of the sunflowermodel with logarithmic growth and give the approximate expression of the periodic so-lution of this model. Local Hopf bifurcation properties between the linear growth modeland the logarithmic growth of the sunflower model are compared. It reveals the diferencebetween period changes of periodic solutions as well as the diference between variousform of approximate expression of the periodic solutions. From theoretical investigationand numerical simulation, we obtain a large-scale existence of periodic solution.
Finally, the Hopf-pitchfork bifurcation of a two-neuron system with discrete and dis-tributed delays is investigated. Through analyzing the characteristic equation, we obtainthe occurrence conditions of Hopf-pitchfork bifurcation. Using the center manifold theo-rem and the Faria and Magalhaes normal form theory, we obtain the normal form and theirunfolding with original parameters of the system near the bifurcation point and refine thebifurcation diagram. It is revealed that when the time delay and the parameter changenear the vicinity of the critical point, some dynamical behaviors of neuron system arefound, such as stable periodic orbit, the coexistence of two stable non-trivial equilibrium,and the coexistence of a stable periodic orbit and two stable equilibrium, by theoreticalinvestigation and numerical simulation.
本文主要利用非线性动力学、泛函微分方程等相关理论和方法，研究离散时滞与分布时滞的非线性生物模型的动力学性质。分析了时滞项对非线性生物模型的一些动力学行为的影响，主要包括系统的稳定性、系统的持久性、局部Hopf分支、全局Hopf分支和两类余维二分支等方面的理论结果。并给出了从模型中得到的数学理论结果代表的生物学意义。
首先，利用中心流形定理、Hassard规范型理论以及Faria和Magalhaes规范型理论，研究了一个具有休眠特征的离散时滞的捕食-被捕食模型的Hopf及Hopf-fold分支。分析了时滞对系统稳定性、持久性、周期解以及混沌的影响。给出了系统产生Hopf分支和Hopf-fold分支的条件，并从中得到了离散时滞的引入可以使系统产生稳定性切换和多周期轨共存等现象。推证出了带有原系统参数在Hopf-fold临界点附近的规范型及分支性质。利用Lyapunov-Razumikhin有界定理等相关知识证明了时滞系统的持久性质。通过数值模拟展示了周期解的稳定性和大范围存在等性质。
其次，研究了具有休眠和脉冲扰动的捕食-被捕食模型的稳定性及分支问题。利用脉冲微分方程理论、小振幅扰动理论和比较原理得到了被捕食者灭绝的周期解的全局渐近稳定条件和系统处于持久状态的条件。利用数值模拟得到了随着脉冲扰动项的增加会导致周期解发生倍周期分支从而导致混沌的现象，分析描述了脉冲扰动项和孵化项对系统固有周期行为的影响。
然后，利用规范型理论和拓扑度等相关理论研究了两个具有分布时滞的向日葵模型的局部Hopf分支和全局Hopf分支性质。利用儒歇定理和辐角原理讨论了呈对数生长的向日葵模型的稳定性及产生局部Hopf分支条件，给出了模型的周期解的近似表达形式。将呈对数生长的向日葵模型的Hopf分支性质与已有的呈线性生长的向日葵模型的Hopf分支性质进行了比较，揭示了周期解的周期变化和近似表达形式之间的差别。分别从理论和数值模拟上，揭示了呈线性生长的向日葵模型的周期解可以大范围存在的现象。
最后，研究了一个带有离散及分布混合时滞的神经元系统的Hopf-pitchfork分支。通过对特征方程的分析，给出了时滞神经元系统发生余维二Hopf-pitchfork分支的条件。利用中心流形定理与Faria和Magalhaes规范型理论得到了系统在Hopf-pitchfork临界点附近的规范型和带有原系统参数的分支集，细化了分支图。从理论和数值模拟两个方面揭示了当时滞项和系数项在临界点附近扰动时，神经元系统会出现两个稳定平衡点共存以及两个稳定平衡点与一个稳定周期轨共存的现象。
The diferential equations with discrete and distributed time delays are widely ap-plied in biological systems for describing biology dynamical behaviors, such as stability,periodic oscillation and chaos of systems.
In this thesis, we study the dynamical behaviors of nonlinear biology models withdiscrete and distributed time-delays by using nonlinear dynamics systems, functional dif-ferential equations and other related theories and methods. We analyze the efects of timedelay on the dynamics behavior of nonlinear biology models, including the stability ofthe system, the permanence of the system, local Hopf bifurcation, global Hopf bifurca-tion, and two types of codimension two bifurcation and other related theoretical results.The corresponding biological results, which is represented by the obtained mathematicaltheoretical results from studying the biological system, are revealed.
First, we investigate Hopf bifurcation and Hopf-fold bifurcation of predator-preymodel with dormancy of predators and discrete delay by using the center manifold theo-rem, the Hassard normal form theory and the Faria and Magalhaes normal form theory.The efects of time delay on the stability, persistence, periodic solution and chaos ofsystem are analyzed. The occurrence conditions of Hopf bifurcation and Hopf-fold bi-furcation are obtained. It shows that time delay in the prey-species growth can lead tothe phenomena of stability switches and the coexistence of multiple periodic orbits. Wededuce the normal form with original parameters in the model and bifurcation propertiesnear the bifurcation point. By using the Lyapunov-Razumikhin theorem and other relatedknowledge, the uniform persistence of predator-prey model with time delay is proved.Numerical simulation shows the stability as well as a large-scale existence of periodicsolution.
Next, we investigate the stability and bifurcation of predator-prey model with dor-mancy of predators and impulsive perturbations. By using the theories of impulsive e-quations, small amplitude perturbation skills and the comparison technique, we get theconditions which guarantee the global asymptotical stability of the prey-eradication peri-odic solution and the permanence of the system. Further, influences of the impulsive per-turbation on the inherent oscillation are studied numerically, which shows rich dynamics,such as period-doubling bifurcation, chaos, and period-halving bifurcation. Moreover, the efects of the impulsive perturbation and hatching rate on the chaos of the system are ob-tained by numerical simulation. In addition, using the theories of the normal form and thetopological degree, we study the local and global Hopf bifurcation properties of two sun-flower equations with distributed delay. Using the Rouche theorem and the principle ofargument, we discuss the occurrence conditions of local Hopf bifurcation of the sunflowermodel with logarithmic growth and give the approximate expression of the periodic so-lution of this model. Local Hopf bifurcation properties between the linear growth modeland the logarithmic growth of the sunflower model are compared. It reveals the diferencebetween period changes of periodic solutions as well as the diference between variousform of approximate expression of the periodic solutions. From theoretical investigationand numerical simulation, we obtain a large-scale existence of periodic solution.
Finally, the Hopf-pitchfork bifurcation of a two-neuron system with discrete and dis-tributed delays is investigated. Through analyzing the characteristic equation, we obtainthe occurrence conditions of Hopf-pitchfork bifurcation. Using the center manifold theo-rem and the Faria and Magalhaes normal form theory, we obtain the normal form and theirunfolding with original parameters of the system near the bifurcation point and refine thebifurcation diagram. It is revealed that when the time delay and the parameter changenear the vicinity of the critical point, some dynamical behaviors of neuron system arefound, such as stable periodic orbit, the coexistence of two stable non-trivial equilibrium,and the coexistence of a stable periodic orbit and two stable equilibrium, by theoreticalinvestigation and numerical simulation.
引文
[1] May R M. Time Delay Versus Stability in Population Models with Two or ThreeTrophic Levels[J]. Ecology,1973,54(2):315–325.
[2]魏俊杰,王洪滨,蒋卫华.时滞微分方程的分支理论及应用[M].北京:科学出版社,2012:1–81.
[3]马知恩.种群生态学的数学建摸与研究[M].合肥:安徽教育出版社,1996:1–58.
[4] Gurney M S, Blythe S P, Nisbet R M. Nicholson’s Blowflies Revisited[J]. Nature,1980,287:17–21.
[5] Nicholson A J. Compensatory Reactions of Populations to Stresses, and TheirEvolutionary Significance[J]. Australian Journal of Zoology,1954,2(1):1–8.
[6]秦元勋.带有时滞的动力系统的运动稳定性[M].北京:科学出版社,1989:117–132.
[7] Cooke K L, Grossman E. Discrete Delay, Distributed Delay and Stability Switch-es[J]. Journal of Mathematical Analysis and Applications,1982,86:592–627.
[8] Cooke K L, van den Driessche P. On Zeroes of Some Transcendental Equations[J].Funkcialaj Ekvacioj,1986,29(1):77–90.
[9] Ruan S, Wei J. On the Zeros of Transcendental Functions with Applications to Sta-bility of Delayed Diferential Equations with Two Delays[J]. Dynamics of Contin-uous, Discrete Impulsive Systems Series A: Mathematical Analysis,2003,10:863–874.
[10] Lee M S, Hsu C S. On the Decomposition Method of Stability Analysis for Re-tarded Dynamical Systems[J]. SIAM Journal on Control,1969,7(2):242–259.
[11] Hu G, Li W, Yan X. Hopf Bifurcations in a Predator-prey System with MultipleDelays[J]. Chaos, Solitons&Fractals,2009,42(2):1273–1285.
[12]陈珊珊.几类时滞反应扩散方程的动力学性质分析[D].哈尔滨:哈尔滨工业大学博士学位论文,2012:75–88.
[13] Beretta E, Kuang Y. Geometric Stability Switch Criteria in Delay Diferential Sys-tems with Delay Dependant Parameters[J]. SIAM Journal on Mathematical Anal-ysis,2002,33:1144–1165.
[14] Hsu S B. A Survey of Constructing Lyapunov Functions for Mathematical Modelsin Population Biology[J]. Taiwanese Journal of Mathematics,2005,9(2):151–173.
[15] Li M Y, Shuai Z, Wang C. Global Stability of Multi-group Epidemic Models withDistributed Delays[J]. Journal of Mathematical Analysis and Applications,2010,361(1):38–47.
[16] Guo H, Li M Y, Shuai Z. A Graph-theoretic Approach to the Method of GlobalLyapunov Functions[J]. Proceedings of the American Mathematical Society,2008,136(8):2793–2802.
[17] Li M Y, Shuai Z. Global-stability Problem for Coupled Systems of DiferentialEquations on Networks[J]. Journal of Diferential Equations,2010,248(1):1–20.
[18] Li M Y, Shu H. Global Dynamics of an In-host Viral Model with IntracellularDelay[J]. Bulletin of Mathematical Biology,2010,72(6):1492–1505.
[19]蒋卫华.时滞微分方程的分支分析博士学位论文[D].哈尔滨:哈尔滨工业大学,2005:2–30.
[20] Hassard B D, Kazarinof N D, Wan Y. Theory and Applications of Hopf Bifurca-tion[M]. Cambridge: Cambridge University Press,1981:36–120.
[21] Faria T, Magalhaes L. Normal Forms for Retarded Functional Diferential Equationwith Parameters and Applications to Hopf Bifurcation[J]. Applied Mathematicsand Computation,1995,122:181–200.
[22] Jiang W, Yuan Y. Bogdanov-Takens Singularity in Van der Pol’s Oscillator withDelayed Feedback[J]. Physica D,2007,227:149–161.
[23] Wang H, Jiang W. Hopf-pitchfork Bifurcation in van der Pol’s Oscillator with Non-linear Delayed Feedback[J]. Journal of Mathematical Analysis and Applications,2010,368:9–18.
[24] Jiang W, Wang H. Hopf-transcritical Bifurcation in Retarded Functional Diferen-tial Equations[J]. Nonlinear Analysis,2010,73:3626–3640.
[25] Jiang W, Niu B. On the Coexistence of Periodic or Quasi-periodic Oscillations neara Hopf-pitchfork Bifurcation in NFDE[J]. Communications in Nonlinear Scienceand Numerical Simulation,2013,18:464–477.
[26]牛犇.中立型泛函微分方程的余维二分支研究[D].哈尔滨:哈尔滨工业大学博士学位论文,2012:45–85.
[27]丁宇婷.时滞微分方程的高余维分支的规范型约化研究[D].哈尔滨:哈尔滨工业大学博士学位论文,2012:45–85.
[28]魏俊杰,黄启昌.泛函微分方程分支理论发展概况[J].科学通报,1997,42(24):2581–2584.
[29] Nussbaum R D. Periodic Solutions of Some Nonlinear Autonomous Function-al Diferential Equations[J]. Annali di Matematica Pura ed Applicata,1974,101(1):263–306.
[30] Browder F E. Nonlinear Monotone Operators and Convex Sets in Banach S-paces[J]. Bulletin of the American Mathematical Society,1965,71(5):780–785.
[31] Krawcewicz W, Wu J. Theory of Degrees with Applications to Bifurcations andDiferential Equations[M]. Canadian: Wiley-Interscience,1997:1–100.
[32] Wu J H. Symmetric Functional Diferential Equations and Neural Networks withMemory[J]. Transactions of the American Mathematical Society,1998,350:4799–4838.
[33] Engelborghs K, Luzyanina T, Samaey G. DDE-BIFTOOL V.2.00: A MatlabPackage for Bifurcation Analysis of Delay Diferential Equations[J]. Techni-cal Report TW330, Department of Computer Science, Department of Comput-er Science, Katholieke Universiteit Leuven, Belgium,2001:1–40[2001]. http://www.cs.kuleuven.ac.be/～koen/delay.
[34] Engelborghs K, Luzyanina T, Roose D. Numerical Bifurcation Analysis of Delay-Diferential Equations Using DDE-BIFTOOL[J]. ACM Transactions on Mathe-matical Software,2002,28(1):1–21.
[35] Mil’man V D, Myshkis A D. On the Stability of Motion in the Presence of Impuls-es[J]. Siberian Mathematical Journal,1960,1:233–237.
[36] Simeonov P S, Bainov D D. Stability with Respect to Part of the Variables in Sys-tems with Impulse Efect[J]. Journal of Mathematical Analysis and Applications,1986,117(1):247–263.
[37] Bainov D D, Simeonov P S. Impulsive Diferential Equations: Periodic Solutionsand Applications[M]. Harlow: Longman Scientific and Technical,1993.
[38] Fu X, Qi J, Liu Y. The Existence of Periodic Orbits for Nonlinear Impulsive Dif-ferential systems[J]. Communications in Nonlinear Science and Numerical Simu-lation,1999,4(1):50–53.
[39] Qi J, Fu X. Existence of Limit Cycle of Impulsive Diferential Equations with Im-pulses at Variable Times[J]. Nonlinear Analysis: Theory, Methods&Applications,2001,44:345–353.
[40] Ballinger G, Liu X. Existence, Uniqueness and Boundedness Results for ImpulsiveDelay Diferential Equations[J]. Applicable Analysis,2000,74:71–93.
[41] Liu X, Ballinger G. Existence and Continuability of Solutions for Diferential E-quations with Delays and State-dependent Impulses[J]. Nonlinear Analysis,2002,51:633–647.
[42] Alwan M S, Liu X, Xie W. Existence, Continuation, and Uniqueness Problems ofStochastic Impulsive Systems with Time Delay[J]. Journal of the Franklin Institute,2010,347(7):1317–1333.
[43] Liu X, Chen L. Complex Dynamics of Holling Type II Lotka-Volterra Predator-prey System with Impulsive Perturbations on the Predator[J]. Chaos, Solitons&Fractals,2003,16(2):311–320.
[44] Jiao J, Meng X, Chen L. Harvesting Policy for a Delayed Stage-structured HollingII Predator-prey Model with Impulsive Stocking Prey[J]. Chaos, Solitons&Frac-tals,2009,41(1):103–112.
[45] He D, Huang W, Xu Q. The Dynamic Complexity of an Impulsive Holling IIPredator-prey Model with Mutual Interference[J]. Applied Mathematical Mod-elling,2010,34(9):2654–2664.
[46] Nie L, Teng Z, Hu L, et al. Qualitative Analysis of a Modified Leslie-Gower andHolling-type II Predator-prey Model with State Dependent Impulsive Efects[J].Nonlinear Analysis: Real World Applications,2010,11(3):1364–1373.
[47] Wang X, Yu H, Zhong S, et al. Analysis of Mathematics and Dynamics in a FoodWeb System with Impulsive Perturbations and Distributed Time Delay[J]. Journalof Computational and Applied Mathematics,2010,34(12):3850–38631.
[48] Liu Z, Wu J, Chen Y, et al. Impulsive Perturbations in a Periodic Delay Diferen-tial Equation Model of Plankton Allelopathy[J]. Nonlinear Analysis: Real WorldApplications,2010,11(1):432–445.
[49] Lakmeche A, Arino O. Bifurcation of Non Trivial Periodic Solutions of Impul-sive Diferential Equations Arising Chemotherapeutic Treatment[J]. Dynamics ofContinuous, Discrete and Impulsive Systems,2000,7:265–280.
[50] Holling C S. The Functional Response of Predator to Prey Density and its Role inMimicry and Population Regulation[J]. Memoirs of the Entomological Society ofCanada,1965,45:5–60.
[51] Kuwamura M, Nakazawa T, Ogawa T. A Minimum Model of Prey-predator Sys-tem with Dormancy of Predators and the Paradox of Enrichment[J]. Journal ofMathematical Biology,2009,58:459–479.
[52] Rosenzweig M L. Paradox of Enrichment: Destabilization of Exploitation Ecosys-tems in Ecological Time[J]. Science,1971,171:385–387.
[53] Chen K. Uniqueness of a Limit Cycle for a Predator-Prey System[J]. SIAM Journalon Mathematical Analysis,1981,12(4):541–548.
[54] McCauley E, Nisbet R M, Murdoch W W, et al. Large-amplitude Cycles of Daphniaand Its Algal Prey in Enriched Environments[J]. Nature,1999,402:653–656.
[55] Kuwamura M, Chiba H. Mixed-mode Oscillations and Chaos in a Prey-predatorSystem with Dormancy of Predators[J]. Chaos,2009,190(4):043121–043121.
[56] Wangersky P J, Cunningham W J. Time Lag in Prey-predator Population Model-s[J]. Ecology,1957,38(1):136–139.
[57] Martin A, Ruan S. Predator-prey Models with Delay and Prey Harvesting[J]. Math-ematical Biology,2001,43(3):247–267.
[58] Ruan S. On Nonlinear Dynamics of Predator-prey Models with Discrete Delay[J].Mathematical Modelling of Natural Phenomena,2009,4(2):140–188.
[59] Gourley S A, Kuang Y. A Stage Structured Predator-prey Model and its Depen-dence on Maturation Delay and Death Rate[J]. Journal of Mathematical Biology,2004,49:188–200.
[60] Yi F, Wei J, Shi J. Bifurcation and Spatiotemporal Patterns in a Homeoge-neous Difusive Predator-prey System[J]. Journal of Diferential Equations,2009,246(5):1944–1977.
[61] Ruan S, Wei J. On the Zeros of a Third Degree Exponential Polynomial withApplications to a Delayed Model for the Control of Testosterone Secretion[J]. IMAJournal of Mathematics Applied in Medicine and Biology,2001,18:41–52.
[62] Li M Y, Shu H. Multiple Stable Periodic Oscillations in a Mathematical Model ofCTL Response to HTLV-I Infection[J]. Bulletin of Mathematical Biology,2011,73:1774–1793.
[63] Nindjin A F, Aziz-Alaoui M A. Persistence and Global Stability in a DelayedLeslie-Gower Type Three Species Food Chain[J]. Journal of Mathematical Analy-sis,2008,340:340–357.
[64] Li M Y, Graef J R, Wang L, et al. Global Dynamics of a SEIR Model with VaryingTotal Population Size[J]. Mathematical Biosciences,1999,160(2):191–213.
[65] Wang W, Ma Z. Harmless Delays for Uniform Persistence[J]. Journal of Mathe-matical Analysis and Applications,1991,158:256–268.
[66] Freedmen H I, Ruan S. Uniform Persistence in Functional Diferential Equation-s[J]. Journal of Diferential Equations,1995,115:173–192.
[67] Lou Y, Zhao X Q. A Reaction-difusion Malaria Model with Incubation Period inthe Vector Population[J]. Journal of Mathematical Biology,2011,62(4):543–568.
[68] Hale J K. Theory of Functional Diferential Equations[M]. New York: Springer-Verlag,1977:11–190.
[69] Guckenheimer J, Holmes P. Nonlinear Oscillations, Dynamical Systems, and Bi-furcations of Vector Fields[M]. New York: Springer,1983.
[70] Carr J. Applications of Centre Manifold Theory[M]. New York: Springer,1981.
[71] Beninca E, Huisman J, Heerkloss R, et al. Chaos in a Long-term Experiment witha Plankton Community[J]. Nature,2008,451:822–825.
[72] Beninca E, Johnk K D, Heerkloss R, et al. Coupled Predator-prey Oscillations in aChaotic Food Web[J]. Nature,2009,12:1367–1378.
[73] Lakshmikantham V, Bainov D D, Simeonov P S. Theory of Impulsive DiferentialEquations[M]. Singapore: World Scientific,1989.
[74] Israelsson D, Johnsson A. A Theory for Circumnutations in Helianthus Annuus[J].Physiologia Plantarum,1967,20:957–976.
[75] Zachariassen E, Johnsson A, Brown A H, et al. Influence of the G-force onthe Circumnutations of Sunflower Hypocotyls[J]. Physiologia Plantarum,1987,70(3):447–452.
[76] Stockus A. Basic Assumptions and Comparison of Three Gravitropic ResponseModels[J]. Advances in Space Research,1994,14(8):145–148.
[77] Stockus A, Moore D. Comparison of Plant and Fungal Gravitropic ResponsesUsing Imitational Modelling[J]. Plant, Cell&Environment,1996,19(7):787–800.
[78] Brown A H, Chapman D K, Lewis R F, et al. Circumnutations of SunflowerHypocotyls in Satellite Orbit[J]. Plant physiology,1990,94(1):233–238.
[79] Luijtelaer J. The Rotating Solar Boiler[J]. Master Thesis March,2006,31.
[80] Kalogirou S. Recent Patents in Solar Energy Collectors and Applications[J]. Re-cent Patents on Engineering,2007,1:23–33.
[81] Somolinos A S. Periodic Solutions of the Sunflower Equation: x¨+(a/r) x˙+(b/r) sin x(t r)=0[J]. Quarterly of Applied Mathematics,1978,35:465–478.
[82] Kulenovic M R S, Ladas G. Oscillations of the Sunflower Equation[J]. Quarterlyof Applied Mathematics,1988,46:23–28.
[83] Lizana M. Existence and Partial Characterization of the Global Attractor for theSunflower Equation[J]. Journal of Mathematical Analysis and Applications,1995,190(1):1–11.
[84] Lizana M. Global Analysis of the Sunflower Equation with Small Delay[J]. Non-linear Analysis: Theory, Methods&Applications,1999,36(6):697–706.
[85] Zhang C, Zheng B. Hopf Bifurcation in Numerical Approximation of the Sunflow-er Equation[J]. Journal of Applied Mathematics and Computing,2006,22:113–124.
[86]魏俊杰.向日葵方程的Hopf分支[J].应用数学学报,1996,19(1):73–79.
[87] Li J. Hopf Bifurcation of the Sunflower Equation[J]. Nonlinear Analysis: RealWorld Applications,2009,10(4):2574–2580.
[88]文贤章,王志成,庾建设.向日葵方程的分支[J].湖南大学学报,1999,26:1–6.
[89]魏俊杰,黄启昌.以滞量为参数的向日葵方程的Hopf分支[J].科学通报,1995,40(3):198–200.
[90] Johnsson A. Geotropic Responses in Helianthus and Their Dependence on theAuxin Ratio-. with a Refined Mathematical Description of the Course of GeotropicMovements[J]. Physiologia Plantarum,1971,24(3):419–125.
[91] Kazarinof N D, Wan Y H. Hopf Bifurcation and Stability of Periodic Solutions ofDiferential-diference and Integro-diferential Equations[J]. Journal of the Instituteof Mathematics and Its Applications,1978,21:461–477.
[92]曲颖.几类具时滞连续动力系统的稳定性和分支分析[D].哈尔滨:哈尔滨工业大学博士学位论文,2010:48–60.
[93] Hopfield J J. Neurons with Graded Response Have Collective Computational Prop-erties Like Those of Two-state Neurons[J]. Proceedings of the National Academyof Sciences,1984,81:3088–3092.
[94] Zeng Z, Huang D, Wang Z. Pattern Memory Analysis Based on Stability Theoryof Cellular Neural Networks[J]. Applied Mathematical Modelling,2008,32:112–121.
[95] Plaza J, Plaza A, Perez R, et al. On the Use of Small Training Sets for NeuralNetwork-based Characterization of Mixed Pixels in Remotely Sensed Hyperspec-tral Images[J]. Pattern Recognition,2009,42(11):3032–3045.
[96] Ahmadkhanlou F, Adeli H. Optimum Cost Design of Reinforced Concrete SlabsUsing Neural Dynamics Model[J]. Engineering Applications of Artificial Intelli-gence,2005,18(1):65–72.
[97] Amari S I, Cichocki A. Adaptive Blind Signal Processing-neural Network Ap-proaches[J]. Proceedings of the IEEE,1998,86:2026–2048.
[98] Wei J, Ruan S. Stability and Bifurcation in a Neural Network Model with TwoDelays[J]. Physica D,1999,130:255–272.
[99] Campbell S A, Ruan S, Wei J. Qualitative Analysis of a Neural Network Modelwith Multiple Time Delays[J]. International Journal of Bifurcation and Chaos,1999,9:1585–1595.
[100] Faria T. On a Planar System Modeling a Neuron Network with Memory[J]. Journalof Diferential Equations,2000,168:129–149.
[101] Gopalsamy K, He X. Stability in Asymmetric Hopfield Nets with TransmissionDelays[J]. Physica D,1994,76:344–358.
[102] Yang Y, Jin Y. Stability and Bifurcation in a Simplified Five-neuron BAM NeuralNetwork with Delays[J]. Chaos, Solitons&Fractals,2009,42(4):2357–2363.
[103] Han Y, Song Y. Stability and Hopf Bifurcation in a Three-neuron UnidirectionalRing with Distributed Delays[J]. Nonlinear Dynamics,2012,69:357–370.
[104] Wang L, Zou X. Convergence of Discrete-time Neural Networks with Delays[J].International Journal of Qualitative Theory of Diferential Equations and Applica-tions,2008,2:24–37.
[105] Zhang C, Zheng B, Wang L. Multiple Hopf Bifurcations of Symmetric BAM Neu-ral Network Model with Delay[J]. Applied Mathematics Letters,2009,22:616–622.
[106] Wu J, Zivari-Piran H, Hunter J D, et al. Projective Clustering Using Neural Net-works with Adaptive Delay and Signal Transmission Loss[J]. Neural Computation,2011,23:1568–1604.
[107] Song Y. Spatio-temporal Patterns of Hopf Bifurcating Periodic Oscillations ina Pair of Identical Tri-neuron Network Loops[J]. Communications in NonlinearScience and Numerical Simulation,2012,17:943–952.
[108] Li Y, Jiang W. Hopf and Bogdanov-Takens Bifurcations in a Coupled FitzHugh-Nagumo Neural System with Delay[J]. Nonlinear Dynamics,2011,65:161–173.
[109] Ding Y, Jiang W, Yu P. Hopf-zero Bifurcation in a Generalized Gopalsamy NeuralNetwork Model[J]. Nonlinear Dynamics,2012,70(2):1037–1050.
[110] Dong T, Liao X, Huang T, et al. Hopf-pitchfork Bifurcation in an Inertial Two-neuron System with Time Delay[J]. Neurocomputing,2012,97:223–232.
[111] He X, Li C, Huang T, et al. Codimension Two Bifurcation in a Delayed NeuralNetwork with Unidirectional Coupling[J]. Nonlinear Analysis: Real World Appli-cations,2013,14:1191–1202.
[112] Tao D, Liao X. Bogdanov-Takens Bifurcation in a Tri-neuron BAM Neural Net-work Model with Multiple Delays[J]. Nonlinear Dynamics,2013,71(3):583–595.
[113] Du Y, Xu R, Liu Q. Stability and Bifurcation Analysis for a Neural Network Modelwith Discrete and Distributed Delays[J]. Mathematical Methods in the AppliedSciences,2013,36(1):49–59.
[114] Ruan S, Filfil R S. Dynamics of a Two-neuron System with Discrete and Distribut-ed Delays[J]. Physica D: Nonlinear Phenomena,2004,191:323–342.
[115] Bi P, Hu Z. Hopf Bifurcation and Stability for a Neural Network Model[J]. AppliedMathematics and Computation,2012,218:6748–6761.
[116] Kuznetsov Y A. Elements of Applied Bifurcation Theory[M]. New York: Springer-Verlag,1998.
[117] Zhang S, Chen L. Chaos in Three Species Food Chain System with ImpulsivePerturbations[J]. Chaos, Solitons&Fractals,2005,24:73–83.
[118] Zhao M, Wang X, Yu H, et al. Dynamics of an Ecological Model with ImpulsiveControl Strategy and Distributed Time Delay[J]. Mathematics and Computers inSimulation,2012,82(8):1432–1444.
[2]魏俊杰,王洪滨,蒋卫华.时滞微分方程的分支理论及应用[M].北京:科学出版社,2012:1–81.
[3]马知恩.种群生态学的数学建摸与研究[M].合肥:安徽教育出版社,1996:1–58.
[4] Gurney M S, Blythe S P, Nisbet R M. Nicholson’s Blowflies Revisited[J]. Nature,1980,287:17–21.
[5] Nicholson A J. Compensatory Reactions of Populations to Stresses, and TheirEvolutionary Significance[J]. Australian Journal of Zoology,1954,2(1):1–8.
[6]秦元勋.带有时滞的动力系统的运动稳定性[M].北京:科学出版社,1989:117–132.
[7] Cooke K L, Grossman E. Discrete Delay, Distributed Delay and Stability Switch-es[J]. Journal of Mathematical Analysis and Applications,1982,86:592–627.
[8] Cooke K L, van den Driessche P. On Zeroes of Some Transcendental Equations[J].Funkcialaj Ekvacioj,1986,29(1):77–90.
[9] Ruan S, Wei J. On the Zeros of Transcendental Functions with Applications to Sta-bility of Delayed Diferential Equations with Two Delays[J]. Dynamics of Contin-uous, Discrete Impulsive Systems Series A: Mathematical Analysis,2003,10:863–874.
[10] Lee M S, Hsu C S. On the Decomposition Method of Stability Analysis for Re-tarded Dynamical Systems[J]. SIAM Journal on Control,1969,7(2):242–259.
[11] Hu G, Li W, Yan X. Hopf Bifurcations in a Predator-prey System with MultipleDelays[J]. Chaos, Solitons&Fractals,2009,42(2):1273–1285.
[12]陈珊珊.几类时滞反应扩散方程的动力学性质分析[D].哈尔滨:哈尔滨工业大学博士学位论文,2012:75–88.
[13] Beretta E, Kuang Y. Geometric Stability Switch Criteria in Delay Diferential Sys-tems with Delay Dependant Parameters[J]. SIAM Journal on Mathematical Anal-ysis,2002,33:1144–1165.
[14] Hsu S B. A Survey of Constructing Lyapunov Functions for Mathematical Modelsin Population Biology[J]. Taiwanese Journal of Mathematics,2005,9(2):151–173.
[15] Li M Y, Shuai Z, Wang C. Global Stability of Multi-group Epidemic Models withDistributed Delays[J]. Journal of Mathematical Analysis and Applications,2010,361(1):38–47.
[16] Guo H, Li M Y, Shuai Z. A Graph-theoretic Approach to the Method of GlobalLyapunov Functions[J]. Proceedings of the American Mathematical Society,2008,136(8):2793–2802.
[17] Li M Y, Shuai Z. Global-stability Problem for Coupled Systems of DiferentialEquations on Networks[J]. Journal of Diferential Equations,2010,248(1):1–20.
[18] Li M Y, Shu H. Global Dynamics of an In-host Viral Model with IntracellularDelay[J]. Bulletin of Mathematical Biology,2010,72(6):1492–1505.
[19]蒋卫华.时滞微分方程的分支分析博士学位论文[D].哈尔滨:哈尔滨工业大学,2005:2–30.
[20] Hassard B D, Kazarinof N D, Wan Y. Theory and Applications of Hopf Bifurca-tion[M]. Cambridge: Cambridge University Press,1981:36–120.
[21] Faria T, Magalhaes L. Normal Forms for Retarded Functional Diferential Equationwith Parameters and Applications to Hopf Bifurcation[J]. Applied Mathematicsand Computation,1995,122:181–200.
[22] Jiang W, Yuan Y. Bogdanov-Takens Singularity in Van der Pol’s Oscillator withDelayed Feedback[J]. Physica D,2007,227:149–161.
[23] Wang H, Jiang W. Hopf-pitchfork Bifurcation in van der Pol’s Oscillator with Non-linear Delayed Feedback[J]. Journal of Mathematical Analysis and Applications,2010,368:9–18.
[24] Jiang W, Wang H. Hopf-transcritical Bifurcation in Retarded Functional Diferen-tial Equations[J]. Nonlinear Analysis,2010,73:3626–3640.
[25] Jiang W, Niu B. On the Coexistence of Periodic or Quasi-periodic Oscillations neara Hopf-pitchfork Bifurcation in NFDE[J]. Communications in Nonlinear Scienceand Numerical Simulation,2013,18:464–477.
[26]牛犇.中立型泛函微分方程的余维二分支研究[D].哈尔滨:哈尔滨工业大学博士学位论文,2012:45–85.
[27]丁宇婷.时滞微分方程的高余维分支的规范型约化研究[D].哈尔滨:哈尔滨工业大学博士学位论文,2012:45–85.
[28]魏俊杰,黄启昌.泛函微分方程分支理论发展概况[J].科学通报,1997,42(24):2581–2584.
[29] Nussbaum R D. Periodic Solutions of Some Nonlinear Autonomous Function-al Diferential Equations[J]. Annali di Matematica Pura ed Applicata,1974,101(1):263–306.
[30] Browder F E. Nonlinear Monotone Operators and Convex Sets in Banach S-paces[J]. Bulletin of the American Mathematical Society,1965,71(5):780–785.
[31] Krawcewicz W, Wu J. Theory of Degrees with Applications to Bifurcations andDiferential Equations[M]. Canadian: Wiley-Interscience,1997:1–100.
[32] Wu J H. Symmetric Functional Diferential Equations and Neural Networks withMemory[J]. Transactions of the American Mathematical Society,1998,350:4799–4838.
[33] Engelborghs K, Luzyanina T, Samaey G. DDE-BIFTOOL V.2.00: A MatlabPackage for Bifurcation Analysis of Delay Diferential Equations[J]. Techni-cal Report TW330, Department of Computer Science, Department of Comput-er Science, Katholieke Universiteit Leuven, Belgium,2001:1–40[2001]. http://www.cs.kuleuven.ac.be/～koen/delay.
[34] Engelborghs K, Luzyanina T, Roose D. Numerical Bifurcation Analysis of Delay-Diferential Equations Using DDE-BIFTOOL[J]. ACM Transactions on Mathe-matical Software,2002,28(1):1–21.
[35] Mil’man V D, Myshkis A D. On the Stability of Motion in the Presence of Impuls-es[J]. Siberian Mathematical Journal,1960,1:233–237.
[36] Simeonov P S, Bainov D D. Stability with Respect to Part of the Variables in Sys-tems with Impulse Efect[J]. Journal of Mathematical Analysis and Applications,1986,117(1):247–263.
[37] Bainov D D, Simeonov P S. Impulsive Diferential Equations: Periodic Solutionsand Applications[M]. Harlow: Longman Scientific and Technical,1993.
[38] Fu X, Qi J, Liu Y. The Existence of Periodic Orbits for Nonlinear Impulsive Dif-ferential systems[J]. Communications in Nonlinear Science and Numerical Simu-lation,1999,4(1):50–53.
[39] Qi J, Fu X. Existence of Limit Cycle of Impulsive Diferential Equations with Im-pulses at Variable Times[J]. Nonlinear Analysis: Theory, Methods&Applications,2001,44:345–353.
[40] Ballinger G, Liu X. Existence, Uniqueness and Boundedness Results for ImpulsiveDelay Diferential Equations[J]. Applicable Analysis,2000,74:71–93.
[41] Liu X, Ballinger G. Existence and Continuability of Solutions for Diferential E-quations with Delays and State-dependent Impulses[J]. Nonlinear Analysis,2002,51:633–647.
[42] Alwan M S, Liu X, Xie W. Existence, Continuation, and Uniqueness Problems ofStochastic Impulsive Systems with Time Delay[J]. Journal of the Franklin Institute,2010,347(7):1317–1333.
[43] Liu X, Chen L. Complex Dynamics of Holling Type II Lotka-Volterra Predator-prey System with Impulsive Perturbations on the Predator[J]. Chaos, Solitons&Fractals,2003,16(2):311–320.
[44] Jiao J, Meng X, Chen L. Harvesting Policy for a Delayed Stage-structured HollingII Predator-prey Model with Impulsive Stocking Prey[J]. Chaos, Solitons&Frac-tals,2009,41(1):103–112.
[45] He D, Huang W, Xu Q. The Dynamic Complexity of an Impulsive Holling IIPredator-prey Model with Mutual Interference[J]. Applied Mathematical Mod-elling,2010,34(9):2654–2664.
[46] Nie L, Teng Z, Hu L, et al. Qualitative Analysis of a Modified Leslie-Gower andHolling-type II Predator-prey Model with State Dependent Impulsive Efects[J].Nonlinear Analysis: Real World Applications,2010,11(3):1364–1373.
[47] Wang X, Yu H, Zhong S, et al. Analysis of Mathematics and Dynamics in a FoodWeb System with Impulsive Perturbations and Distributed Time Delay[J]. Journalof Computational and Applied Mathematics,2010,34(12):3850–38631.
[48] Liu Z, Wu J, Chen Y, et al. Impulsive Perturbations in a Periodic Delay Diferen-tial Equation Model of Plankton Allelopathy[J]. Nonlinear Analysis: Real WorldApplications,2010,11(1):432–445.
[49] Lakmeche A, Arino O. Bifurcation of Non Trivial Periodic Solutions of Impul-sive Diferential Equations Arising Chemotherapeutic Treatment[J]. Dynamics ofContinuous, Discrete and Impulsive Systems,2000,7:265–280.
[50] Holling C S. The Functional Response of Predator to Prey Density and its Role inMimicry and Population Regulation[J]. Memoirs of the Entomological Society ofCanada,1965,45:5–60.
[51] Kuwamura M, Nakazawa T, Ogawa T. A Minimum Model of Prey-predator Sys-tem with Dormancy of Predators and the Paradox of Enrichment[J]. Journal ofMathematical Biology,2009,58:459–479.
[52] Rosenzweig M L. Paradox of Enrichment: Destabilization of Exploitation Ecosys-tems in Ecological Time[J]. Science,1971,171:385–387.
[53] Chen K. Uniqueness of a Limit Cycle for a Predator-Prey System[J]. SIAM Journalon Mathematical Analysis,1981,12(4):541–548.
[54] McCauley E, Nisbet R M, Murdoch W W, et al. Large-amplitude Cycles of Daphniaand Its Algal Prey in Enriched Environments[J]. Nature,1999,402:653–656.
[55] Kuwamura M, Chiba H. Mixed-mode Oscillations and Chaos in a Prey-predatorSystem with Dormancy of Predators[J]. Chaos,2009,190(4):043121–043121.
[56] Wangersky P J, Cunningham W J. Time Lag in Prey-predator Population Model-s[J]. Ecology,1957,38(1):136–139.
[57] Martin A, Ruan S. Predator-prey Models with Delay and Prey Harvesting[J]. Math-ematical Biology,2001,43(3):247–267.
[58] Ruan S. On Nonlinear Dynamics of Predator-prey Models with Discrete Delay[J].Mathematical Modelling of Natural Phenomena,2009,4(2):140–188.
[59] Gourley S A, Kuang Y. A Stage Structured Predator-prey Model and its Depen-dence on Maturation Delay and Death Rate[J]. Journal of Mathematical Biology,2004,49:188–200.
[60] Yi F, Wei J, Shi J. Bifurcation and Spatiotemporal Patterns in a Homeoge-neous Difusive Predator-prey System[J]. Journal of Diferential Equations,2009,246(5):1944–1977.
[61] Ruan S, Wei J. On the Zeros of a Third Degree Exponential Polynomial withApplications to a Delayed Model for the Control of Testosterone Secretion[J]. IMAJournal of Mathematics Applied in Medicine and Biology,2001,18:41–52.
[62] Li M Y, Shu H. Multiple Stable Periodic Oscillations in a Mathematical Model ofCTL Response to HTLV-I Infection[J]. Bulletin of Mathematical Biology,2011,73:1774–1793.
[63] Nindjin A F, Aziz-Alaoui M A. Persistence and Global Stability in a DelayedLeslie-Gower Type Three Species Food Chain[J]. Journal of Mathematical Analy-sis,2008,340:340–357.
[64] Li M Y, Graef J R, Wang L, et al. Global Dynamics of a SEIR Model with VaryingTotal Population Size[J]. Mathematical Biosciences,1999,160(2):191–213.
[65] Wang W, Ma Z. Harmless Delays for Uniform Persistence[J]. Journal of Mathe-matical Analysis and Applications,1991,158:256–268.
[66] Freedmen H I, Ruan S. Uniform Persistence in Functional Diferential Equation-s[J]. Journal of Diferential Equations,1995,115:173–192.
[67] Lou Y, Zhao X Q. A Reaction-difusion Malaria Model with Incubation Period inthe Vector Population[J]. Journal of Mathematical Biology,2011,62(4):543–568.
[68] Hale J K. Theory of Functional Diferential Equations[M]. New York: Springer-Verlag,1977:11–190.
[69] Guckenheimer J, Holmes P. Nonlinear Oscillations, Dynamical Systems, and Bi-furcations of Vector Fields[M]. New York: Springer,1983.
[70] Carr J. Applications of Centre Manifold Theory[M]. New York: Springer,1981.
[71] Beninca E, Huisman J, Heerkloss R, et al. Chaos in a Long-term Experiment witha Plankton Community[J]. Nature,2008,451:822–825.
[72] Beninca E, Johnk K D, Heerkloss R, et al. Coupled Predator-prey Oscillations in aChaotic Food Web[J]. Nature,2009,12:1367–1378.
[73] Lakshmikantham V, Bainov D D, Simeonov P S. Theory of Impulsive DiferentialEquations[M]. Singapore: World Scientific,1989.
[74] Israelsson D, Johnsson A. A Theory for Circumnutations in Helianthus Annuus[J].Physiologia Plantarum,1967,20:957–976.
[75] Zachariassen E, Johnsson A, Brown A H, et al. Influence of the G-force onthe Circumnutations of Sunflower Hypocotyls[J]. Physiologia Plantarum,1987,70(3):447–452.
[76] Stockus A. Basic Assumptions and Comparison of Three Gravitropic ResponseModels[J]. Advances in Space Research,1994,14(8):145–148.
[77] Stockus A, Moore D. Comparison of Plant and Fungal Gravitropic ResponsesUsing Imitational Modelling[J]. Plant, Cell&Environment,1996,19(7):787–800.
[78] Brown A H, Chapman D K, Lewis R F, et al. Circumnutations of SunflowerHypocotyls in Satellite Orbit[J]. Plant physiology,1990,94(1):233–238.
[79] Luijtelaer J. The Rotating Solar Boiler[J]. Master Thesis March,2006,31.
[80] Kalogirou S. Recent Patents in Solar Energy Collectors and Applications[J]. Re-cent Patents on Engineering,2007,1:23–33.
[81] Somolinos A S. Periodic Solutions of the Sunflower Equation: x¨+(a/r) x˙+(b/r) sin x(t r)=0[J]. Quarterly of Applied Mathematics,1978,35:465–478.
[82] Kulenovic M R S, Ladas G. Oscillations of the Sunflower Equation[J]. Quarterlyof Applied Mathematics,1988,46:23–28.
[83] Lizana M. Existence and Partial Characterization of the Global Attractor for theSunflower Equation[J]. Journal of Mathematical Analysis and Applications,1995,190(1):1–11.
[84] Lizana M. Global Analysis of the Sunflower Equation with Small Delay[J]. Non-linear Analysis: Theory, Methods&Applications,1999,36(6):697–706.
[85] Zhang C, Zheng B. Hopf Bifurcation in Numerical Approximation of the Sunflow-er Equation[J]. Journal of Applied Mathematics and Computing,2006,22:113–124.
[86]魏俊杰.向日葵方程的Hopf分支[J].应用数学学报,1996,19(1):73–79.
[87] Li J. Hopf Bifurcation of the Sunflower Equation[J]. Nonlinear Analysis: RealWorld Applications,2009,10(4):2574–2580.
[88]文贤章,王志成,庾建设.向日葵方程的分支[J].湖南大学学报,1999,26:1–6.
[89]魏俊杰,黄启昌.以滞量为参数的向日葵方程的Hopf分支[J].科学通报,1995,40(3):198–200.
[90] Johnsson A. Geotropic Responses in Helianthus and Their Dependence on theAuxin Ratio-. with a Refined Mathematical Description of the Course of GeotropicMovements[J]. Physiologia Plantarum,1971,24(3):419–125.
[91] Kazarinof N D, Wan Y H. Hopf Bifurcation and Stability of Periodic Solutions ofDiferential-diference and Integro-diferential Equations[J]. Journal of the Instituteof Mathematics and Its Applications,1978,21:461–477.
[92]曲颖.几类具时滞连续动力系统的稳定性和分支分析[D].哈尔滨:哈尔滨工业大学博士学位论文,2010:48–60.
[93] Hopfield J J. Neurons with Graded Response Have Collective Computational Prop-erties Like Those of Two-state Neurons[J]. Proceedings of the National Academyof Sciences,1984,81:3088–3092.
[94] Zeng Z, Huang D, Wang Z. Pattern Memory Analysis Based on Stability Theoryof Cellular Neural Networks[J]. Applied Mathematical Modelling,2008,32:112–121.
[95] Plaza J, Plaza A, Perez R, et al. On the Use of Small Training Sets for NeuralNetwork-based Characterization of Mixed Pixels in Remotely Sensed Hyperspec-tral Images[J]. Pattern Recognition,2009,42(11):3032–3045.
[96] Ahmadkhanlou F, Adeli H. Optimum Cost Design of Reinforced Concrete SlabsUsing Neural Dynamics Model[J]. Engineering Applications of Artificial Intelli-gence,2005,18(1):65–72.
[97] Amari S I, Cichocki A. Adaptive Blind Signal Processing-neural Network Ap-proaches[J]. Proceedings of the IEEE,1998,86:2026–2048.
[98] Wei J, Ruan S. Stability and Bifurcation in a Neural Network Model with TwoDelays[J]. Physica D,1999,130:255–272.
[99] Campbell S A, Ruan S, Wei J. Qualitative Analysis of a Neural Network Modelwith Multiple Time Delays[J]. International Journal of Bifurcation and Chaos,1999,9:1585–1595.
[100] Faria T. On a Planar System Modeling a Neuron Network with Memory[J]. Journalof Diferential Equations,2000,168:129–149.
[101] Gopalsamy K, He X. Stability in Asymmetric Hopfield Nets with TransmissionDelays[J]. Physica D,1994,76:344–358.
[102] Yang Y, Jin Y. Stability and Bifurcation in a Simplified Five-neuron BAM NeuralNetwork with Delays[J]. Chaos, Solitons&Fractals,2009,42(4):2357–2363.
[103] Han Y, Song Y. Stability and Hopf Bifurcation in a Three-neuron UnidirectionalRing with Distributed Delays[J]. Nonlinear Dynamics,2012,69:357–370.
[104] Wang L, Zou X. Convergence of Discrete-time Neural Networks with Delays[J].International Journal of Qualitative Theory of Diferential Equations and Applica-tions,2008,2:24–37.
[105] Zhang C, Zheng B, Wang L. Multiple Hopf Bifurcations of Symmetric BAM Neu-ral Network Model with Delay[J]. Applied Mathematics Letters,2009,22:616–622.
[106] Wu J, Zivari-Piran H, Hunter J D, et al. Projective Clustering Using Neural Net-works with Adaptive Delay and Signal Transmission Loss[J]. Neural Computation,2011,23:1568–1604.
[107] Song Y. Spatio-temporal Patterns of Hopf Bifurcating Periodic Oscillations ina Pair of Identical Tri-neuron Network Loops[J]. Communications in NonlinearScience and Numerical Simulation,2012,17:943–952.
[108] Li Y, Jiang W. Hopf and Bogdanov-Takens Bifurcations in a Coupled FitzHugh-Nagumo Neural System with Delay[J]. Nonlinear Dynamics,2011,65:161–173.
[109] Ding Y, Jiang W, Yu P. Hopf-zero Bifurcation in a Generalized Gopalsamy NeuralNetwork Model[J]. Nonlinear Dynamics,2012,70(2):1037–1050.
[110] Dong T, Liao X, Huang T, et al. Hopf-pitchfork Bifurcation in an Inertial Two-neuron System with Time Delay[J]. Neurocomputing,2012,97:223–232.
[111] He X, Li C, Huang T, et al. Codimension Two Bifurcation in a Delayed NeuralNetwork with Unidirectional Coupling[J]. Nonlinear Analysis: Real World Appli-cations,2013,14:1191–1202.
[112] Tao D, Liao X. Bogdanov-Takens Bifurcation in a Tri-neuron BAM Neural Net-work Model with Multiple Delays[J]. Nonlinear Dynamics,2013,71(3):583–595.
[113] Du Y, Xu R, Liu Q. Stability and Bifurcation Analysis for a Neural Network Modelwith Discrete and Distributed Delays[J]. Mathematical Methods in the AppliedSciences,2013,36(1):49–59.
[114] Ruan S, Filfil R S. Dynamics of a Two-neuron System with Discrete and Distribut-ed Delays[J]. Physica D: Nonlinear Phenomena,2004,191:323–342.
[115] Bi P, Hu Z. Hopf Bifurcation and Stability for a Neural Network Model[J]. AppliedMathematics and Computation,2012,218:6748–6761.
[116] Kuznetsov Y A. Elements of Applied Bifurcation Theory[M]. New York: Springer-Verlag,1998.
[117] Zhang S, Chen L. Chaos in Three Species Food Chain System with ImpulsivePerturbations[J]. Chaos, Solitons&Fractals,2005,24:73–83.
[118] Zhao M, Wang X, Yu H, et al. Dynamics of an Ecological Model with ImpulsiveControl Strategy and Distributed Time Delay[J]. Mathematics and Computers inSimulation,2012,82(8):1432–1444.