退化时滞微分系统的Hopf分支问题的若干研究
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摘要
在经济、工程、生物等实际系统中，时滞现象和退化现象是非常普遍的现象。关于这两种现象，许多学者做出了很多的成果。但是，据我们所知，有许多系统同时具有退化和时滞的现象，称这类系统为退化时滞微分系统。也有许多系统具有多个时滞，称这类系统为多时滞系统。
实际模型中某些参数的变化的变化会引起解的稳定性的变化，从而产生周期解或者(极限环)，即所谓的分支现象。这种现象在退化系统和生物系统中都很明显。本文主要研究三维退化时滞微分方程和多时滞捕食与食饵系统的稳定性与Hopf分支。我们以时滞τ做为分支参数研究了两类系统的hopf分支现象，通过分析系统的特征超越方程，发现当时滞穿越某些值的时出现了分支。结合利用Hopf分支定理获得了系统的Hopf分支存在的条件；利用中心流型定理和正规形方法分析了Hopf分支的性质，包括分支的方向和分支周期解的稳定性。
第一章叙述了问题的产生背景与意义和本文所要做的工作。
第二章讨论了三维退化时滞微分系统的Hopf分支存在性。
第三章讨论了多时滞中立型捕食与食饵系统的Hopf分支存在性。
第四章研究了多时滞捕食与食饵系统的Hopf分支的性质。
In many practical fields, delay and degenerate phenomena are existing extensively, such as economy, engineering, biology and so on.About these phenomena many results have been obtained. But, as we know, delay and degenerate phenomena appear in many systems together, these systems are called degenerate differential system with delay. And some systems have two delay, we call these systems system with multiple delay.
Some parameters'variation of models will cause the variation of stability of solutions, which can produce periodic solutions(or limit circle). Bifurcaion phenomenon is kind of it. This phenomenon is common in the degenerate differential system and ecolgical system This paper discusses the problems of Hopf bifurcation for three-neuron degenerate differential equations with delay and a neutral predator- prey model with two delay. We take delayτas a bifurcation parameter to investigate the hopf bifucation phenomena of thes two systems. By analyzing the associated characteristic transcendental equations of systems, it is found that Hopf bifurcation occurs when delay pass through a value, and using Hopf bifucation theorem, we obtain one condition for the existence of Hopf bifucation in systems. Furthermore, based on the center manifold theorem and the method of normal form, some results about properties of Hopf bifucation are obtained including the direction of Hopf bifucation and stablity of Hopf bifucating periodic solutions.
In chapter one, the backgrounds and significance of the problems are given. The main work done in this paper is introduced.
In chapter two, discusses the existence of Hopf bifurcation in three-neuron degenerate differential equations with delay.
In chapter three, discusses the existence of Hopf bifurcation in a neutral predatorprdy model with two delay.
In chapter four, studies the characters of Hopf bifucation.
实际模型中某些参数的变化的变化会引起解的稳定性的变化，从而产生周期解或者(极限环)，即所谓的分支现象。这种现象在退化系统和生物系统中都很明显。本文主要研究三维退化时滞微分方程和多时滞捕食与食饵系统的稳定性与Hopf分支。我们以时滞τ做为分支参数研究了两类系统的hopf分支现象，通过分析系统的特征超越方程，发现当时滞穿越某些值的时出现了分支。结合利用Hopf分支定理获得了系统的Hopf分支存在的条件；利用中心流型定理和正规形方法分析了Hopf分支的性质，包括分支的方向和分支周期解的稳定性。
第一章叙述了问题的产生背景与意义和本文所要做的工作。
第二章讨论了三维退化时滞微分系统的Hopf分支存在性。
第三章讨论了多时滞中立型捕食与食饵系统的Hopf分支存在性。
第四章研究了多时滞捕食与食饵系统的Hopf分支的性质。
In many practical fields, delay and degenerate phenomena are existing extensively, such as economy, engineering, biology and so on.About these phenomena many results have been obtained. But, as we know, delay and degenerate phenomena appear in many systems together, these systems are called degenerate differential system with delay. And some systems have two delay, we call these systems system with multiple delay.
Some parameters'variation of models will cause the variation of stability of solutions, which can produce periodic solutions(or limit circle). Bifurcaion phenomenon is kind of it. This phenomenon is common in the degenerate differential system and ecolgical system This paper discusses the problems of Hopf bifurcation for three-neuron degenerate differential equations with delay and a neutral predator- prey model with two delay. We take delayτas a bifurcation parameter to investigate the hopf bifucation phenomena of thes two systems. By analyzing the associated characteristic transcendental equations of systems, it is found that Hopf bifurcation occurs when delay pass through a value, and using Hopf bifucation theorem, we obtain one condition for the existence of Hopf bifucation in systems. Furthermore, based on the center manifold theorem and the method of normal form, some results about properties of Hopf bifucation are obtained including the direction of Hopf bifucation and stablity of Hopf bifucating periodic solutions.
In chapter one, the backgrounds and significance of the problems are given. The main work done in this paper is introduced.
In chapter two, discusses the existence of Hopf bifurcation in three-neuron degenerate differential equations with delay.
In chapter three, discusses the existence of Hopf bifurcation in a neutral predatorprdy model with two delay.
In chapter four, studies the characters of Hopf bifucation.
引文
[1] 郑祖庥，泛函微分方程理论[M]，安徽教育出版社，1994．
[2] Hale.J,Theory of funcational differential equations[M], New York:Springer-verlag,1989.
[3] 蒋威，退化、时滞微分系统[M]，安徽大学出版社，1998．
[4] 蒋威，郑祖庥，退化时滞差分系统的通解[J]．数学研究，1998，31(1)．
[5] 蒋威，退化中立型微分方程的常数变易公式和通解[J]，应用数学学报，1998，21(4)．
[6] Jinang Wei,Zheng Zu-xin,The Solvability of the Degenerate Differential Systems with delay[J],Chinese quarterly Journal of mathematics,2000,15(3).
[7] Zhou Xian-feng,Jiang Wei,Eigenvalue Distribution of Degenerate NFDE with Delay[J],Mathmatica Applicata,2002,15(2):48-51.
[8] 蒋威，退化时滞微分方程的周期解问题[J]．应用数学学报，2003，26(2)．
[9] 蒋威，时变退化时滞微分系统的变易公式[J]．数学年刊，2003,24A(2)：161—166．
[10] Dai,L.,Singular Control Systems,Berlin[M],New York,Springer-Verlag,1989.
[11] B.D.Hassard,N.D.Kazarinoff, Y.H.Wan,Theory and Applicaction of Hopf bifucation[M],Cambridge University Press,Cambridge,1981.
[12] G.Iooss,D.D.Joiseph,Elementary Stability and Bifucaion Theory[M],seconded Springer,New York,1989.
[13] X.Liao,K.Wong,Z.Wu,Hopf bifurcation and stability of periodic solutions for van dri pol equaion with distributed delay[J],Nonlinear Dyn. 2001,(26):23-44.
[14] X.Liao,Z.Wu,J.Yu,Stability swithes and bifurcation analysis of a neural net-workwith continuously delay[J],IEEE Trans.Systems Man Cybernet .-PartA1999,(29):692-696.
[15] J.E.Marsden,M.McCracken,The Hopf Bifurcation and its Application[J], Applied MathematicsSciences,vol.19,Springer,Heidelberg,1976.
[16] K.Murakaimi,Bifurcated periodic solutions for delayed van der Pol equaion[J],Neural Parallel Sci.Comput,1999,(7):1-16.
[17] H.G.Schuster,Deterministic Chaos,Physik-Verlag,Weinheim,1984.
[18] V.Venkatasubramanian,Singularity induced bifurcation and the van der Poloscillator[J] ,IEEE Trans.Circuits System-1994,(I41):765-769.
[19] Wenwu Yu,Jinde.Cao ,Hopf bifurcation and stability of periodic solutions for van der pol equation with time delay[J],Nonlinear Analysis,2005,(362):141-165.
[20] Wenwu Yu,Jinde.Cao , Stability and Hopf analysis on a four neuron BAM neural network with time delays[J], Physics letters , 2006,(351):64-78.
[21] S.Ruan,J,wei, On the zero of transcendental functions with applications to stability of delay differential equations with to delay[J], Dynam.Contin.Discrete Implus.System,2003,(10):863.
[22] Leung.A.,Periodic solutions for a prey-predator delay equation[J],J.Diff. Eqns.,1977,(26),391-403.
[23] Lu,Z.Wang.W.,Global stability for two-species Lotka-Volterra systems with delay[J],J.Diff.Eqns.,1995,(115):173-192.
[24] Beretta.E.,Kuang.Y.,Convergence results in a well-know delayed prey-predator system[J],J.Math.Anal.Appl,1996,(204) :840-853.
[25] Faria.T.,Stability and Bifurcation for a delay prey-predator model and the effect of dissusion[J] ,J.Math.Anal.Appl,2001,(254):433-463.
[26] He.X.,Stability and delays in a prey-predator systems[J], J.Math.Anal.Appl, 1996,(198):355-370.
[27] 帅智圣，苗春梅等，具有庇护所的三种群捕食-食饵模型[J])生物数学学报，2004，19(1)：65-71．
[28] 陈福来，文贤章，具有扩散和放养的时滞竞争系统的正周期解[J]，生物数学学报，2006，21(1)：57-67．
[29] LIU Zhi-jun,On Positive Periodic Solution of a Class of Neutral DelayTwo Species Competitive Systems[J],Journal of Biomathematics,2004,19(1):9-16.
[30] Takeuchi.Y,Cooperative systems theory and global stability of diffusion models[J],Acta Appl Math,1989(19):963-975.
[31] Zheng. J.G,Chen.L,Chen. J,Persistence and perodic orbits for two-species nonautomous diffusion Lotka-Volterra models[J],Math.Comput.Modelling, 1984,20(2) :69-80.
[32] Zheng.J.G,Chen.L.S,Chen.X.D,Persistence and global stability for two-species nonautomous competition Lotka-Volterra with time delay[J],Nonlinear Anal, 1999,(37) 1019-1028.
[33] 王东达，孙纪方，沈景清，具时迟Volterra方程周期解分歧现象[J]，数学年刊，2001,22A(6)：773-782．
[34] 宋永利，韩茂安，魏俊杰，多时滞捕食-食饵系统正平衡点的稳定性及全局Hopf分支[J]，数学年刊，2004,25A(6)：783-790．
[35] 段文英，具时滞的捕食-食饵系统系统的稳定的稳定性与Hopf分支[J]，生物数学学报，2004,19(1)：87-92．
[36] 周玉平，黄迅成，一个三维chemostat竞争系统的Hopf分支和周期解[J]，2006,19(2)：388-394．
[37] 刘婧，郑斯宁，Chemostat系统中的Hopf分支[J]，大连理工大学学报，2002,42(9)：377-390．
[38] Crooke.P.S,Tanner.R.D,Hopf birfucation for a variable yield contionous fermentation model[J],Int.J.Engineering Science,1982,20:439-443.
[39] 王元明，黄迅成，一类捕食者-被捕食者种群模型的Hopf分支问题[J]．甘肃科学学报，2006(1)71—76．
[2] Hale.J,Theory of funcational differential equations[M], New York:Springer-verlag,1989.
[3] 蒋威，退化、时滞微分系统[M]，安徽大学出版社，1998．
[4] 蒋威，郑祖庥，退化时滞差分系统的通解[J]．数学研究，1998，31(1)．
[5] 蒋威，退化中立型微分方程的常数变易公式和通解[J]，应用数学学报，1998，21(4)．
[6] Jinang Wei,Zheng Zu-xin,The Solvability of the Degenerate Differential Systems with delay[J],Chinese quarterly Journal of mathematics,2000,15(3).
[7] Zhou Xian-feng,Jiang Wei,Eigenvalue Distribution of Degenerate NFDE with Delay[J],Mathmatica Applicata,2002,15(2):48-51.
[8] 蒋威，退化时滞微分方程的周期解问题[J]．应用数学学报，2003，26(2)．
[9] 蒋威，时变退化时滞微分系统的变易公式[J]．数学年刊，2003,24A(2)：161—166．
[10] Dai,L.,Singular Control Systems,Berlin[M],New York,Springer-Verlag,1989.
[11] B.D.Hassard,N.D.Kazarinoff, Y.H.Wan,Theory and Applicaction of Hopf bifucation[M],Cambridge University Press,Cambridge,1981.
[12] G.Iooss,D.D.Joiseph,Elementary Stability and Bifucaion Theory[M],seconded Springer,New York,1989.
[13] X.Liao,K.Wong,Z.Wu,Hopf bifurcation and stability of periodic solutions for van dri pol equaion with distributed delay[J],Nonlinear Dyn. 2001,(26):23-44.
[14] X.Liao,Z.Wu,J.Yu,Stability swithes and bifurcation analysis of a neural net-workwith continuously delay[J],IEEE Trans.Systems Man Cybernet .-PartA1999,(29):692-696.
[15] J.E.Marsden,M.McCracken,The Hopf Bifurcation and its Application[J], Applied MathematicsSciences,vol.19,Springer,Heidelberg,1976.
[16] K.Murakaimi,Bifurcated periodic solutions for delayed van der Pol equaion[J],Neural Parallel Sci.Comput,1999,(7):1-16.
[17] H.G.Schuster,Deterministic Chaos,Physik-Verlag,Weinheim,1984.
[18] V.Venkatasubramanian,Singularity induced bifurcation and the van der Poloscillator[J] ,IEEE Trans.Circuits System-1994,(I41):765-769.
[19] Wenwu Yu,Jinde.Cao ,Hopf bifurcation and stability of periodic solutions for van der pol equation with time delay[J],Nonlinear Analysis,2005,(362):141-165.
[20] Wenwu Yu,Jinde.Cao , Stability and Hopf analysis on a four neuron BAM neural network with time delays[J], Physics letters , 2006,(351):64-78.
[21] S.Ruan,J,wei, On the zero of transcendental functions with applications to stability of delay differential equations with to delay[J], Dynam.Contin.Discrete Implus.System,2003,(10):863.
[22] Leung.A.,Periodic solutions for a prey-predator delay equation[J],J.Diff. Eqns.,1977,(26),391-403.
[23] Lu,Z.Wang.W.,Global stability for two-species Lotka-Volterra systems with delay[J],J.Diff.Eqns.,1995,(115):173-192.
[24] Beretta.E.,Kuang.Y.,Convergence results in a well-know delayed prey-predator system[J],J.Math.Anal.Appl,1996,(204) :840-853.
[25] Faria.T.,Stability and Bifurcation for a delay prey-predator model and the effect of dissusion[J] ,J.Math.Anal.Appl,2001,(254):433-463.
[26] He.X.,Stability and delays in a prey-predator systems[J], J.Math.Anal.Appl, 1996,(198):355-370.
[27] 帅智圣，苗春梅等，具有庇护所的三种群捕食-食饵模型[J])生物数学学报，2004，19(1)：65-71．
[28] 陈福来，文贤章，具有扩散和放养的时滞竞争系统的正周期解[J]，生物数学学报，2006，21(1)：57-67．
[29] LIU Zhi-jun,On Positive Periodic Solution of a Class of Neutral DelayTwo Species Competitive Systems[J],Journal of Biomathematics,2004,19(1):9-16.
[30] Takeuchi.Y,Cooperative systems theory and global stability of diffusion models[J],Acta Appl Math,1989(19):963-975.
[31] Zheng. J.G,Chen.L,Chen. J,Persistence and perodic orbits for two-species nonautomous diffusion Lotka-Volterra models[J],Math.Comput.Modelling, 1984,20(2) :69-80.
[32] Zheng.J.G,Chen.L.S,Chen.X.D,Persistence and global stability for two-species nonautomous competition Lotka-Volterra with time delay[J],Nonlinear Anal, 1999,(37) 1019-1028.
[33] 王东达，孙纪方，沈景清，具时迟Volterra方程周期解分歧现象[J]，数学年刊，2001,22A(6)：773-782．
[34] 宋永利，韩茂安，魏俊杰，多时滞捕食-食饵系统正平衡点的稳定性及全局Hopf分支[J]，数学年刊，2004,25A(6)：783-790．
[35] 段文英，具时滞的捕食-食饵系统系统的稳定的稳定性与Hopf分支[J]，生物数学学报，2004,19(1)：87-92．
[36] 周玉平，黄迅成，一个三维chemostat竞争系统的Hopf分支和周期解[J]，2006,19(2)：388-394．
[37] 刘婧，郑斯宁，Chemostat系统中的Hopf分支[J]，大连理工大学学报，2002,42(9)：377-390．
[38] Crooke.P.S,Tanner.R.D,Hopf birfucation for a variable yield contionous fermentation model[J],Int.J.Engineering Science,1982,20:439-443.
[39] 王元明，黄迅成，一类捕食者-被捕食者种群模型的Hopf分支问题[J]．甘肃科学学报，2006(1)71—76．