Z_4等变系统的极限环个数和一类四点边值问题
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摘要
本文第一章为引言,主要内容是介绍所研究课题的来源,现状,以及本文的研究方法和主要结论.
第二章主要研究近哈密顿系统在Z_4等变七次扰动下产生极限环的个数.利用Hopf分支和异宿环分支的方法,扰动可得16个极限环,并且给出了它们的分布.本章中我们应用了最新的理论和方法,涉及相当复杂的计算.
第三章主要研究近哈密顿系统在Z_4等变五次扰动下产生极限环的个数及近哈密顿系统在Z_4等变三次扰动下产生极限环的个数,利用第二章的有关结果和方法,并应用了一些技巧,分别得到13个和5个极限环,并且给出了它们的分布.
第四章主要研究了方程(u丨¨)+q(t)f(t)=0,t∈(0,1),边值条件为(?)(0)=0,u(1)=a_1u(ξ)+a_2u(η),其中0＜ξ,η＜1,a_1+a_2＜1.应用锥上的不动点定理可得正解存在.本章主要创新是将已有的一类三点边值问题复杂化至四点边值问题,三点边值问题的所得结果在四点边值问题下同样成立.
As an introduction,in the first chapter we introduce the background of our research and main topics that we will study in the following chapters.We also give a description of our methods and results detained in this thesis in the first chapter.
In the second chapter,our main purpose is to concern with the number of limit cycles of a near-Hamiltonian system under Z_4-equivariant septic perturbation.Using the methord of Hopf and heteroclinic bifurcation,we found that the perturbed system can have 16 limit cycles.In this chapter,we use the latest theorems and methods, concerning with quite complicated computation,and the distributions are given.
In the third chapter,we study the number of limit cycles of a near-Hamiltonian system under Z_4-equivariant quintic perturbation and the number of limit cycles of a near-Hamiltonian system under Z_4-equivariant cubic perturbation.Using the methods and results of the second chapter,then by some skills,we found that the perturbed systems respectively have 13 and 5 limit cycles,the distributions are given.
In the fourth chapter,we study equation(u|¨)+q(t)f(t) = 0,t∈(0,1) with boundary conditions u(0) = 0,u(1) = a_u(ξ) + a_2u(η),where 0 <ξ,η< 1,a_1 + a_2 < 1.The existence result of positive solution is obtained by applying the fixed point theorem in cones.The approaches developed here extend the ideas and techniques derived in recent literaures.The main innovative point of this chapter,we complicate a class of threepoint boundary problem into four-point boundary problem,the results of three-point boundary problem still holds in four-point boundary problem.
第二章主要研究近哈密顿系统在Z_4等变七次扰动下产生极限环的个数.利用Hopf分支和异宿环分支的方法,扰动可得16个极限环,并且给出了它们的分布.本章中我们应用了最新的理论和方法,涉及相当复杂的计算.
第三章主要研究近哈密顿系统在Z_4等变五次扰动下产生极限环的个数及近哈密顿系统在Z_4等变三次扰动下产生极限环的个数,利用第二章的有关结果和方法,并应用了一些技巧,分别得到13个和5个极限环,并且给出了它们的分布.
第四章主要研究了方程(u丨¨)+q(t)f(t)=0,t∈(0,1),边值条件为(?)(0)=0,u(1)=a_1u(ξ)+a_2u(η),其中0＜ξ,η＜1,a_1+a_2＜1.应用锥上的不动点定理可得正解存在.本章主要创新是将已有的一类三点边值问题复杂化至四点边值问题,三点边值问题的所得结果在四点边值问题下同样成立.
As an introduction,in the first chapter we introduce the background of our research and main topics that we will study in the following chapters.We also give a description of our methods and results detained in this thesis in the first chapter.
In the second chapter,our main purpose is to concern with the number of limit cycles of a near-Hamiltonian system under Z_4-equivariant septic perturbation.Using the methord of Hopf and heteroclinic bifurcation,we found that the perturbed system can have 16 limit cycles.In this chapter,we use the latest theorems and methods, concerning with quite complicated computation,and the distributions are given.
In the third chapter,we study the number of limit cycles of a near-Hamiltonian system under Z_4-equivariant quintic perturbation and the number of limit cycles of a near-Hamiltonian system under Z_4-equivariant cubic perturbation.Using the methods and results of the second chapter,then by some skills,we found that the perturbed systems respectively have 13 and 5 limit cycles,the distributions are given.
In the fourth chapter,we study equation(u|¨)+q(t)f(t) = 0,t∈(0,1) with boundary conditions u(0) = 0,u(1) = a_u(ξ) + a_2u(η),where 0 <ξ,η< 1,a_1 + a_2 < 1.The existence result of positive solution is obtained by applying the fixed point theorem in cones.The approaches developed here extend the ideas and techniques derived in recent literaures.The main innovative point of this chapter,we complicate a class of threepoint boundary problem into four-point boundary problem,the results of three-point boundary problem still holds in four-point boundary problem.
引文
[1]Hilbert,D.,Mathematische Probleme,ArchivdeMath.u Phys.,1:213-237(1901).
[2]S.Smale,Mathematical problems for the next century.Math.Intelligencer,20:7-15(1998).
[3]W.H.Yao,P.Yu,Bifurcation of small limit cycles in Z_5-equivariant planar vector fields of order 5.J.Math.Anal.Appl.,328:400-413(2007).
[4]H.S.Y.Chan,K.W.Chung,J.Li,Bifurcations of limit cycles in a Z_3-equivariant planar vector field of degree 5.Int.J.Bifurcat.Chaos,11:2287-98(2001).
[5]H.S.Y.Chan,K.W.Chung,J.Li,Bifurcations of limit cycles in a Z_q-equivariant planar vector field of degree 5.In:Proc Int Conf Foundations of Computational Mathematics in Honor of Professor Steve Smale's 70th Birthday,2002,p.61-83.
[6]J.Li,H.S.Y.Chan,K.W.Chung,Investigations of bifurcations of limit cycles in Z_2-equivariant planar vector fields of degree 5.Int.J.Bifurcat.Chaos,12:2137-2157(2002).
[7]J.Li,Chaos and Melnikov method.Chongqing University Press,1989.
[8]P.Yu,M.Han,Small limit cycles bifurcating from fine points in cubic order Z_2-equivariant vector fields.Chaos Solitons Fractals,24:329-348(2005).
[9]P.Yu,M.Han,Y.Yuan,Analysis on limit cycles of Z_q-equivariant polynomial vector fields with degree 3 or 4.J.Math.Anal.Appl.,322:51-65(2006).
[10]Y.H.Wu,M.Han,X.L.Liu,On the study of limit cycles of a cubic polynomials system under Z_4-equivariant quintic perturbation.Chaos,Solitons and Fractals,24:999-1012(2005).
[11]T.H.Zhang,M.Han,H.Zang,Bifurcations of limit cycles for a cubic Hamiltonian system under quartic perturbations.Chaos,Solitons and Fractals,22:1127-1138(2004).
[12]J.Li,H.S.Y.Chan,K.W.Chung,Bifurcations of limit cycles in a Z_6-equivariant planar vector field of degree.5.Sci.China.Ser.A,45(7):817-826(2002).
[13]H.J.Cao,Z.R.Liu,Bifurcation set and distribution of limit cycles for a class of cubic Hamiltonian system with higher-order perturbed terms.Chaos,Solitons and Fractals,11:2293-2304(2000).
[14]Y.Hou,M.Han,Melnikov functions for planar near-Hamiltonian systems and Hopf bifurcations,J.Shanghai Normal University(Natural Sciences),35:1-10(2006).
[15]M.Han,J.Yang,A.Tarta and Y.Gao,Limit cycles near homoclinic and heteroclinic loops,J.Dyn.Diff.Equat.,20:923-944(2008).
[16]M.Han,T.Zhang,H.Zang,Bifurcation and limit cycles forming compound eyes in Z_q-equivariant perturbed Hamiltonian system.Science in China,36(11):1267-1278(2006).
[17]M.Han,Z.Wang,H.Zang,Limit cycles by Hopf and homoclinic bifurcations for near-Hamiltonian systems.Chinese Ann.Math.Ser.A,28:679-690(2007)(in Chinese).
[18]J.Li.Hilbert's 16th problem and bifurcations of planar polynomial vector fields.Int.J.Bifurcat.Chaos,13(1):47-106(2003).
[19]J.Li,Z.Liu,Bifurcation set and limit cycles forming compound eyes in a perturbed Hamiltonian system.Publ.Math.,35:487-506(1991).
[20]J.Li,H.S.Y.Chan,K.W.Chung,Investigations of bifurcations of limit cycles in Z_2-equivariant planar vector fields of degree 5.Int.J.Bifurcat.Chaos,12:2137-2157(2002).
[21]M.Han,Z.Zhang,Cyclicity 1 and 2 conditions for a 2-polycycle of integrable systems on the plane.J.Differ.Equat.,155:245-261(1999).
[22]W.G Ge,Boundary value problems for nonlinear ordinary differential equations,Science press,2007.
[23]W.G Ge,J.L Ren,A extension of Mawhin's continuation theorem and its application to boundary value problem with a p-laplacian,Nonlinear Anal.,58,477-488(2004).
[24]Z.B Bai,B.J Huang,W.G ge,The iterative solutions for some fourth-order p-Laplace equation boundary value prolems,Applied Math.Lett.,19,8-14(2006).
[25]J.Hale,Functional Differential Equations,Springer - Verlag,1971.
[26]J.Hale,Theory of Functional Differential Equations,Springer - Verlag,1977.
[27]J.Hale and S.M.V.Lunel,Introduction to Functional Differential Equations,Springer -Verlag,1993.
[28]V.Lakshmikantham,D.D.Bainov,and P.S.Simeonov,Theory of impulsive differential equations,World Scientific,Singapore,1989.
[29]S.Hu and V.Lakshmikantham,Periodic boundary value problems for the second impulsive.differential equations,Nonlinear Anal.,1:75-85(1989).
[30]R.Y Ma,Existence theorems for a second-order three-point boundary value problems,J.Math.Appli.Anal.,212:430-442(1997).
[31]R.Y Ma,Multiple results for a three-point boundary value problems at resonance,Nonlinear Anal.,53:777-789(2003).
[32]R.Y Ma,Nonlocal problem of nonliear ordinary differential equation,Science press,2004.
[33]R.Y Ma,Positive.solutions of a nonlinear three-point boundary-value problem,Elec.Journal of Equations,34,1-8(1998).
[34]H.L Ma,R.Y Ma,The existence of solutions of second order value problem for ordinary differential equation,Journal of Southwest China Normal University(Natural Science),1000-5471(2005).
[35]R.Y Ma,Existence theorems for a second order m-point boundary value problem,J.Math.Anal.Appl.,211,545-555(1997).
[36]C.P.Gupta,A Sharper condition for the solvability of a three-point second order boundary value problem,J.Math.Anal.Appl.,205,586-579(1997).
[37]韩茂安,平面近哈密顿系统极限环分支的Melnikov函数方法.2007暑期讲习班讲义.
[38]M.Han,Introduction to Limit Cycle Bifurcations for near-Hamiltonian Systems.
[39]J.Jiang,J.Zhang,M.Han,Limit cycles for a class of quintic near-Hamiltonian systems near a nilpotent center,preprint,2007.
[40]R.Roussarie,On the number of limit cycles which appear by perturbation of separatrix loop of planar vector fields,Bol.Soc.Bras.Mat.,17:67-101(1986).
[41]I.D.Iliev,On the limit cycles obtainable from polynomial perturbations of the Bogdanov-Takens Hamiltonian,Israel J.Math.115:269-284(2000).
[42]B.Li,Z.Zhang,A Note on a Result of G.S.Petrov about the Weakened 16th Hilbert Problem,Journal of Mathmatical Analysis and Applications.190:489-516(1995).
[43]P.Mardesic,The number of limit cycles of polynomial deformations of Hamiltonian vector field,Ergodic Theory Dynamical Systems.10:523-529(1990).
[44]G.S.Petrov,The Chebyshev property of elliptic integrals,Functional Anal.Appl.22:72-73(1988).
[45]Y.Wu,M.Han.On the study of limit cycles of the generalized Rayleigh-Lienard oscillator.Internat.J.Bifur.ChaosAppl.Sci.Engrg.14:8,2905-2914(2004).
[46]P.Yu,M.Han,Twelve limit cycles in a cubic case of the 16th Hilbert problem.Int.J.Bifurcations and Chaos,15:7,2191-29-05(2005).
[47]T.Zhang,H.Zang,M.Han.Bifurcations of limit cycles in a cubic system.Chaos Solitons Fractals 20:3,629-638(2004).
[2]S.Smale,Mathematical problems for the next century.Math.Intelligencer,20:7-15(1998).
[3]W.H.Yao,P.Yu,Bifurcation of small limit cycles in Z_5-equivariant planar vector fields of order 5.J.Math.Anal.Appl.,328:400-413(2007).
[4]H.S.Y.Chan,K.W.Chung,J.Li,Bifurcations of limit cycles in a Z_3-equivariant planar vector field of degree 5.Int.J.Bifurcat.Chaos,11:2287-98(2001).
[5]H.S.Y.Chan,K.W.Chung,J.Li,Bifurcations of limit cycles in a Z_q-equivariant planar vector field of degree 5.In:Proc Int Conf Foundations of Computational Mathematics in Honor of Professor Steve Smale's 70th Birthday,2002,p.61-83.
[6]J.Li,H.S.Y.Chan,K.W.Chung,Investigations of bifurcations of limit cycles in Z_2-equivariant planar vector fields of degree 5.Int.J.Bifurcat.Chaos,12:2137-2157(2002).
[7]J.Li,Chaos and Melnikov method.Chongqing University Press,1989.
[8]P.Yu,M.Han,Small limit cycles bifurcating from fine points in cubic order Z_2-equivariant vector fields.Chaos Solitons Fractals,24:329-348(2005).
[9]P.Yu,M.Han,Y.Yuan,Analysis on limit cycles of Z_q-equivariant polynomial vector fields with degree 3 or 4.J.Math.Anal.Appl.,322:51-65(2006).
[10]Y.H.Wu,M.Han,X.L.Liu,On the study of limit cycles of a cubic polynomials system under Z_4-equivariant quintic perturbation.Chaos,Solitons and Fractals,24:999-1012(2005).
[11]T.H.Zhang,M.Han,H.Zang,Bifurcations of limit cycles for a cubic Hamiltonian system under quartic perturbations.Chaos,Solitons and Fractals,22:1127-1138(2004).
[12]J.Li,H.S.Y.Chan,K.W.Chung,Bifurcations of limit cycles in a Z_6-equivariant planar vector field of degree.5.Sci.China.Ser.A,45(7):817-826(2002).
[13]H.J.Cao,Z.R.Liu,Bifurcation set and distribution of limit cycles for a class of cubic Hamiltonian system with higher-order perturbed terms.Chaos,Solitons and Fractals,11:2293-2304(2000).
[14]Y.Hou,M.Han,Melnikov functions for planar near-Hamiltonian systems and Hopf bifurcations,J.Shanghai Normal University(Natural Sciences),35:1-10(2006).
[15]M.Han,J.Yang,A.Tarta and Y.Gao,Limit cycles near homoclinic and heteroclinic loops,J.Dyn.Diff.Equat.,20:923-944(2008).
[16]M.Han,T.Zhang,H.Zang,Bifurcation and limit cycles forming compound eyes in Z_q-equivariant perturbed Hamiltonian system.Science in China,36(11):1267-1278(2006).
[17]M.Han,Z.Wang,H.Zang,Limit cycles by Hopf and homoclinic bifurcations for near-Hamiltonian systems.Chinese Ann.Math.Ser.A,28:679-690(2007)(in Chinese).
[18]J.Li.Hilbert's 16th problem and bifurcations of planar polynomial vector fields.Int.J.Bifurcat.Chaos,13(1):47-106(2003).
[19]J.Li,Z.Liu,Bifurcation set and limit cycles forming compound eyes in a perturbed Hamiltonian system.Publ.Math.,35:487-506(1991).
[20]J.Li,H.S.Y.Chan,K.W.Chung,Investigations of bifurcations of limit cycles in Z_2-equivariant planar vector fields of degree 5.Int.J.Bifurcat.Chaos,12:2137-2157(2002).
[21]M.Han,Z.Zhang,Cyclicity 1 and 2 conditions for a 2-polycycle of integrable systems on the plane.J.Differ.Equat.,155:245-261(1999).
[22]W.G Ge,Boundary value problems for nonlinear ordinary differential equations,Science press,2007.
[23]W.G Ge,J.L Ren,A extension of Mawhin's continuation theorem and its application to boundary value problem with a p-laplacian,Nonlinear Anal.,58,477-488(2004).
[24]Z.B Bai,B.J Huang,W.G ge,The iterative solutions for some fourth-order p-Laplace equation boundary value prolems,Applied Math.Lett.,19,8-14(2006).
[25]J.Hale,Functional Differential Equations,Springer - Verlag,1971.
[26]J.Hale,Theory of Functional Differential Equations,Springer - Verlag,1977.
[27]J.Hale and S.M.V.Lunel,Introduction to Functional Differential Equations,Springer -Verlag,1993.
[28]V.Lakshmikantham,D.D.Bainov,and P.S.Simeonov,Theory of impulsive differential equations,World Scientific,Singapore,1989.
[29]S.Hu and V.Lakshmikantham,Periodic boundary value problems for the second impulsive.differential equations,Nonlinear Anal.,1:75-85(1989).
[30]R.Y Ma,Existence theorems for a second-order three-point boundary value problems,J.Math.Appli.Anal.,212:430-442(1997).
[31]R.Y Ma,Multiple results for a three-point boundary value problems at resonance,Nonlinear Anal.,53:777-789(2003).
[32]R.Y Ma,Nonlocal problem of nonliear ordinary differential equation,Science press,2004.
[33]R.Y Ma,Positive.solutions of a nonlinear three-point boundary-value problem,Elec.Journal of Equations,34,1-8(1998).
[34]H.L Ma,R.Y Ma,The existence of solutions of second order value problem for ordinary differential equation,Journal of Southwest China Normal University(Natural Science),1000-5471(2005).
[35]R.Y Ma,Existence theorems for a second order m-point boundary value problem,J.Math.Anal.Appl.,211,545-555(1997).
[36]C.P.Gupta,A Sharper condition for the solvability of a three-point second order boundary value problem,J.Math.Anal.Appl.,205,586-579(1997).
[37]韩茂安,平面近哈密顿系统极限环分支的Melnikov函数方法.2007暑期讲习班讲义.
[38]M.Han,Introduction to Limit Cycle Bifurcations for near-Hamiltonian Systems.
[39]J.Jiang,J.Zhang,M.Han,Limit cycles for a class of quintic near-Hamiltonian systems near a nilpotent center,preprint,2007.
[40]R.Roussarie,On the number of limit cycles which appear by perturbation of separatrix loop of planar vector fields,Bol.Soc.Bras.Mat.,17:67-101(1986).
[41]I.D.Iliev,On the limit cycles obtainable from polynomial perturbations of the Bogdanov-Takens Hamiltonian,Israel J.Math.115:269-284(2000).
[42]B.Li,Z.Zhang,A Note on a Result of G.S.Petrov about the Weakened 16th Hilbert Problem,Journal of Mathmatical Analysis and Applications.190:489-516(1995).
[43]P.Mardesic,The number of limit cycles of polynomial deformations of Hamiltonian vector field,Ergodic Theory Dynamical Systems.10:523-529(1990).
[44]G.S.Petrov,The Chebyshev property of elliptic integrals,Functional Anal.Appl.22:72-73(1988).
[45]Y.Wu,M.Han.On the study of limit cycles of the generalized Rayleigh-Lienard oscillator.Internat.J.Bifur.ChaosAppl.Sci.Engrg.14:8,2905-2914(2004).
[46]P.Yu,M.Han,Twelve limit cycles in a cubic case of the 16th Hilbert problem.Int.J.Bifurcations and Chaos,15:7,2191-29-05(2005).
[47]T.Zhang,H.Zang,M.Han.Bifurcations of limit cycles in a cubic system.Chaos Solitons Fractals 20:3,629-638(2004).