高维空间鞍点同宿环的稳定性
摘要
本文考虑高维空间连接双曲鞍点的同宿环的稳定性问题,在可定义回复映射的条件下,给出了同宿环在其部分邻域是渐近稳定的判据,将文献[1]给出的关于3维系统鞍点同宿环的稳定性结果先后推广到了(m+2)维空间和(m+n+2)维空间中的鞍点同宿环。
全文共分两章,在第一章回顾了有关同宿环的稳定性研究的历史和现状,然后概括叙述了本文所研究的内容。
第二章研究高维空间连接双曲鞍点的同宿环的稳定性问题.首先在第1节中考虑(m+2)维空间系统:假设系统(1.1)满足以下几点:
(H_1)F(0)=0,A=D_xF(0)的特征值为λ~-,λ~+及λ_i,i=1,2,…,m,且满足λ~-<0<λ~+,Reλ_i<λ~-.
(H_2)系统(1.1)存在一个同宿于鞍点0的轨道Γ,且Γ在0点分别与相应于λ~-和λ~+的特征向量张成的直线相切.
(H_3)对充分小ε>0,存在Γ的正向不变的ε-部分邻域U_ε.
然后在第2节中考虑了(m+n+2)维空间系统:
若记鞍点O的稳定流形和不稳定流形为W~s和W~u,则对(2.1)假设dimW~s=m+1,dimW~u=n+1.
我们解决问题的主要思路和方法如下:通过在鞍点充分小邻域内,给出系统在适当的线性变换下的第一个规范型,接着采用将局部稳定流形和不稳定流形拉直的变换建立了第二个规范型。然后,在鞍点的小邻域内适当选取同宿轨道的横截面,并分两部分来构造回复映射。在鞍点的小邻域内,本质上我们利用线性近似系统的流来构造奇异流映射的主部,而在鞍点的邻域外的同宿轨道的小管状邻域内,则用近似于一个非奇异矩阵的微分同胚来获得正则流映射,将两者复合即得到所需要的回复映射。最后,我们通过技巧性地估计向量的模,给出了在横截面上回复映射初始点与首次回归点离同宿点的距离之比,由此得到关于非共振双曲鞍点同宿环的非常简洁的稳定性判据。本文得到的关于同宿环的稳定性判别准则与文[10]关于由同宿环分支出的唯一周期轨道在相同条件下的稳定性准则完全一致。
In this paper,the stability of cycles homoclinic to hyperbolic saddle point is considered in high-dimensional space. Under the condition that the recurrent map is well defined, the criterion is given for the asymptotic stability of the homoclinic cycle confined in its partial neighborhood. The stability results obtained in [1] for 3-dimensional system are extend to the (m+2)-dimensional space and (m+n+2)-dimensional space, respectively.
The full paper is divided into two chapters. The first one is devoted to the summarization of the research history and the current situation concerned with the stability of the homoclinic cycle, and to the brief introduction of the contents of the paper.
Chapter 2 studies of the stability problem for the homoclinic cycle to the hyperbolic saddle point in high-dimensional space.
First consider the (m+2)-dimensional system:
We assume that the following are true:
(H_1) F(0)=0,A=D_xF(0) has the eigenvaluesλ~-,λ~+,λ_i,i=1,2,…,m, satisfyingλ~-<0<λ~+,Reλ_i<λ~-.
(H_2) System (1.1) has an orbitΓhomoclinic to 0, which is tangent to the lines spanned by the eigenvectors corresponding to the leading eigenvaluesλ~-,λ~+ at 0.
(H_3) For any smallε>0,there exists anε- partial neighborhood ofΓ,which is invariant under the flow defined by (1.1) in positive time.
Secondly, the following (m+n+2)-dimensional system is considered:
Denote the stable manifold and the unstable manifold of the hyperbolic saddle O by W~s and W~u,then we assume for system (2.1) that dimW~s=m+1,dimW~u=n+1.
Our strategy is as follows. By taking a suitable linear transformation, we get the first normal form, by a coordinate change to straighten the local stable manifold and the local unstable manifold, we establish the second normal form. Then, in the small neighborhood of the saddle O, we select two cross sections transversal to the homoclinic orbitΓ,and construct the recurrent map by two steps: in the small neighborhood of the saddle, we build the main part of the singular flow map by the linearlyapproximate system, in the tubular neighborhood ofΓoutside of the small neighborhood of O,we use a differential homeomorphism to express the regular flow map. The recurrent map is achieved by compose the singular flow map and the regular flow map. At last, by estimating the modules of some vectors rather skilled, we obtain the ratio of the distance between the first recurrent point and the homoclinic point to the distance between the initial point and the homoclinic point. As a direct consequence, we derive two quite concise stability criteria for the non-resonant homoclinic cycle to hyperbolic saddle. The stability criteria demonstrated here are exactly the same as the one obtained in [10] for the unique periodic orbit bifurcated from the homoclinic cycle in the same conditions.
全文共分两章,在第一章回顾了有关同宿环的稳定性研究的历史和现状,然后概括叙述了本文所研究的内容。
第二章研究高维空间连接双曲鞍点的同宿环的稳定性问题.首先在第1节中考虑(m+2)维空间系统:假设系统(1.1)满足以下几点:
(H_1)F(0)=0,A=D_xF(0)的特征值为λ~-,λ~+及λ_i,i=1,2,…,m,且满足λ~-<0<λ~+,Reλ_i<λ~-.
(H_2)系统(1.1)存在一个同宿于鞍点0的轨道Γ,且Γ在0点分别与相应于λ~-和λ~+的特征向量张成的直线相切.
(H_3)对充分小ε>0,存在Γ的正向不变的ε-部分邻域U_ε.
然后在第2节中考虑了(m+n+2)维空间系统:
若记鞍点O的稳定流形和不稳定流形为W~s和W~u,则对(2.1)假设dimW~s=m+1,dimW~u=n+1.
我们解决问题的主要思路和方法如下:通过在鞍点充分小邻域内,给出系统在适当的线性变换下的第一个规范型,接着采用将局部稳定流形和不稳定流形拉直的变换建立了第二个规范型。然后,在鞍点的小邻域内适当选取同宿轨道的横截面,并分两部分来构造回复映射。在鞍点的小邻域内,本质上我们利用线性近似系统的流来构造奇异流映射的主部,而在鞍点的邻域外的同宿轨道的小管状邻域内,则用近似于一个非奇异矩阵的微分同胚来获得正则流映射,将两者复合即得到所需要的回复映射。最后,我们通过技巧性地估计向量的模,给出了在横截面上回复映射初始点与首次回归点离同宿点的距离之比,由此得到关于非共振双曲鞍点同宿环的非常简洁的稳定性判据。本文得到的关于同宿环的稳定性判别准则与文[10]关于由同宿环分支出的唯一周期轨道在相同条件下的稳定性准则完全一致。
In this paper,the stability of cycles homoclinic to hyperbolic saddle point is considered in high-dimensional space. Under the condition that the recurrent map is well defined, the criterion is given for the asymptotic stability of the homoclinic cycle confined in its partial neighborhood. The stability results obtained in [1] for 3-dimensional system are extend to the (m+2)-dimensional space and (m+n+2)-dimensional space, respectively.
The full paper is divided into two chapters. The first one is devoted to the summarization of the research history and the current situation concerned with the stability of the homoclinic cycle, and to the brief introduction of the contents of the paper.
Chapter 2 studies of the stability problem for the homoclinic cycle to the hyperbolic saddle point in high-dimensional space.
First consider the (m+2)-dimensional system:
We assume that the following are true:
(H_1) F(0)=0,A=D_xF(0) has the eigenvaluesλ~-,λ~+,λ_i,i=1,2,…,m, satisfyingλ~-<0<λ~+,Reλ_i<λ~-.
(H_2) System (1.1) has an orbitΓhomoclinic to 0, which is tangent to the lines spanned by the eigenvectors corresponding to the leading eigenvaluesλ~-,λ~+ at 0.
(H_3) For any smallε>0,there exists anε- partial neighborhood ofΓ,which is invariant under the flow defined by (1.1) in positive time.
Secondly, the following (m+n+2)-dimensional system is considered:
Denote the stable manifold and the unstable manifold of the hyperbolic saddle O by W~s and W~u,then we assume for system (2.1) that dimW~s=m+1,dimW~u=n+1.
Our strategy is as follows. By taking a suitable linear transformation, we get the first normal form, by a coordinate change to straighten the local stable manifold and the local unstable manifold, we establish the second normal form. Then, in the small neighborhood of the saddle O, we select two cross sections transversal to the homoclinic orbitΓ,and construct the recurrent map by two steps: in the small neighborhood of the saddle, we build the main part of the singular flow map by the linearlyapproximate system, in the tubular neighborhood ofΓoutside of the small neighborhood of O,we use a differential homeomorphism to express the regular flow map. The recurrent map is achieved by compose the singular flow map and the regular flow map. At last, by estimating the modules of some vectors rather skilled, we obtain the ratio of the distance between the first recurrent point and the homoclinic point to the distance between the initial point and the homoclinic point. As a direct consequence, we derive two quite concise stability criteria for the non-resonant homoclinic cycle to hyperbolic saddle. The stability criteria demonstrated here are exactly the same as the one obtained in [10] for the unique periodic orbit bifurcated from the homoclinic cycle in the same conditions.
引文
[1] 冯贝叶，空间同宿环和异宿环的稳定性[J]．数学学报，39：5(1996)，649-658．
[2] 冯贝叶，钱敏，鞍点分界线圈的稳定性及其分支极限环的条件[J]．数学学报，1985，28：53-70．，
[3] 冯贝叶，临界情况下奇环的稳定性[J]．数学学报，1990，33：113-134．
[4] Cherkas, L.A.,On the stability of a singular cycle[J].Differencial' nye Uravenja,1968, (4):1012-1017.
[5] 韩茂安，罗定军，朱德明，奇闭轨分支出极限环的唯一性(Ⅱ)、(Ⅲ)[J]．数学学报，1992，35(4)：541-548；1992，35(5)：673-684．
[6] Hirsch, M. W., Pugh, c. c., and Shub, M., Invariant Manifolds. Lect.Notes,in Math.,583[M],Berlin, Sringer-Verlag , 1977.
[7] 罗定军，朱德明，分界线环的稳定性和分支极限环的唯一性[J]．数学年刊，1990，11A(1)：95-103．
[8] 马知恩，汪儿年，鞍点分界线环的稳定性及产生极限环的条件[J]．数学年刊，1983，4A：105-110．
[9] Reyn, J. W., A stability criterion for separatrix polygous in the phase plane[J]. Nieuw Archief Voor Wiskonde,1979,27: 238-254.
[10] Zhu D.,Stability and uniqueness of periodic orbits produced during homoclinic bifurcation[J]. Acta Math. Sinica, New Ser., 1995, 11(3):267-277.
[11] Wiggins, S., Global Bifurcations and Chaos[M]. Sringer-Verlag, New York, 1988.
[12] Andronov, A. A ., Theory of Bifurcations of Dynamic Systems on a Plane. New york, Wiley, 1975.
[13] 叶谚谦，极限环论，上海：上海科学技术出版社，1965.
[14] Shui-Nee Chow and Hale, J. K., Methods of Bifurcation Theory, Berlin,Springer-Verlag, 1982.
[15] 李继彬，冯贝叶，稳定性，分歧与混沌，昆明：云南科技出版社，1995．
[16] Melnikov, B. K., On the stability of the center for periodic perturbation of time,Trans. Moscow Math. Soc, 12(1963),1-57.
[17] Cherkas, L. A., On the stability of a singular cycle, Differencial*nye Uravnenija,4(1968),1012-1017.
[18] 袁小凤，二阶Melnikov函数及其应用，数学学报，37：1(1994)，135-144．
[19] 孙建华，超临界情形的鞍点分界线环分支，数学年刊，12A(1991)，第5期，636-643．
[20] 韩茂安，朱德明，微分方程分支理论，北京：煤炭工业出版社，1994
[21] M.C.Irwin, Smooth Dynamical Systems, London, Academic Press, Inc,1980.
[22] Bo Deng, The Shilnikov problem, exponential expansion, strong λ-lemma,C~1-linearization and homoclinic bifurcation, J. D. E., 79(1989), 189-231.
[23] P. Hartman Ordinary Differential Equations, Second Edition, Boston, Birkhauser,1982.
[24] 朱德明，Melnikov型向量函数和奇异轨道的主法向，中国科学，37A(1994)，814-822．
[25] 韩茂安，李良应，同宿，异宿环分支出极限环的唯一性，数学年刊，16A(1995)，第5期，645-651．
[26] Andronov, A. A., Theory of Bifurcations of Dynamic Systems on a plane. New York, Wiley, 1975.
[27] Zhu Deming, Persistence of the nongeneric heteroclinic orbit with a saddle-node equilibrium, Science in China , Chinese Edition, 24A:9(1994),911-916.
[28] Zhu Deming, Transversal heteroclinic orbits in general degenerate cases, Science in China, 39 A:2( 1996), 112-121.
[29] Zhu Deming, Exponential trichotomy and heteroclinic bifurcation, Nonlinear Analysis, 28:3(1997), 547-557.
[30] Zhu Deming, Xia Zhihong, Bifurcations of heteroclinic loops, Science in China, 41A:8(1998), 837-848.
[31] Zhu Deming, Xu Ming, Exponentional trichotomy, orthogonality condition and their application, Chin. Ann. Math., 18B: 1(1997), 55-64.
[32] Kokubu H., Homoclinic and heteroclinic bifurcations of vector fields, Japan J. Appl. Math., 5:3(1998), 455-501.
[33]Tian Qingping, Zhu Deming, Bifurcations of nontwisted heteroclinic loops, Science in China, 30A:3(2000), 193-202.
[34] Zhu Deming, Milnikov vector and heteroclinic manifold, Science in China, 37A:6(1994), 673-682.
[35] Gruendler J., Homoclinic solutions for autonomous ordinary differential equations with nonautonomons perturbations, J. Diff. Equs., 122:1(1995), 1-26.
[2] 冯贝叶，钱敏，鞍点分界线圈的稳定性及其分支极限环的条件[J]．数学学报，1985，28：53-70．，
[3] 冯贝叶，临界情况下奇环的稳定性[J]．数学学报，1990，33：113-134．
[4] Cherkas, L.A.,On the stability of a singular cycle[J].Differencial' nye Uravenja,1968, (4):1012-1017.
[5] 韩茂安，罗定军，朱德明，奇闭轨分支出极限环的唯一性(Ⅱ)、(Ⅲ)[J]．数学学报，1992，35(4)：541-548；1992，35(5)：673-684．
[6] Hirsch, M. W., Pugh, c. c., and Shub, M., Invariant Manifolds. Lect.Notes,in Math.,583[M],Berlin, Sringer-Verlag , 1977.
[7] 罗定军，朱德明，分界线环的稳定性和分支极限环的唯一性[J]．数学年刊，1990，11A(1)：95-103．
[8] 马知恩，汪儿年，鞍点分界线环的稳定性及产生极限环的条件[J]．数学年刊，1983，4A：105-110．
[9] Reyn, J. W., A stability criterion for separatrix polygous in the phase plane[J]. Nieuw Archief Voor Wiskonde,1979,27: 238-254.
[10] Zhu D.,Stability and uniqueness of periodic orbits produced during homoclinic bifurcation[J]. Acta Math. Sinica, New Ser., 1995, 11(3):267-277.
[11] Wiggins, S., Global Bifurcations and Chaos[M]. Sringer-Verlag, New York, 1988.
[12] Andronov, A. A ., Theory of Bifurcations of Dynamic Systems on a Plane. New york, Wiley, 1975.
[13] 叶谚谦，极限环论，上海：上海科学技术出版社，1965.
[14] Shui-Nee Chow and Hale, J. K., Methods of Bifurcation Theory, Berlin,Springer-Verlag, 1982.
[15] 李继彬，冯贝叶，稳定性，分歧与混沌，昆明：云南科技出版社，1995．
[16] Melnikov, B. K., On the stability of the center for periodic perturbation of time,Trans. Moscow Math. Soc, 12(1963),1-57.
[17] Cherkas, L. A., On the stability of a singular cycle, Differencial*nye Uravnenija,4(1968),1012-1017.
[18] 袁小凤，二阶Melnikov函数及其应用，数学学报，37：1(1994)，135-144．
[19] 孙建华，超临界情形的鞍点分界线环分支，数学年刊，12A(1991)，第5期，636-643．
[20] 韩茂安，朱德明，微分方程分支理论，北京：煤炭工业出版社，1994
[21] M.C.Irwin, Smooth Dynamical Systems, London, Academic Press, Inc,1980.
[22] Bo Deng, The Shilnikov problem, exponential expansion, strong λ-lemma,C~1-linearization and homoclinic bifurcation, J. D. E., 79(1989), 189-231.
[23] P. Hartman Ordinary Differential Equations, Second Edition, Boston, Birkhauser,1982.
[24] 朱德明，Melnikov型向量函数和奇异轨道的主法向，中国科学，37A(1994)，814-822．
[25] 韩茂安，李良应，同宿，异宿环分支出极限环的唯一性，数学年刊，16A(1995)，第5期，645-651．
[26] Andronov, A. A., Theory of Bifurcations of Dynamic Systems on a plane. New York, Wiley, 1975.
[27] Zhu Deming, Persistence of the nongeneric heteroclinic orbit with a saddle-node equilibrium, Science in China , Chinese Edition, 24A:9(1994),911-916.
[28] Zhu Deming, Transversal heteroclinic orbits in general degenerate cases, Science in China, 39 A:2( 1996), 112-121.
[29] Zhu Deming, Exponential trichotomy and heteroclinic bifurcation, Nonlinear Analysis, 28:3(1997), 547-557.
[30] Zhu Deming, Xia Zhihong, Bifurcations of heteroclinic loops, Science in China, 41A:8(1998), 837-848.
[31] Zhu Deming, Xu Ming, Exponentional trichotomy, orthogonality condition and their application, Chin. Ann. Math., 18B: 1(1997), 55-64.
[32] Kokubu H., Homoclinic and heteroclinic bifurcations of vector fields, Japan J. Appl. Math., 5:3(1998), 455-501.
[33]Tian Qingping, Zhu Deming, Bifurcations of nontwisted heteroclinic loops, Science in China, 30A:3(2000), 193-202.
[34] Zhu Deming, Milnikov vector and heteroclinic manifold, Science in China, 37A:6(1994), 673-682.
[35] Gruendler J., Homoclinic solutions for autonomous ordinary differential equations with nonautonomons perturbations, J. Diff. Equs., 122:1(1995), 1-26.