平面动力系统极限集的新性质
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摘要
本文主要考虑平面系统极限集的性质,利用平衡点的稳定性和平面动力系统的一般性质,引入了Dulac函数和正(负)不变集构造方法,给出V+和V-中ω极限集和α极限集的性态,研究了Poincare-Bendixson定理,得到如下一些结果,这些结果都是新的.
定理2.3.1若在系统(1.1.1)的平衡点处满足(?)P/(?)x+(?)Q/(?)y不变号,则有界区域内半轨线的极限集只可能是以下二类型之一(1)单平衡点集(2)闭轨线.
定理3.2.1对于平面系统(1.1.1),设单连通区域V+∈Ω+-(V-∈Ω-+)在V+,V-中,div(V-1P,V-1Q)V+(V-)在任何子域中恒等于零,且于(1.1.1)的平衡点不等于零,则对于V+(V-)中任何非平衡点P的极限集ω(p),有
(a)若ω(P)(?)V+(V-),则ω(p)为单点集或空集；
(b)若ω(p)∩(?)V+≠φ(ω(p)∩(?)V-≠φ),贝ω(p)∩(?)V+由奇点构成；
定理3.2.2对于平面系统(1.1.1),设单连通区域V+∈Ω++(V-∈Ω--)在V+,V-中,div(V-1P,V-1Q)V+(V-)在任何子域中恒等于零,且于(1.1.1)的平衡点不等于零,则对于V+(V-)中任何非平衡点P的极限集ω(p),有
(c)若ω(p)(?)V+(V-),则ω(p)为单点集或空集；
(d)若ω(p)∩(?)V+≠φ(ω(p)∩(?)V-≠φ),则ω(p)∩(?)V+由奇点构成；
本文正文分三部分：第一章,我们主要介绍了本文的研究背景与意义,引入了一些基本概念及引理；第二章是本文主体内容的第一部分,我们介绍了关于平面上Poincare-Bendixson定理的一个注记,讨论并证明了定理2.3.1和定理2.3.2,给出了极限集的类型；第三章我们主要给出了平面系统极限集的一个新的性质.
In this paper, we consider the properties of the omega limit set of a planar system, using the stability of an equilibrium and the general nature of the plane dynamic system, the introduction of a Dulac function and positive(negative) invariant set constructed,given the state of theωlimit set and a limit set. we study the Poincare-Bendixson theorem and obtain the following some results,these results are news.
Theorem 2.3.1 If the equilibrium point of the system (1.1.1) keeps (?)P/(?)x+(?)Q/(?)y having the same sign,then the limit set of the semiorbit within the bound region could only be one of the following two types (1) a single equilibrium set (2) a closed orbit.
Theorem3.2.1 For the plane system(1.1.1), Let simple connected region V+∈Ω+-(V-∈Ω-+) in V+(V-),,div[V-1P,V-1Q) V+(V-) equals to zero in any sub-domain, and the equilibrium of (1.1.1) is not equal to zero, then for any non-equilibrium point the limit set of P, there
(a) Ifω(p)(?)V+(V-), thenω(p) is a single-point set or empty set;
(b) ifω(p)∩(?)V+≠φ(ω)(p)∩(?)V-≠φ, thenφ(p)∩(?)V+ consist of singular point.
Theorem3.2.2 For the plane system(1.1.1), Let simple connected region V+∈Ω++(V-∈Ω--) in V+(V-),, div(V-lP,V-1Q)V+(V-) equals to zero in any sub-domain, and the equilibrium of (1.1.1) is not equal to zero, then for any non-equilibrium point the limit set of P, there
(a) Ifω(p)(?)V+(V-), thenω(p) is a single-point set or empty set;
(b) ifω(p)∩(?)V+≠φ(ω)(p)∩(?)V-≠φ), thenω(p)∩(?)V+ consist of singular point.
The paper is divided into three chapters. The first chapter,we introduce the background and the value of studying this paper and give the basic definitions and lemma. The second chapter is the part of the major content,we introduce on a note of the Poincar6-Bendixson theorem in the plane,discussing and proving theorem 2.3.1 and theorem 2.3.2, giving the type of the omega limit set.The third chapter introduces a new property of the omega limit set in the plane.
定理2.3.1若在系统(1.1.1)的平衡点处满足(?)P/(?)x+(?)Q/(?)y不变号,则有界区域内半轨线的极限集只可能是以下二类型之一(1)单平衡点集(2)闭轨线.
定理3.2.1对于平面系统(1.1.1),设单连通区域V+∈Ω+-(V-∈Ω-+)在V+,V-中,div(V-1P,V-1Q)V+(V-)在任何子域中恒等于零,且于(1.1.1)的平衡点不等于零,则对于V+(V-)中任何非平衡点P的极限集ω(p),有
(a)若ω(P)(?)V+(V-),则ω(p)为单点集或空集；
(b)若ω(p)∩(?)V+≠φ(ω(p)∩(?)V-≠φ),贝ω(p)∩(?)V+由奇点构成；
定理3.2.2对于平面系统(1.1.1),设单连通区域V+∈Ω++(V-∈Ω--)在V+,V-中,div(V-1P,V-1Q)V+(V-)在任何子域中恒等于零,且于(1.1.1)的平衡点不等于零,则对于V+(V-)中任何非平衡点P的极限集ω(p),有
(c)若ω(p)(?)V+(V-),则ω(p)为单点集或空集；
(d)若ω(p)∩(?)V+≠φ(ω(p)∩(?)V-≠φ),则ω(p)∩(?)V+由奇点构成；
本文正文分三部分：第一章,我们主要介绍了本文的研究背景与意义,引入了一些基本概念及引理；第二章是本文主体内容的第一部分,我们介绍了关于平面上Poincare-Bendixson定理的一个注记,讨论并证明了定理2.3.1和定理2.3.2,给出了极限集的类型；第三章我们主要给出了平面系统极限集的一个新的性质.
In this paper, we consider the properties of the omega limit set of a planar system, using the stability of an equilibrium and the general nature of the plane dynamic system, the introduction of a Dulac function and positive(negative) invariant set constructed,given the state of theωlimit set and a limit set. we study the Poincare-Bendixson theorem and obtain the following some results,these results are news.
Theorem 2.3.1 If the equilibrium point of the system (1.1.1) keeps (?)P/(?)x+(?)Q/(?)y having the same sign,then the limit set of the semiorbit within the bound region could only be one of the following two types (1) a single equilibrium set (2) a closed orbit.
Theorem3.2.1 For the plane system(1.1.1), Let simple connected region V+∈Ω+-(V-∈Ω-+) in V+(V-),,div[V-1P,V-1Q) V+(V-) equals to zero in any sub-domain, and the equilibrium of (1.1.1) is not equal to zero, then for any non-equilibrium point the limit set of P, there
(a) Ifω(p)(?)V+(V-), thenω(p) is a single-point set or empty set;
(b) ifω(p)∩(?)V+≠φ(ω)(p)∩(?)V-≠φ, thenφ(p)∩(?)V+ consist of singular point.
Theorem3.2.2 For the plane system(1.1.1), Let simple connected region V+∈Ω++(V-∈Ω--) in V+(V-),, div(V-lP,V-1Q)V+(V-) equals to zero in any sub-domain, and the equilibrium of (1.1.1) is not equal to zero, then for any non-equilibrium point the limit set of P, there
(a) Ifω(p)(?)V+(V-), thenω(p) is a single-point set or empty set;
(b) ifω(p)∩(?)V+≠φ(ω)(p)∩(?)V-≠φ), thenω(p)∩(?)V+ consist of singular point.
The paper is divided into three chapters. The first chapter,we introduce the background and the value of studying this paper and give the basic definitions and lemma. The second chapter is the part of the major content,we introduce on a note of the Poincar6-Bendixson theorem in the plane,discussing and proving theorem 2.3.1 and theorem 2.3.2, giving the type of the omega limit set.The third chapter introduces a new property of the omega limit set in the plane.
引文
[1]H. Poincare, Jour. Math. Pures et Appl [J]. (3),7 (1881),375-442; 8 (1882),251-296; (4),1(1885),167-244; 2(1886),157-217.
[2]I. Bendixson, Acta Math,24(1901),1-88或数学进展[J],3卷1期(1957).
[3]H. L. Smith, Moonotone Dynamical systems:An Introduction to the Theory of Competitive and Cooperative Systems, American Mathematical Society, Providence, Rhode Island[J],1995
[4]C. C. MCCLUSKEY AND J. S. MULDOWNEY, Stability Implications of Bendixson's criterion[J], SIAM REV. Vol40. No4.930-934.1998.
[5]杨芸等,关于平面上Poincare-Bendixson定理的一个注记[J],合肥：合肥师范学院学报2008(06)18-20.
[6]张芷芬等,微分方程稳定性理论[M],北京：科技出版社1985.
[7]张锦炎,常微分方程定性理论[M],北京：北京大学出版社1980.
[8]P.HARTMAN, Ordinary Differential Equations, John Wiley Sons [M], New York, 1964.
[9]M. Y. L AND J. S MuLDowney, ON R. A. Smith's autonomous convergence theorem[J], Rocky Mountain J.Math,25(1995),365-379.
[10]叶彦谦,极限环论,上海：上海科学技术出版社1965.
[11]陈纪修等,数学分析[M],北京：高等教育出版社,2000.
[12]周毓荣,邓晋宜.Dulac函数在研究极限环个数中的应用.数学年刊(A)[J],1992,13(6)：692-698.
[13]肖箭,盛立人,Dulac的构造及其应用[J],北京：系统科学与数学1999,19(4),420-425.
[14]叶彦谦著.多项式微分系统定性理论[M],上海科学技术出版社,1995.
[15]张椂,Poincare型微分方程式的极限环线[J],西北大学学报,2(1957),23-30.
[16]赵彦利,平面动力系统的ω(α)极限集在轨道有界时的性质[J],河北职业技术学院学报,3(2004),21-25.
[2]I. Bendixson, Acta Math,24(1901),1-88或数学进展[J],3卷1期(1957).
[3]H. L. Smith, Moonotone Dynamical systems:An Introduction to the Theory of Competitive and Cooperative Systems, American Mathematical Society, Providence, Rhode Island[J],1995
[4]C. C. MCCLUSKEY AND J. S. MULDOWNEY, Stability Implications of Bendixson's criterion[J], SIAM REV. Vol40. No4.930-934.1998.
[5]杨芸等,关于平面上Poincare-Bendixson定理的一个注记[J],合肥：合肥师范学院学报2008(06)18-20.
[6]张芷芬等,微分方程稳定性理论[M],北京：科技出版社1985.
[7]张锦炎,常微分方程定性理论[M],北京：北京大学出版社1980.
[8]P.HARTMAN, Ordinary Differential Equations, John Wiley Sons [M], New York, 1964.
[9]M. Y. L AND J. S MuLDowney, ON R. A. Smith's autonomous convergence theorem[J], Rocky Mountain J.Math,25(1995),365-379.
[10]叶彦谦,极限环论,上海：上海科学技术出版社1965.
[11]陈纪修等,数学分析[M],北京：高等教育出版社,2000.
[12]周毓荣,邓晋宜.Dulac函数在研究极限环个数中的应用.数学年刊(A)[J],1992,13(6)：692-698.
[13]肖箭,盛立人,Dulac的构造及其应用[J],北京：系统科学与数学1999,19(4),420-425.
[14]叶彦谦著.多项式微分系统定性理论[M],上海科学技术出版社,1995.
[15]张椂,Poincare型微分方程式的极限环线[J],西北大学学报,2(1957),23-30.
[16]赵彦利,平面动力系统的ω(α)极限集在轨道有界时的性质[J],河北职业技术学院学报,3(2004),21-25.