几类平面微分系统的Hopf分支与可积性
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摘要
本篇博士论文主要研究平面微分自治系统的可积性、等时性与极限环分支问题,全文由七章组成.
第一章全面综述了平面多项式微分自治系统的极限环分支、中心与可积性、等时中心与可线性化等问题的历史背景和研究现状,并简单介绍了一下本文的特色工作.
第二章研究了复平面拟解析四次系统的中心与等时中心问题.所采用的技巧是通过同胚变换把拟解析四次系统转换为解析系统来处理.运用计算机代数系统Mathematica,计算了新系统原点的焦点量和周期常数并且得到了其为中心与等时中心的必要条件.最后,我们通过多种方法证明了这些条件的充分性.已有的一些四次系统原点的中心与等时中心条件是本章结果的特例.
第三章研究了一类拟解析的七次系统的原点的中心条件与拟等时中心条件.首先通过同胚变换和复变换将系统的原点化为复域中的初等奇点,然后借助于计算机代数系统Mathematica推导出了该系统原点的前55个奇点量,得到了系统原点的中心条件.最后通过其周期常数的计算,得到了系统原点为拟等时中心的判据.并利用一些有效途径一一证明了这些条件的充分性.
第四章研究了一类拟解析的七次系统的无穷远点的中心条件与等时中心条件.首先通过同胚变换和复变换将拟解析系统的无穷远点化为复域中的初等原点,然后借助于计算机代数系统Mathematica推导出了该系统无穷远点的前77个奇点量,从而导出了无穷奇点为中心的条件.最后,通过计算系统的周期常数得到了系统的拟等时中心条件,并利用一些有效途径一一证明了这些条件的充分性.
在第五章中,分别研究了具有三次幂零奇点的四次、五次、以及七次的平面多项式微分系统的中心条件与极限环分支.应用计算机代数系统Mathematica,分别计算了系统的前11,12,14个拟Lyapunov常数,在此基础上得到了原点为中心的充分必要条件,并且证明了从三类系统的幂零奇点分别可以分支出11,12,14个极限环.
在第六章中,研究了一类Lienard系统的焦点量的计算方法,给出了一种计算这类Lienard系统的焦点量的新方法,并运用该方法计算了一类特殊多项式Lienard系统的焦点量,研究了该系统的中心条件与极限环分支.
在第七章中,研究一类三次Kolmogrov系统,计算出其前5个奇点量,得到了奇点为中心的必要条件.并利用一些有效途径一一证明了这些条件的充分性,同时证明了其可以分支出5个极限环.
This Ph.D. thesis is devoted to the problems of center-focus、isochronicity and bifurcation of limit cycles for planar polynomial autonomous differential systems. It is composed of seven chapters.
In Chapter1, the historical background and the present progress of problems con-cerned with centers, integrability, isochronous centers, linearizability and bifurcation of limit cycles for planar polynomial autonomous differential systems are introduced and summarized. Meanwhile, the main work of this paper is simply concluded.
In Chapter2, we deal with the problem of characterizing center and isochronous centers for complex planar quasi-analytic quardraic system. The technique is based on transforming the quasi-analytic quardraic analytic system into an analytic system. With the help of the computer algebra system-Mathematica, we compute the singular values and period constants of the origin and obtain the necessary center and isochronous center conditions for the transformed system. Finally, we give a proof of the sufficiency by various methods. Our work consists of the existing results related to cubic polynomial system as a special case.
In Chapter3, the conditions of center and pseudo-isochronous center conditions at origin for a class of non-analytic septic system are investigated. Firstly, the origin of non-analytic septic system is transferred into the origin of an analytic system by a homeomorphic transformation and a complex transformation. Furthermore, with the help of computer algebra system-Mathematica, we derive the first55singular point quantities at origin of new system and get the center conditions at the origin. Finally, we find necessary conditions for pseudo-isochronous centers by computing its period constants, then the sufficiency of these conditions are proved by some effective methods.
In Chapter4, the conditions of center and pseudo-isochronous center conditions at infinity for a class of non-analytic septic system are investigated. Firstly, the infinity is transferred into the origin by a homeomorphic transformation and a complex transfor-mation. Furthermore, with the help of computer algebra system-Mathematica, we derive the first77singular point quantities at infinity and get the center conditions at infinity. Finally, we find necessary conditions for pseudo-isochronous centers by computing its period constants, then the sufficiency of these conditions are proved by some effective methods.
In Chapter5, for the three-order nilpotent critical point of classes of quardraic,quintic, septic Lyapunov systems, the center problem and bifurcation of limit cycles are inves-tigated. With the help of computer algebra system-Mathematica, the first11,12,14quasi-Lyapunov constants are deduced respectively. As a result, sufficient and necessary conditions in order to have a center are obtained. The fact of there exist11,12,14small amplitude limit cycles created from the three-order nilpotent critical point is also proved respectively.
In Chapter6, we give a method to compute the singular values for a class of Lienard system. By using this new method, the center conditions and bifurcations of limit circles are investigated for a special Lienard system.
In Chapter7, we are interested in a cubic Kolmogrov system, the first five singular values are calculated, and the the center conditions are obtained and proved by some technical transformations. Furthermore, five limit circles could be bifurcated from the neighborhood of the origin.
第一章全面综述了平面多项式微分自治系统的极限环分支、中心与可积性、等时中心与可线性化等问题的历史背景和研究现状,并简单介绍了一下本文的特色工作.
第二章研究了复平面拟解析四次系统的中心与等时中心问题.所采用的技巧是通过同胚变换把拟解析四次系统转换为解析系统来处理.运用计算机代数系统Mathematica,计算了新系统原点的焦点量和周期常数并且得到了其为中心与等时中心的必要条件.最后,我们通过多种方法证明了这些条件的充分性.已有的一些四次系统原点的中心与等时中心条件是本章结果的特例.
第三章研究了一类拟解析的七次系统的原点的中心条件与拟等时中心条件.首先通过同胚变换和复变换将系统的原点化为复域中的初等奇点,然后借助于计算机代数系统Mathematica推导出了该系统原点的前55个奇点量,得到了系统原点的中心条件.最后通过其周期常数的计算,得到了系统原点为拟等时中心的判据.并利用一些有效途径一一证明了这些条件的充分性.
第四章研究了一类拟解析的七次系统的无穷远点的中心条件与等时中心条件.首先通过同胚变换和复变换将拟解析系统的无穷远点化为复域中的初等原点,然后借助于计算机代数系统Mathematica推导出了该系统无穷远点的前77个奇点量,从而导出了无穷奇点为中心的条件.最后,通过计算系统的周期常数得到了系统的拟等时中心条件,并利用一些有效途径一一证明了这些条件的充分性.
在第五章中,分别研究了具有三次幂零奇点的四次、五次、以及七次的平面多项式微分系统的中心条件与极限环分支.应用计算机代数系统Mathematica,分别计算了系统的前11,12,14个拟Lyapunov常数,在此基础上得到了原点为中心的充分必要条件,并且证明了从三类系统的幂零奇点分别可以分支出11,12,14个极限环.
在第六章中,研究了一类Lienard系统的焦点量的计算方法,给出了一种计算这类Lienard系统的焦点量的新方法,并运用该方法计算了一类特殊多项式Lienard系统的焦点量,研究了该系统的中心条件与极限环分支.
在第七章中,研究一类三次Kolmogrov系统,计算出其前5个奇点量,得到了奇点为中心的必要条件.并利用一些有效途径一一证明了这些条件的充分性,同时证明了其可以分支出5个极限环.
This Ph.D. thesis is devoted to the problems of center-focus、isochronicity and bifurcation of limit cycles for planar polynomial autonomous differential systems. It is composed of seven chapters.
In Chapter1, the historical background and the present progress of problems con-cerned with centers, integrability, isochronous centers, linearizability and bifurcation of limit cycles for planar polynomial autonomous differential systems are introduced and summarized. Meanwhile, the main work of this paper is simply concluded.
In Chapter2, we deal with the problem of characterizing center and isochronous centers for complex planar quasi-analytic quardraic system. The technique is based on transforming the quasi-analytic quardraic analytic system into an analytic system. With the help of the computer algebra system-Mathematica, we compute the singular values and period constants of the origin and obtain the necessary center and isochronous center conditions for the transformed system. Finally, we give a proof of the sufficiency by various methods. Our work consists of the existing results related to cubic polynomial system as a special case.
In Chapter3, the conditions of center and pseudo-isochronous center conditions at origin for a class of non-analytic septic system are investigated. Firstly, the origin of non-analytic septic system is transferred into the origin of an analytic system by a homeomorphic transformation and a complex transformation. Furthermore, with the help of computer algebra system-Mathematica, we derive the first55singular point quantities at origin of new system and get the center conditions at the origin. Finally, we find necessary conditions for pseudo-isochronous centers by computing its period constants, then the sufficiency of these conditions are proved by some effective methods.
In Chapter4, the conditions of center and pseudo-isochronous center conditions at infinity for a class of non-analytic septic system are investigated. Firstly, the infinity is transferred into the origin by a homeomorphic transformation and a complex transfor-mation. Furthermore, with the help of computer algebra system-Mathematica, we derive the first77singular point quantities at infinity and get the center conditions at infinity. Finally, we find necessary conditions for pseudo-isochronous centers by computing its period constants, then the sufficiency of these conditions are proved by some effective methods.
In Chapter5, for the three-order nilpotent critical point of classes of quardraic,quintic, septic Lyapunov systems, the center problem and bifurcation of limit cycles are inves-tigated. With the help of computer algebra system-Mathematica, the first11,12,14quasi-Lyapunov constants are deduced respectively. As a result, sufficient and necessary conditions in order to have a center are obtained. The fact of there exist11,12,14small amplitude limit cycles created from the three-order nilpotent critical point is also proved respectively.
In Chapter6, we give a method to compute the singular values for a class of Lienard system. By using this new method, the center conditions and bifurcations of limit circles are investigated for a special Lienard system.
In Chapter7, we are interested in a cubic Kolmogrov system, the first five singular values are calculated, and the the center conditions are obtained and proved by some technical transformations. Furthermore, five limit circles could be bifurcated from the neighborhood of the origin.
引文
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