R~3中非强1-共振映射的多项式正规形研究
发布时间：2019-03-12 10:54
【摘要】：正规形理论是研究非线性问题时广泛采用的一种手段,无论是它自身的理论还是其应用都具有特别重要的意义.近些年来,该理论在Hilbert第十六问题、分岔理论、动力系统的分类问题以及微分同胚嵌入流问题等很多方面都得到了广泛的应用,这就使得它越来越引起人们的关注.本论文的研究内容主要由两部分组成:第一部分研究了R3空间中在通有条件下映射在非强1-共振不动点附近的多项式正规形和有限确定性;第二部分研究了R3空间中在退化条件下映射在非强1-共振不动点附近的多项式正规形和有限确定性.本论文的第一部分主要对R3空间中在通有条件下线性部分系数矩阵含有重根的非强1-共振映射进行研究.我们利用经典的Poincare-Dulac正规形定理,主要通过讨论映射的线性部分系数矩阵特征根的共振关系得到该类映射所对应的经典共振正规形.在Ichikawa关于映射的有限确定性理论基础上,通过引进一系列共振变换,结合Belistkii定理,讨论了其有限确定性并得到了多项式正规形.本论文的第二部分主要对R3空间中在退化条件下线性部分系数矩阵不含有重根的非强1-共振映射进行研究.我们同样利用引入共振变换的方法,结合Ichikawa和Belistkii定理得到了该类映射在满足一定的退化条件时,可以与一个多项式正规形光滑等价并能有限确定.
[Abstract]:The formal theory is a kind of means to study the non-linear problem, whether it is its own theory or its application. In recent years, the theory has been widely used in many aspects such as Hilbert's sixteenth problem, the bifurcation theory, the classification of the power system and the problem of the embedded flow of the differential and the embryo, which makes it more and more concerned. The research contents of this paper are mainly composed of two parts: the first part studies the formal and finite certainty of the polynomial which is mapped on the non-strong 1-resonance fixed point in the R3 space; The second part studies the polynomial normal form and the finite certainty that are mapped on the non-strong 1-resonance fixed point in the R3 space under the condition of degradation. The first part of this paper mainly studies the non-strong 1-resonance mapping of the linear partial coefficient matrix containing the heavy root in the R3 space. In this paper, we use the classical Poincare-Dulac normal-shape theorem to obtain the classical resonance normal form corresponding to this kind of mapping by discussing the resonance relation of the linear part coefficient matrix characteristic root of the mapping. On the basis of Ichikawa's finite certainty theory about mapping, by introducing a series of resonance transforms, the finite certainty is discussed and the polynomial normal form is obtained by combining the Belistkii theorem. The second part of this paper mainly studies the non-strong 1-resonance mapping of the linear partial coefficient matrix without heavy root in the R3 space. In the same way, we use the method of introducing the resonance transformation, and in combination with the Ichikawa and the Belistkii theorem, this kind of mapping can be equivalent to the normal shape of a polynomial and can be determined in a finite way when certain degradation conditions are satisfied.
【学位授予单位】：北京工业大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O174.14
本文编号：2438710
[Abstract]:The formal theory is a kind of means to study the non-linear problem, whether it is its own theory or its application. In recent years, the theory has been widely used in many aspects such as Hilbert's sixteenth problem, the bifurcation theory, the classification of the power system and the problem of the embedded flow of the differential and the embryo, which makes it more and more concerned. The research contents of this paper are mainly composed of two parts: the first part studies the formal and finite certainty of the polynomial which is mapped on the non-strong 1-resonance fixed point in the R3 space; The second part studies the polynomial normal form and the finite certainty that are mapped on the non-strong 1-resonance fixed point in the R3 space under the condition of degradation. The first part of this paper mainly studies the non-strong 1-resonance mapping of the linear partial coefficient matrix containing the heavy root in the R3 space. In this paper, we use the classical Poincare-Dulac normal-shape theorem to obtain the classical resonance normal form corresponding to this kind of mapping by discussing the resonance relation of the linear part coefficient matrix characteristic root of the mapping. On the basis of Ichikawa's finite certainty theory about mapping, by introducing a series of resonance transforms, the finite certainty is discussed and the polynomial normal form is obtained by combining the Belistkii theorem. The second part of this paper mainly studies the non-strong 1-resonance mapping of the linear partial coefficient matrix without heavy root in the R3 space. In the same way, we use the method of introducing the resonance transformation, and in combination with the Ichikawa and the Belistkii theorem, this kind of mapping can be equivalent to the normal shape of a polynomial and can be determined in a finite way when certain degradation conditions are satisfied.
【学位授予单位】：北京工业大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O174.14
【共引文献】
相关硕士学位论文 前1条
1 杨柳芳;平面映射的线性化及相关函数方程的C~1解问题[D];重庆师范大学;2015年
，本文编号：2438710
本文链接：https://www.wllwen.com/kejilunwen/yysx/2438710.html
