具有相同反射函数的三角型微分系统研究
发布时间：2018-12-11 23:24
【摘要】：自Mironenko[1]创建微分系统的反射函数以来,许多专家就纷纷利用这一理论研究微分系统的解的定性性态.特别地,当一个微分系统的反射函数已知,且该系统是2ω-周期系统时,其Poincare映射就可以借助反射函数来建立,从而周期系统周期解的个数及稳定性态就迎刃而解.利用微分系统的等价性,那就可知道与该周期系统等价的周期微分系统周期解的性态.但遗憾的是一般情况下我们是很难求出任给一个微分系统的反射函数.因此如何在反射函数未知情况下判定两个系统的等价性问题,这是一个非常有趣的问题.对于微分系统Mironenko在[7]中给出,若△(t,x)满足则扰动系统与(1)等价,这里α(t)为t的奇的纯量函数,由此可推出也与(1)等价,这里αi(t)为奇的纯量函数,△i(t,x)为(2)的解.由此可见,求出(2)的解△(t,x)即反射积分,对判定两个微分系统的等价性尤为重要.Bel'skii在[31]中给出Riccati方程 和Abel方程及一般多项式方程:(?)的反射积分的结构形式,及这些方程具有这些反射积分的充分条件.Bel'skii在[32]中通过寻找二次多项式形式的反射积分,研究了二次微分系统与二次三角型微分系统的等价性,并利用该系统解的性态研究了一般时变二次多项式微分系统解的性态.本文在前人研究的基础上,主要运用Mironenko的反射函数方法研究了三次三角型微分系统的反射积分的结构形式及与其扰动微分系统之间的等价关系.在本文中,首先我们着重研究一般的三次微分系统等价于三角型三次微分系统的条件以及Aij(t),Bij(t)所应具备的特性.我们先从研究微分系统(4)具有三次多项式型的反射积分入手,讨论了(4)的反射积分所具备的结构形式,其次研究了(4)具有这些结构形式的反射积分的充分条件,进而讨论了(3)等价于(4)的必要条件,以及当它们均为t的周期系统时其周期解的性态.其次,我们还研究了微分系统(3)何时等价于微分系统:得出了(3)所具有的特征,此时(3)可以不是三角型系统,以及此时(4)的反射积分的结构形式,及(4)具有这些反射积分的充分条件。
[Abstract]:Since Mironenko [1] created the reflection function of the differential system, many experts have used this theory to study the qualitative behavior of the solution of the differential system. In particular, when the reflection function of a differential system is known and the system is a 2 蠅 -periodic system, its Poincare map can be established by means of the reflection function, thus the number of periodic solutions and the stability of the periodic system can be easily solved. By using the equivalence of the differential system, we can know the behavior of the periodic solution of the periodic differential system which is equivalent to the periodic system. Unfortunately, in general, it is difficult to obtain a reflection function for a differential system. Therefore, how to determine the equivalence of two systems under the unknown reflection function is a very interesting problem. For the differential system Mironenko in [7], if (TX) satisfies, the perturbed system is equivalent to (1), where 伪 (t) is an odd scalar function of t, which is also equivalent to (1). Here 伪 i (t) is an odd scalar function and I (t0 x) is the solution of (2). Thus, it is very important to find the solution (TX) of (2), that is, reflection integral, to determine the equivalence of two differential systems. In [31], Bel'skii gives the Riccati equation, Abel equation and general polynomial equation: (?) Bel'skii in [32] by searching for the reflection integral in quadratic polynomial form, In this paper, we study the equivalence between quadratic differential systems and quadratic triangular differential systems, and study the solutions of general time-varying quadratic polynomial differential systems by using the behavior of the solutions of the systems. On the basis of previous studies, this paper mainly studies the structural form of the reflection integral of the cubic triangular differential system and the equivalent relation between the reflection integral and the perturbed differential system by using Mironenko's reflection function method. In this paper, we first focus on the condition that the general cubic differential system is equivalent to the triangular cubic differential system and the properties of Aij (t), Bij (t). Starting with the study of the reflection integral of the differential system (4) with cubic polynomial type, we discuss the structural forms of the reflection integral of (4), and then study the sufficient conditions for the reflection integral with these structural forms. Then we discuss the necessary conditions for (3) to be equivalent to (4) and the behavior of the periodic solution when they are all periodic systems with t. Secondly, we also study when the differential system (3) is equivalent to the differential system: we obtain the characteristics of (3), where (3) may not be a trigonometric system, and (4) the structural form of the reflection integral. And (4) having sufficient conditions for these reflection integrals.
【学位授予单位】：扬州大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175
，
本文编号：2373416
[Abstract]:Since Mironenko [1] created the reflection function of the differential system, many experts have used this theory to study the qualitative behavior of the solution of the differential system. In particular, when the reflection function of a differential system is known and the system is a 2 蠅 -periodic system, its Poincare map can be established by means of the reflection function, thus the number of periodic solutions and the stability of the periodic system can be easily solved. By using the equivalence of the differential system, we can know the behavior of the periodic solution of the periodic differential system which is equivalent to the periodic system. Unfortunately, in general, it is difficult to obtain a reflection function for a differential system. Therefore, how to determine the equivalence of two systems under the unknown reflection function is a very interesting problem. For the differential system Mironenko in [7], if (TX) satisfies, the perturbed system is equivalent to (1), where 伪 (t) is an odd scalar function of t, which is also equivalent to (1). Here 伪 i (t) is an odd scalar function and I (t0 x) is the solution of (2). Thus, it is very important to find the solution (TX) of (2), that is, reflection integral, to determine the equivalence of two differential systems. In [31], Bel'skii gives the Riccati equation, Abel equation and general polynomial equation: (?) Bel'skii in [32] by searching for the reflection integral in quadratic polynomial form, In this paper, we study the equivalence between quadratic differential systems and quadratic triangular differential systems, and study the solutions of general time-varying quadratic polynomial differential systems by using the behavior of the solutions of the systems. On the basis of previous studies, this paper mainly studies the structural form of the reflection integral of the cubic triangular differential system and the equivalent relation between the reflection integral and the perturbed differential system by using Mironenko's reflection function method. In this paper, we first focus on the condition that the general cubic differential system is equivalent to the triangular cubic differential system and the properties of Aij (t), Bij (t). Starting with the study of the reflection integral of the differential system (4) with cubic polynomial type, we discuss the structural forms of the reflection integral of (4), and then study the sufficient conditions for the reflection integral with these structural forms. Then we discuss the necessary conditions for (3) to be equivalent to (4) and the behavior of the periodic solution when they are all periodic systems with t. Secondly, we also study when the differential system (3) is equivalent to the differential system: we obtain the characteristics of (3), where (3) may not be a trigonometric system, and (4) the structural form of the reflection integral. And (4) having sufficient conditions for these reflection integrals.
【学位授予单位】：扬州大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175
，
本文编号：2373416
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