具有相同反射函數(shù)的三角型微分系統(tǒng)研究
發(fā)布時(shí)間：2018-12-11 23:24
【摘要】：自Mironenko[1]創(chuàng)建微分系統(tǒng)的反射函數(shù)以來,許多專家就紛紛利用這一理論研究微分系統(tǒng)的解的定性性態(tài).特別地,當(dāng)一個(gè)微分系統(tǒng)的反射函數(shù)已知,且該系統(tǒng)是2ω-周期系統(tǒng)時(shí),其Poincare映射就可以借助反射函數(shù)來建立,從而周期系統(tǒng)周期解的個(gè)數(shù)及穩(wěn)定性態(tài)就迎刃而解.利用微分系統(tǒng)的等價(jià)性,那就可知道與該周期系統(tǒng)等價(jià)的周期微分系統(tǒng)周期解的性態(tài).但遺憾的是一般情況下我們是很難求出任給一個(gè)微分系統(tǒng)的反射函數(shù).因此如何在反射函數(shù)未知情況下判定兩個(gè)系統(tǒng)的等價(jià)性問題,這是一個(gè)非常有趣的問題.對于微分系統(tǒng)Mironenko在[7]中給出,若△(t,x)滿足則擾動系統(tǒng)與(1)等價(jià),這里α(t)為t的奇的純量函數(shù),由此可推出也與(1)等價(jià),這里αi(t)為奇的純量函數(shù),△i(t,x)為(2)的解.由此可見,求出(2)的解△(t,x)即反射積分,對判定兩個(gè)微分系統(tǒng)的等價(jià)性尤為重要.Bel'skii在[31]中給出Riccati方程 和Abel方程及一般多項(xiàng)式方程:(?)的反射積分的結(jié)構(gòu)形式,及這些方程具有這些反射積分的充分條件.Bel'skii在[32]中通過尋找二次多項(xiàng)式形式的反射積分,研究了二次微分系統(tǒng)與二次三角型微分系統(tǒng)的等價(jià)性,并利用該系統(tǒng)解的性態(tài)研究了一般時(shí)變二次多項(xiàng)式微分系統(tǒng)解的性態(tài).本文在前人研究的基礎(chǔ)上,主要運(yùn)用Mironenko的反射函數(shù)方法研究了三次三角型微分系統(tǒng)的反射積分的結(jié)構(gòu)形式及與其擾動微分系統(tǒng)之間的等價(jià)關(guān)系.在本文中,首先我們著重研究一般的三次微分系統(tǒng)等價(jià)于三角型三次微分系統(tǒng)的條件以及Aij(t),Bij(t)所應(yīng)具備的特性.我們先從研究微分系統(tǒng)(4)具有三次多項(xiàng)式型的反射積分入手,討論了(4)的反射積分所具備的結(jié)構(gòu)形式,其次研究了(4)具有這些結(jié)構(gòu)形式的反射積分的充分條件,進(jìn)而討論了(3)等價(jià)于(4)的必要條件,以及當(dāng)它們均為t的周期系統(tǒng)時(shí)其周期解的性態(tài).其次,我們還研究了微分系統(tǒng)(3)何時(shí)等價(jià)于微分系統(tǒng):得出了(3)所具有的特征,此時(shí)(3)可以不是三角型系統(tǒng),以及此時(shí)(4)的反射積分的結(jié)構(gòu)形式,及(4)具有這些反射積分的充分條件。
[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 偽 (t) is an odd scalar function of t, which is also equivalent to (1). Here 偽 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.
【學(xué)位授予單位】：揚(yáng)州大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2017
【分類號】：O175
，
本文編號：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 偽 (t) is an odd scalar function of t, which is also equivalent to (1). Here 偽 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.
【學(xué)位授予單位】：揚(yáng)州大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2017
【分類號】：O175
，
本文編號：2373416
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