有兩個(gè)周期外力的Josephson系統(tǒng)的擬周期解
發(fā)布時(shí)間：2018-10-14 17:51
【摘要】：目前,對(duì)具有兩個(gè)周期外力的Josephson方程的動(dòng)態(tài)研究的論文很少.而Josephson方程所產(chǎn)生的效應(yīng),在很多領(lǐng)域都有廣泛的應(yīng)用,例如：地質(zhì)學(xué)中,用地磁儀來測量地球表面磁場的波動(dòng)；測量人體心臟和大腦磁場的變化等等.本文主要運(yùn)用KAM理論和牛頓方法來研究帶有兩個(gè)周期外力的Josephson系統(tǒng)的擬周期解的存在性.我們主要是研究原系統(tǒng)在其平衡解附近,是否存在擬周期解?本文通過可逆變換把系統(tǒng)化為可用KAM理論分析的標(biāo)準(zhǔn)型,然后再運(yùn)用牛頓方法作一系列可逆的變換將標(biāo)準(zhǔn)型約化.在對(duì)標(biāo)準(zhǔn)型約化過程中,需要解同調(diào)方程,解將會(huì)出現(xiàn)小分母問題,這時(shí)我們要求參數(shù)滿足Diophantine條件,并且還需要進(jìn)行一些測度估值.這樣將得到迭代后的系統(tǒng)的收斂形式是存在擬周期解的.由于所作的一系列變換都是可逆變換,那么原系統(tǒng)對(duì)于在一定參數(shù)范圍內(nèi)的大多數(shù)參數(shù),其標(biāo)準(zhǔn)型在平衡解附近存在擬周期解.
[Abstract]:At present, there are few studies on the dynamics of Josephson equations with two periodic forces. The effect of Josephson equation has been widely used in many fields, for example, geomagnetic instrument is used to measure the fluctuation of magnetic field on the earth's surface, the magnetic field of human heart and brain is measured and so on. In this paper, the existence of quasi periodic solutions for Josephson systems with two periodic external forces is studied by using KAM theory and Newton method. We mainly study whether the quasi periodic solution exists near the equilibrium solution of the original system. In this paper, the system is transformed into a standard form which can be analyzed by KAM theory by invertible transformation, and then the canonical form is reduced by a series of reversible transformations by Newton's method. In the process of reduction of canonical forms we need to solve the homology equation and the solution will have a small denominator problem. In this case we need the parameters to satisfy the Diophantine condition and some measure estimates. In this way, it is obtained that the convergence form of the iterative system is quasi periodic solution. Since the series of transformations are invertible, there is a quasi-periodic solution in the normal form of the original system for most of the parameters in a certain parameter range near the equilibrium solution.
【學(xué)位授予單位】：湖南師范大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2016
【分類號(hào)】：O175
，
本文編號(hào)：2271186
[Abstract]:At present, there are few studies on the dynamics of Josephson equations with two periodic forces. The effect of Josephson equation has been widely used in many fields, for example, geomagnetic instrument is used to measure the fluctuation of magnetic field on the earth's surface, the magnetic field of human heart and brain is measured and so on. In this paper, the existence of quasi periodic solutions for Josephson systems with two periodic external forces is studied by using KAM theory and Newton method. We mainly study whether the quasi periodic solution exists near the equilibrium solution of the original system. In this paper, the system is transformed into a standard form which can be analyzed by KAM theory by invertible transformation, and then the canonical form is reduced by a series of reversible transformations by Newton's method. In the process of reduction of canonical forms we need to solve the homology equation and the solution will have a small denominator problem. In this case we need the parameters to satisfy the Diophantine condition and some measure estimates. In this way, it is obtained that the convergence form of the iterative system is quasi periodic solution. Since the series of transformations are invertible, there is a quasi-periodic solution in the normal form of the original system for most of the parameters in a certain parameter range near the equilibrium solution.
【學(xué)位授予單位】：湖南師范大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2016
【分類號(hào)】：O175
，
本文編號(hào)：2271186
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