有两个周期外力的Josephson系统的拟周期解
发布时间：2018-10-14 17:51
【摘要】：目前,对具有两个周期外力的Josephson方程的动态研究的论文很少.而Josephson方程所产生的效应,在很多领域都有广泛的应用,例如：地质学中,用地磁仪来测量地球表面磁场的波动；测量人体心脏和大脑磁场的变化等等.本文主要运用KAM理论和牛顿方法来研究带有两个周期外力的Josephson系统的拟周期解的存在性.我们主要是研究原系统在其平衡解附近,是否存在拟周期解?本文通过可逆变换把系统化为可用KAM理论分析的标准型,然后再运用牛顿方法作一系列可逆的变换将标准型约化.在对标准型约化过程中,需要解同调方程,解将会出现小分母问题,这时我们要求参数满足Diophantine条件,并且还需要进行一些测度估值.这样将得到迭代后的系统的收敛形式是存在拟周期解的.由于所作的一系列变换都是可逆变换,那么原系统对于在一定参数范围内的大多数参数,其标准型在平衡解附近存在拟周期解.
[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.
【学位授予单位】：湖南师范大学
【学位级别】：硕士
【学位授予年份】：2016
【分类号】：O175
，
本文编号：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.
【学位授予单位】：湖南师范大学
【学位级别】：硕士
【学位授予年份】：2016
【分类号】：O175
，
本文编号：2271186
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