【作者】 邹兰军；

【导师】 周伟灿；

【作者基本信息】 南京气象学院 ， 应用数学， 2004， 硕士

【摘要】 本文由层结大气运动基本方程导出Liénard型方程，进而考虑Liénard及Duffing型方程周期解的存在性问题。对于这类非线性的二阶常微分方程，我们首先考虑其一维的特殊的形式，运用先验估计，进而利用Schauder不动点定理，证明了其两点边值问题解的存在性，并给出具体方程的解作为例子，然后用Mathematica作出数值解的图。其次，我们推广上述方程为一般情形，得到周期解存在的充分条件。再次，我们考虑n维的带有阻尼项的Duffing型方程，运用同胚延拓及不动点方法，讨论了非共振条件下周期解的存在性问题。最后，我们给出有限维向量空间中对称双线性型非退化的简单条件。我们应用这个条件和傅立叶级数的基本性质证明了二阶非线性常微系统周期解的唯一性定理。更多还原

【Abstract】 In this paper, a Lienard type equation is deduced from the basic equation of atmospheric motion in stratified atmosphere, and then we consider the existence of the periodic solutions for Lienard and Duffing type equations. For the class of nonlinear second order ordinary differential equations (ODEs), we firstly consider one of their special one-dimensional forms, and then prove the existence of solution to its two-point boundary value problem in the light of prior estimate and Schauder fixed point theorem. We also show the exact solutions for the special equations as an example and plot the numerical solutions by Mathematica. Secondly, we extend the above equation to a general form and obtain the sufficient condition for the existence of the periodic solution. Thirdly, we consider the n-dimensional Duffing type equation with damping term. By virtue of homeomorphism, extension and fixed point method, we discuss the existence of periodic solution under the nonresonance condition. Finally, we give a simple condition for nondegeneracy of symmetric bilinear forms on infinite dimensional vector spaces. We apply this condition and elementary properties of Fourier series to prove a uniqueness theorem for periodic solutions of a class of second order nonlinear ordinary differential systems.更多还原