二阶非线性微分方程的周期解与无界解

项目名称： 二阶非线性微分方程的周期解与无界解

项目编号： No.11501381

项目类型： 青年科学基金项目

立项/批准年度： 2016

项目学科： 数理科学和化学

项目作者： 马田田

作者单位： 首都师范大学

项目金额： 18万元

中文摘要： 对于共振奇异Duffing方程，研究表明当Lazer-Leach条件成立时，该方程存在周期解。本项目拟研究当Lazer-Leach条件不成立时，共振奇异Duffing方程周期解的存在性。同时借助于时间映射研究当恢复项跨越共振点时Liénard方程周期解的存在性。.径向对称系统的周期解是近来引人关注的问题，该问题本质上是二阶纯量奇异方程周期解的存在性问题。本项目将研究在半线性条件下双边接触共振点时，径向对称系统周期解的存在性。另外，拟研究具有不对称非线性项的耦合系统周期解与无界解的共存性问题。

中文关键词： 周期解；存在性；多重性；无界解

英文摘要： It is proved that if Lazer-Leach condition holds, the resonant Duffing equations with singularities have periodic solutions. In this project, we shall further study the existence of periodic solutions of resonant Duffing equations with singularities when the Lazer-Leach condition does not hold. Meanwhile, we shall study the periodic solutions of Liénard equations with singularities by using time map when the restoring terms cross resonant points..Recently, the problem of periodic solutions of singular radially symmetric systems has attracted much more attention. This problem is essentially the existence of periodic solutions of the second order scalar differential equations with singularities. We shall study the periodic solutions of singular radially symmetric systems when the nonlinear term is semilinear and touches double adjacent resonant points. Moreover, we shall study the periodic solutions and unbounded solutions of coupled systems with asymmetric nonlinearities.

英文关键词： periodic solutions;the existence of periodic solutions;multiplicity of periodic solutions;unbounded solutions