二阶非线性微分方程的周期解与无界解

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.

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。

微分方程周期解的存在性与多解性是常微分方程定性理论与动力系统的重要研究内容。具有奇异性微分方程（系统）在力学、电子学及工程技术等领域中有广泛的应用。本项目一方面研究了具有奇异性Rayleigh方程的周期解问题，通过证明一个连续性定理，借助于时间映射证明了当阻尼项满足次线性条件时，方程至少存在一个周期解。同时，还证明了共振条件下具有奇异性Duffing方程当恢复项在奇异点处满足强奇异条件，在无穷远处满足次二次条件时，该方程周期解的存在性。另一方面，本项目还研究了平面微分系统周期解的存在性与多解性。对于一类满足半线性条件的平面哈密顿系统，应用Poincare-Birkhoff扭转定理证明了当非线性项满足不对称条件，相关的时间映射满足振动条件时，系统具有无限多个周期解。还证明了一类超线性的平面哈密顿系统存在无限多个调和解与无限多个次调和解。对于一类满足次二次条件的平面微分系统，应用重合度理论在新的条件下证明了系统至少存在一个周期解。