论文标题
平面哈密顿系统：索引理论和对亚亨马克的存在的应用
Planar Hamiltonian systems: index theory and applications to the existence of subharmonics
论文作者
论文摘要
我们考虑$ jz'= \ nabla_z h（t，z）$的平面哈密顿系统，其中$ h：\ mathbb {r} \ times \ times \ mathbb {r}^2 \ to \ mathbb {r} $在$ $ \ nabla_z h（nabla_z h（0）中， h（t，z）$是渐近线性的，对于$ \ vert z \ vert \ to +\ infty $。在对线性平面哈密顿系统的索引理论重新验证后，通过使用Poincaré-Birkhoff固定点定理，我们证明，上述非线性系统具有足够大的任何顺序的$ k $的亚谐波解决方案，每当旋转数字（或等效地，平均的Conley-Zhhnder resceices of System at System at Zero and Ats at a in Infintial and art Zero and Ant Ant Antery and arthement of。申请是对来自二阶标量ODES的平面汉密尔顿系统的案例。
We consider a planar Hamiltonian system of the type $Jz' = \nabla_z H(t,z)$, where $H: \mathbb{R} \times \mathbb{R}^2 \to \mathbb{R}$ is a function periodic in the time variable, such that $\nabla_z H(t,0) \equiv 0$ and $\nabla_z H(t,z)$ is asymptotically linear for $\vert z \vert \to +\infty$. After revisiting the index theory for linear planar Hamiltonian systems, by using the Poincaré-Birkhoff fixed point theorem we prove that the above nonlinear system has subharmonic solutions of any order $k$ large enough, whenever the rotation numbers (or, equivalently, the mean Conley-Zehnder indices) of the linearizations of the system at zero and at infinity are different. Applications are given to the case of planar Hamiltonian systems coming from second order scalar ODEs.
