Degenerate Lower Dimensional Tori in Hamiltonian Systems
Degenerate Lower Dimensional Tori in Hamiltonian Systems
复制标题
DOI：
10.1016/j.jde.2006.02.006
复制
发表时间：
2006-08
影响因子：
2.4
通讯作者：
Yuecai Han;Yong Li;Y. Yi
中科院分区：
文献类型：
--
作者：
Yuecai Han;Yong Li;Y. Yi
本文研究了H（x，y，z）= n ω，y <$+12 <$z，M（ω）z <$+εP（x，y，z，ω）的Hamilton系统中低维环面的持久性，其中（x，y，z）∈Tn×Rn× R2 m，ε是一个小参数，M（ω）可以是奇异的.我们表明，在弱Melnikov非共振条件和某些奇异性去除条件下的扰动，大多数未扰动的n-环面仍然可以生存的小扰动。作为一个应用，我们将考虑的持久性不变环面的某些共振表面的近可积，适当退化的哈密顿系统，既不满足Kolmogorov也不g-非退化条件。
We study the persistence of lower-dimensional tori in Hamiltonian systems of the form H(x,y,z)=〈ω,y〉+12〈z,M(ω)z〉+εP(x,y,z,ω), where (x,y,z)∈Tn×Rn×R2m, ε is a small parameter, and M(ω) can be singular. We show under a weak Melnikov nonresonant condition and certain singularity-removing conditions on the perturbation that the majority of unperturbed n-tori can still survive from the small perturbation. As an application, we will consider the persistence of invariant tori on certain resonant surfaces of a nearly integrable, properly degenerate Hamiltonian system for which neither the Kolmogorov nor the g-nondegenerate condition is satisfied.