We consider the existence of responsive tori for the completely degenerate Hamiltonian system with the following Hamiltonian ##IMG## [http://ej.iop.org/images/0951-7715/33/11/6072/nonaba093ieqn1.gif] {$H\left(\theta ,I,x,y,{\epsilon}\right)=\langle \omega ,I\rangle +\lambda \frac{{x}^{n}}{n}+\frac{{y}^{m}}{m}+{\epsilon}P\left(\theta ,x,y,{\epsilon}\right),\enspace \left(\theta ,I,x,y\right)\in {\mathbb{T}}^{d}{\times}{\mathbb{R}}^{d+2},$} which is associated with the standard symplectic structure, where λ = ±1 and n > 2, n ⩾ m ⩾ 2 are integers. With P satisfying certain non-degenerate conditions, we obtain the following results: (1) For λ = −1 and ϵ sufficiently small, responsive tori exist for each ω satisfying a weak non-resonant condition; (2) For λ = 1 and ϵ * sufficiently small, there exists a Cantor set ##IMG## {$\mathcal{E}\subset \left(0,{{\epsilon}}_...}

中文翻译：

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哈密​​顿系统中的完全退化响应托里

我们考虑使用以下哈密顿量## IMG ## [http://ej.iop.org/images/0951-7715/33/11/6072/nonaba093ieqn1.gif]对完全退化的哈密顿量系统存在响应托里$ H \ left（\ theta，I，x，y，{\ epsilon} \ right）= \ langle \ omega，I \ rangle + \ lambda \ frac {{x} ^ {n}} {n} + \ frac {{y} ^ {m}} {m} + {\ epsilon} P \ left（\ theta，x，y，{\ epsilon} \ right），\ enspace \ left（\ theta，I，x，y \右）\ in {\ mathbb {T}} ^ {d} {\ times} {\ mathbb {R}} ^ {d + 2}，$}中，它与标准辛结构相关，其中λ=±1并且n> 2，n⩾m⩾2是整数。当P满足某些非简并条件时，我们得到以下结果：（1）对于λ= -1和ϵ足够小，每个满足弱非谐振条件的ω都存在响应托里。（2）对于λ= 1且ϵ *足够小，