Completely degenerate responsive tori in Hamiltonian systems

We consider the existence of responsive tori for the completely degenerate Hamiltonian system with the following Hamiltonian $H(\theta ,I,x,y,ϵ)=⟨\omega ,I⟩+\lambda \frac{x^n}{n}+\frac{y^m}{m}+ϵP(\theta ,x,y,ϵ),(\theta ,I,x,y)\in T^d\times R^{d+2},$ 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 $E\subset (0,{ϵ}_{\ast })$ with almost full Lebesgue measure such that responsive tori exist for each $ϵ\in E$ if ω satisfies a Diophantine condition. Non-existence of responsive tori are also discussed when P fails to satisfy the non-degenerate condition. Our results are directly applicable to the existence problem of quasi-periodic responsive solutions of degenerate harmonic oscillators. (paper)