We consider the existence of responsive tori for the completely degenerate Hamiltonian system with the following Hamiltonian H ( θ , I , x , y , ϵ ) = ⟨ ω , I ⟩ + λ x n n + y m m + ϵ P ( θ , x , y , ϵ ) , ( θ , I , x , y ) ∈ T d × 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 ⊂ ( 0 , ϵ * ) with almost full Lebesgue measure such that responsive tori exist for each ϵ ∈ 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.

Consider the following completely degenerate quasi-periodically forced skew-product maps of the form $ \begin{eqnarray*} \left\{ \begin{array}{l} \bar{x} = x+y^m+\epsilon f_1(x,y,\theta,\epsilon)+h_1(x,y,\theta,\epsilon),\\ \bar{y} = y+ \lambda x^n+\epsilon f_2(x,y,\theta,\epsilon)+h_2(x,y,\theta,\epsilon),\\ \bar{\theta} = \theta+\omega, \end{array} \right. \end{eqnarray*} $ where $ (x,y,\theta)\in \mathbb{R}\times\mathbb{R}\times \mathbb{T}^d $, $ \lambda = \pm 1 $, $ \omega\in\mathbb{R}^d, $ $ n $ and $ m $ are positive integers satisfying $ n\geq m,\ mn>1, $ $ f_1,f_2,h_1,h_2 $ are real analytic on $ (x,y,\theta) $ and $ C^1 $-Whitney smooth on $ \epsilon, $ and $ h_1, h_2 = \mathcal{O}(|(x,y)|^{n+1}) $. The existence of weak-hyperbolic (weak-elliptic) invariant tori for the above maps with $ \lambda = 1 $ ($ \lambda = -1 $) has been proved in [25,30]([25]). In this paper, we prove the above completely degenerate skew-product map both in the case $ \lambda = 1 $ and in the case $ \lambda = -1 $ not only admits weak-hyperbolic invariant tori but also weak-elliptic invariant tori under the suitable conditions. Moreover, the number of invariant tori is investigated in both cases. See Theorem 2.1 and Theorem 2.2. The results of this paper are the situations that are not discussed in the existing literature.

We consider the existence of response solutions for the quasi-periodic perturbation of degenerate reversible harmonic oscillators $ \ddot{x}-\lambda x^n = \epsilon f(\omega t, x, \dot x, \epsilon), \; \; x\in \mathbb{R}, $ where $ \lambda = \pm 1 $, $ n>1 $ is an integer and $ f(-\omega t, x, -\dot x, \epsilon) = f(\omega t, x, \dot x, \epsilon) $. With $ f $ satisfying certain non-degenerate conditions, we obtain the following results: (1) For $ \lambda = 1 $ and $ \epsilon $ sufficiently small, response solutions exist for each $ \omega $ satisfying a weak non-resonant condition; (2) For $ \lambda = -1 $ and $ \epsilon_* $ sufficiently small, there exists a Cantor set $ \mathcal{E}\in(0, \epsilon_*) $ with almost full Lebesgue measure such that response solutions exist for each $ \epsilon\in\mathcal{E} $ if $ \omega $ satisfies a Diophantine condition. Non-existence of response solutions is also discussed when $ f $ fails to satisfy the non-degenerate conditions.