Response Solutions for Degenerate Reversible Harmonic Oscillators with Zero-average Perturbation

Abstract

In this paper, we consider a class of normally degenerate quasi-periodically forced reversible systems, obtained as perturbations of a set of harmonic oscillators,

$$\left\{ {\matrix{{\dot x = y + {f_1}(\omega t,x,y),} \hfill \cr {\dot y = \lambda {x^l} + {f_2}(\omega t,x,y),} \hfill \cr } } \right.$$

where 0 ≠ λ ∈ ℝ, l > 1 is an integer and the corresponding involution G is (−θ, x, −y) → (θ, x, y). The existence of response solutions of the above reversible systems has already been proved in [22] if [f₂(ωt, 0, 0)] satisfies some non-zero average conditions (See the condition (H) in [22]), here [ · ] denotes the average of a continuous function on ${\mathbb{T}^d}$. However, discussing the existence of response solutions for the above systems encounters difficulties when [f₂(ωt, 0, 0)] = 0, due to a degenerate implicit function must be solved. This article will be doing work in this direction. The purpose of this paper is to consider the case where [f₂(ωt, 0, 0)] = 0. More precisely, with 2p < l, if f₂ satisfies $[{f_2}(\omega t,0,0)] = [{{\partial {f_2}(\omega t,0,0)} \over {\partial x}}] = [{{{\partial ^2}{f_2}(\omega t,0,0)} \over {\partial {x^2}}}] = \cdots = [{{{\partial ^{p - 1}}{f_2}(\omega t,0,0)} \over {\partial {x^{p - 1}}}}] = 0$, either ${\lambda ^{ - 1}}[{{{\partial ^p}{f_2}(\omega t,0,0)} \over {\partial {x^p}}}] < 0$ as l − p is even or ${\lambda ^{ - 1}}[{{{\partial ^p}{f_2}(\omega t,0,0)} \over {\partial {x^p}}}] \ne 0$ as l − p is odd, we obtain the following results: (1) For $\tilde \lambda < 0$ (see ${\tilde \lambda }$ in (2.2)) and ϵ sufficiently small, response solutions exist for each ω satisfying a weak non-resonant condition; (2) For $\tilde \lambda < 0$ and ϵ_* sufficiently small, there exists a Cantor set ${\cal E} \in (0,{_ * })$ with almost full Lebesgue measure such that response solutions exist for each $\in {\cal E}$ if ω satisfies a Diophantine condition. In the remaining case where ${\lambda ^{ - 1}}[{{{\partial ^p}{f_2}(\omega t,0,0)} \over {\partial {x^p}}}] > 0$ and l − p is even, we prove the system admits no response solutions in most regions.