For a quasi-periodically forced differential equation, response solutions are quasi-periodic ones whose frequency vector coincides with that of the forcing function and they are known to play a fundamental role in the harmonic and synchronizing behaviors of quasi-periodically forced oscillators. These solutions are well-understood in quasi-periodically perturbed nonlinear oscillators either in the presence of large damping or in the non-degenerate cases with small or free damping. In this paper, we consider the existence of response solutions in quasi-periodically perturbed, second order differential equations, including nonlinear oscillators, of the form

where \(\lambda \) is a constant, \(0<\epsilon \ll 1\) is a small parameter, \(l>1\) is an integer, \(\omega \in \mathbb {R}^d\) is a frequency vector, and \(f: \mathbb {T}^d\times \mathbb {R}^2\rightarrow \mathbb {R}^1\) is real analytic and non-degenerate in x up to a given order \(p\ge 0\), i.e., \([f(\cdot ,0,0)]=[\frac{\partial f(\cdot ,0,0)}{\partial x}]=[\frac{\partial ^2 f(\cdot ,0,0)}{\partial x^2}]=\cdots =[\frac{\partial ^{p-1} f(\cdot ,0,0)}{\partial x^{p-1}}]=0\) and \([\frac{\partial ^{p} f(\cdot ,0,0)}{\partial x^{p}}]\ne 0\), where \([\ \ ]\) denotes the average value of a continuous function on \(\mathbb {T}^d\). In the case that \(\lambda =0\) and f is independent of \(\dot{x}\), the existence of response solutions was first shown by Gentile (Ergod Theory Dyn Syst 27:427–457, 2007) when \(p=1\). This result was later generalized by Corsi and Gentile (Commun Math Phys 316:489–529, 2012; Ergod Theory Dyn Syst 35:1079–1140, 2015; Nonlinear Differ Equ Appl 24(1):article 3, 2017) to the case that \(p>1\) is odd. In the case \(\lambda \ne 0\), the existence of response solutions is studied by the authors Si and Yi (Nonlinearity 33(11):6072–6099, 2020) when \(p=0\). The present paper is devoted to the study of response solutions of the above quasi-periodically perturbed differential equations for the case \(\lambda \ne 0\) by allowing \(p>0\). Under the conditions that \(0\le p<l/2\) and \(\lambda [\frac{\partial ^{p} f(\cdot ,0,0)}{\partial x^{p}}]> 0\) when \(l-p\) is even, we obtain a general result which particularly implies the following: (1) If either l is odd and \(\lambda <0\) or l is even and \([\frac{\partial ^{p} f(\cdot ,0,0)}{\partial x^{p}}]>0\), then as \(\epsilon \) sufficiently small response solutions exist for each \(\omega \) satisfying a Brjuno-like non-resonant condition; (2) If either l is odd and \(\lambda >0\) or l is even and \([\frac{\partial ^{p} f(\cdot ,0,0)}{\partial x^{p}}]<0\), then there exists an \(\epsilon _*>0\) sufficiently small and a Cantor set \(\mathcal {E}\in (0,\epsilon _*)\) with almost full Lebesgue measure such that response solutions exist for each \(\epsilon \in \mathcal {E}\) and \(\omega \) satisfying a Diophantine condition. Similar results are also obtained in the case \(\lambda =\pm \epsilon \) which particularly concern the existence of large amplitude response solutions.

对于准周期受迫微分方程，响应解是准周期方程，其频率向量与受迫函数的频率向量重合，并且已知它们在准周期性受迫振荡器的谐波和同步行为中起基本作用。这些解在准周期性扰动非线性振荡器中很好理解，无论是在大阻尼存在的情况下，还是在具有小阻尼或自由阻尼的非退化情况下。在本文中，我们考虑了在准周期扰动的二阶微分方程（包括非线性振子）中响应解的存在，形式为

其中\(\lambda \)是一个常数，\(0<\epsilon \ll 1\)是一个小参数，\(l>1\)是一个整数，\(\omega \in \mathbb {R}^ d\)是一个频率向量，并且\(f: \mathbb {T}^d\times \mathbb {R}^2\rightarrow \mathbb {R}^1\)是实解析且在x上是非退化的到给定的顺序\(p\ge 0\)，即\([f(\cdot ,0,0)]=[\frac{\partial f(\cdot ,0,0)}{\partial x} ]=[\frac{\partial ^2 f(\cdot ,0,0)}{\partial x^2}]=\cdots =[\frac{\partial ^{p-1} f(\cdot ,0 ,0)}{\partial x^{p-1}}]=0\)和\([\frac{\partial ^{p} f(\cdot ,0,0)}{\partial x^{p }}]\ne 0\)，其中\([\ \ ]\)表示\(\mathbb {T}^d\)上连续函数的平均值。在\(\lambda =0\)和f独立于\(\dot{x}\) 的情况下，响应解的存在首先由 Gentile 证明（Ergod Theory Dyn Syst 27:427–457, 2007）当\(p=1\)。这一结果后来被 Corsi 和 Gentile（Commun Math Phys 316:489–529, 2012; Ergod Theory Dyn Syst 35:1079–1140, 2015; Nonlinear Differ Equ Appl 24(1):article 3, 2017）推广到案例中即\（p> 1 \）为奇数。在\(\lambda \ne 0\)的情况下，当\(p=0\)时，作者 Si 和 Yi (Nonlinearity 33(11):6072–6099, 2020) 研究了响应解的存在性. 本文致力于通过允许\(p>0\)来研究上述准周期微扰微分方程在\(\lambda \ne 0\)情况下的响应解。在\(0\le p<l/2\)和\(\lambda [\frac{\partial ^{p} f(\cdot ,0,0)}{\partial x^{p}} ]> 0\)当\(lp\)为偶数时，我们得到一个一般结果，特别暗示以下内容： (1) 如果l为奇数且\(\lambda <0\)或l为偶数且\([ \frac{\partial ^{p} f(\cdot ,0,0)}{\partial x^{p}}]>0\)，然后因为\(\epsilon \)存在足够小的响应解对于每个\(\omega \)满足类似 Brjuno 的非共振条件；(2) 如果l为奇数且\(\lambda >0\)或l为偶数且\([\frac{\partial ^{p} f(\cdot ,0,0)}{\partial x^{ p}}]<0\)，那么存在一个足够小的\(\epsilon _*>0\)和一个康托集\(\mathcal {E}\in (0,\epsilon _*)\)几乎完整的 Lebesgue 度量使得对于满足丢番图条件的每个\(\epsilon \in \mathcal {E}\)和\(\omega \)都存在响应解。在\(\lambda =\pm \epsilon \)的情况下也得到了类似的结果 这尤其涉及大振幅响应解的存在。