Response Solutions in Degenerate Oscillators Under Degenerate Perturbations

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 $$\begin{aligned} \ddot{x}+\lambda x^l=\epsilon f(\omega t,x,\dot{x}),\;\qquad \;x\in \mathbb {R}, \end{aligned}$$ 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 $$0\le p 0$$ 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.