Response Solutions in Degenerate Oscillators Under Degenerate Perturbations

Abstract

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 $0\le p<l/2$ 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.