Consider a quasi-periodically forced nonlinear oscillator equation as follows, 0.1 \(\begin{aligned} \ddot{x}+\lambda x^{l}=\varepsilon f(\omega t,x,\dot{x}),\quad \quad x\in {\mathbb R}, \end{aligned}\) where \(\lambda \ne 0\) is a fixed constant and \(l\ge 2\) is an integer, \(\omega \in {\mathbb R}^d\) is a Diophantine frequency vector, \(\varepsilon \) is a small parameter. It is well known that the main difficulty in proof of the existence of response solutions is caused by lack of linear terms in equation (0.1) and the way to overcome this is to compensate it with certain non-degeneracy among the forced function f. For instance, the authors in Si and Yi (Anna Heri Poincaré 23:333-360, 2022) assume the average of the forced function f over d-dimensional torus is non-degenerate at the p-th order, i.e. \([\frac{\partial ^{i} f(\cdot ,0,0)}{\partial x^i}]=0\) , \(i=0,\cdots ,p-1\) and \([\frac{\partial ^{p} f(\cdot ,0,0)}{\partial x^p}]~\ne 0\) for certain \(0\le p<l/2\) , then prove the response solutions exist around relative equilibria that satisfy certain compensated non-degeneracy condition. In this paper, we consider the existence of response solutions for the equation in high order degeneracy, i.e., \([f(\cdot ,x,\dot{x})]\) is degenerate up to the p-th order for any integer \(p\ge \frac{l}{2}\) . Due to this, one can not solve for the non-degenerate relative equilibria from the average equation with forced function f in \(O(\varepsilon )\) order. However, we can normalize equation (0.1) to the new ones till the new forced function of the normalized equations satisfies compensated non-degeneracy condition such that the non-degenerate relative equilibria can be determined by the new forced function of at least \(O(\varepsilon ^2)\) order. Then we can still prove the existence of response solutions around the perturbed relative equilibria which reveals the mechanism for existence of response solutions when the forced function f is in high order degeneracy. For the sake of generality, we will firstly claim our main result for a normalized equation in general form and prove the existence of response solutions by finding relative equilibria, improving the order of perturbation and KAM iterations. Then we will show the recursive normalization procedure that transforms equation (0.1) to the general one with compensated non-degenerate condition. We will also give an explicitly checkable condition for the existence of response solution of (0.1) in terms of Fourier coefficients of lower order terms of f.
