Response solutions in degenerate oscillators under degenerate perturbations (Q2068375)

The authors consider quasi-periodically perturbed differential equations in the form \[ {\ddot{x}} + \lambda x^l = \varepsilon f \left(\omega t, x, \dot{x} \right), \qquad x \in \mathbb{R}, \] where \(\lambda \neq 0\), \(l \neq 1\) is a positive integer, \(\varepsilon > 0\) is sufficiently small, \(\omega \in \mathbb{R}^d\), \(f: \mathbb{T}^d \times \mathbb{R}^2 \to \mathbb{R}\) is analytic and non-degenerate in \(x\) up to the positive order \(p \) which means that \[ \left[ f (\cdot, 0, 0) \right] = \left[ \frac{\partial f (\cdot, 0, 0)}{\partial x} \right] = \cdots = \left[ \frac{\partial^{p-1} f (\cdot, 0, 0)}{\partial x^{p-1}} \right] = 0 \] and that \[ \left[ \frac{\partial^{p} f (\cdot, 0, 0)}{\partial x^{p}} \right] \neq 0. \] Note that \(\left[ \cdot \right] \) denotes the average value on \(\mathbb{T}^d\). For these equations, the authors study the existence of response solutions. Similarly, for \(p=0\), the existence of response solutions is discussed in [\textit{W. Si} and \textit{Y. Yi}, Nonlinearity 33, No. 11, 6072--6098 (2020; Zbl 1455.37048)].