Examples of nearly integrable systems on ${\mathbb{A}}^3$ with asymptotically dense projected orbits

Given an integer $\kappa \ge 2$, we introduce a class of nearly integrable systems on ${\mathbb{A}}^3$, of the form $$H_n(\theta ,r)=\frac{1}{2}\mathrm{\parallel }r{\mathrm{\parallel }}^2+\frac{1}{n}U({\theta }_2,{\theta }_3)+f_n(\theta ,r)$$ where $U\in C^\kappa($\mathbb{T}$^2)$isagenericpotentialfunctionand$f_n$a$C^{\kappa-1}$additionalperturbationsuchthat$\Vert f_n\Vert_{C^{\kappa-1}($\mathbb{A}$^3)}\leq \tfrac{1}{n}$,sothat$H_n$ is a perturbation of the completely integrable system $h(r)=\frac{1}{2}\mathrm{\parallel }r{\mathrm{\parallel }}^2$. Let $\mathrm{\Pi }:{\mathbb{A}}^3\to {\mathbb{R}}^3$ be the canonical projection. We prove that for each $\delta >0$, there exists $n_0$ such that for $n\ge n_0$, the system $H_n$ admits an orbit ${\mathrm{\Gamma }}_n$ at energy $\frac{1}{2}$ whose projection $\mathrm{\Pi }({\mathrm{\Gamma }}_n)$ is $\delta$-dense in $\mathrm{\Pi }(H_n^{-1}(\frac{1}{2}))$, in the sense that the $\delta$-neighborhood of $\mathrm{\Pi }({\mathrm{\Gamma }}_n)$ in ${\mathbb{R}}^3$ covers $\mathrm{\Pi }(H_n^{-1}(\frac{1}{2}))$.