Given the function $p(t)$, when restricted to a region of the phase space of *K* where *z* is small enough and $\tau >0$ is bounded away from zero, the system $K(z,w,t,\tau )$ is a perturbation of the integrable system

${K}_{0}={z}^{2}\tau +\frac{{w}^{2}}{8}-1.$

The “integrable approximating system” ${K}_{0}$ reads

$\frac{\sqrt{2}}{2}{\tau}^{1/2}I-1$

in action-angle form by calculating the action-angle coordinates $(I,\theta ,\tau ,{t}^{\prime})$ defined by the relations

$\{\begin{array}{cc}\hfill z& ={2}^{-1/4}{I}^{1/2}{\tau}^{-1/4}\mathrm{cos}\theta ,\hfill \\ \hfill w& =-2\cdot {2}^{1/4}{I}^{1/2}{\tau}^{1/4}\mathrm{sin}\theta ,\hfill \\ \hfill \tau & =\tau ,\hfill \\ \hfill t& ={t}^{\prime}+{4}^{-1}I{\tau}^{-1}\mathrm{sin}2\theta .\hfill \end{array}$

The calculation goes in the following way. We first fix τ and reduce the system ${K}_{0}$ by the ${\mathbb{S}}^{1}$-symmetry of shifting *t*, and then we calculate the action variable $I=A(h)/2\pi $ by calculating the enclosed area $A(h)$ of the ellipse $\{{K}_{0}=h\}$ in the $(z,w)$-plane. Inverting the mapping $h\mapsto I(h)$ gives

${K}_{0}=h=\frac{\sqrt{2}}{2}{\tau}^{1/2}I-1.$

We thus set

$z={2}^{-1/4}{I}^{1/2}{\tau}^{-1/4}\mathrm{cos}\theta ,w=-2\cdot {2}^{1/4}{I}^{1/2}{\tau}^{1/4}\mathrm{sin}\theta .$

In the unreduced phase space with variables $(I,\theta ,\tau ,t)$, we have

$dw\wedge dz+d\tau \wedge dt=dI\wedge d\theta +d\tau \wedge d(t-{4}^{-1}I{\tau}^{-1}\mathrm{sin}2\theta ).$

Therefore, the set of variables $(I,\theta ,\tau ,{t}^{\prime}=t-{4}^{-1}I{\tau}^{-1}\mathrm{sin}2\theta )$ forms a set of action-angle coordinates of ${K}_{0}$.

The subset $\{\tau >0,I>0\}$ of the phase space of ${K}_{0}$ is seen to be foliated by the invariant 2-tori of ${K}_{0}$ obtained by fixing *I* and τ, and the associated Poincaré map $\stackrel{\u02c7}{\mathcal{\mathcal{S}}}={\mathcal{\mathcal{S}}|}_{D}$ is an exact twist map. The perturbation takes the form

$P(I,\theta ,\tau ,{t}^{\prime})=p\left(t(I,\theta ,\tau ,{t}^{\prime})\right){I}^{2}{\tau}^{-1}{\mathrm{cos}}^{4}\theta .$

We now have a choice to study the dynamics of *K* as a perturbation of ${K}_{0}$ (which is the case for given $p(t)$ when $|z|$ is small enough). We may either work directly with the Hamiltonian or with the Poincaré map $\stackrel{\u02c7}{\mathcal{\mathcal{S}}}$. The objects between the two approaches are naturally related. Fixed and periodic points of $\stackrel{\u02c7}{\mathcal{\mathcal{S}}}$ give rise to periodic solutions of *K* and invariant curves of $\stackrel{\u02c7}{\mathcal{\mathcal{S}}}$ give rise to invariant tori of *K*. They give rise to generalized periodic and quasi-periodic solutions of (1.1), respectively. Recall that by “generalized solutions” we simply refer to those collision solutions along which the collisions are regularized by elastic bouncing, which is exactly what the Levi-Civita regularization implies for the initial system.

We now prove Theorem 1.1.

#### Proof of Theorem 1.1.

We opt to work with the Hamiltonian function. The reader is invited to compare with [5] for results obtained from analyzing the mapping $\stackrel{\u02c7}{\mathcal{\mathcal{S}}}$.

We introduce a small parameter ε and we shall apply the KAM theorem to invariant tori of ${K}_{0}$ in a region where $\tau \sim {\epsilon}^{-2}$, $I\sim \epsilon $, and where the unperturbed energy of ${K}_{0}$ lies in $(-\delta ,\delta )$ for a certain small $\delta >0$ independent of ε.

We observe that the unperturbed system ${K}_{0}$ is non-degenerate in the sense that its Hessian with respect to τ and *I* is non-degenerate. Indeed, since ${K}_{0}$ depends linearly on *I*, we have ${\partial}^{2}{K}_{0}/\partial {I}^{2}=0$ and the determinant of this Hessian matrix equals $-{({\partial}^{2}{K}_{0}/\partial I\partial \tau )}^{2}$, which, up to a constant, is equal to ${\tau}^{-1}\sim {\epsilon}^{2}$.

Since the non-degeneracy of the unperturbed system depends non-trivially on the small parameter ε, in order to show that a standard KAM theorem holds (a simple version that applies to our case is [1, Theorem 6.16], in which it is also remarked that it is enough to have $r>4$ to apply [6, Theorem 2.1]), we have to make yet another rescaling. Set

$(I,\theta ,\tau ,{t}^{\prime})=({\epsilon}^{-2}{I}^{\prime},\theta ,{\epsilon}^{-2}{\tau}^{\prime},{t}^{\prime}).$

We have

$dI\wedge d\theta +d\tau \wedge d{t}^{\prime}={\epsilon}^{-2}(d{I}^{\prime}\wedge d\theta +d{\tau}^{\prime}\wedge d{t}^{\prime}),{K}_{0}(I,\tau )={\epsilon}^{-3}{K}_{0}({I}^{\prime},{\tau}^{\prime}).$

Moreover, by its explicit expression, we also have

$P(I,\theta ,\tau ,{t}^{\prime})={\epsilon}^{-2}P({I}^{\prime},\theta ,{\tau}^{\prime},{t}^{\prime}).$

The rescaled complete system is thus equivalent to the system with Hamiltonian ${K}_{0}({I}^{\prime},{\tau}^{\prime})+\epsilon P({I}^{\prime},\theta ,{\tau}^{\prime},{t}^{\prime})$ and the (standard) symplectic form $d{I}^{\prime}\wedge d\theta +d{\tau}^{\prime}\wedge dt$ after a further rescaling of time. In the region ${\tau}^{\prime}\sim 1$, ${I}^{\prime}\sim {\epsilon}^{3}$, the unperturbed function ${K}_{0}({I}^{\prime},{\tau}^{\prime})$ is ${C}^{\mathrm{\infty}}$-smooth. The determinant of the Hessian of ${K}_{0}({I}^{\prime},{\tau}^{\prime})$ reads $-{({\partial}^{2}{K}_{0}({I}^{\prime},{\tau}^{\prime})/\partial {I}^{\prime}\partial {\tau}^{\prime})}^{2}\sim 1$ and is now independent of the small parameter ε. For $r>4$, the ${C}^{r}$-norm of the perturbation $\epsilon P({I}^{\prime},\theta ,{\tau}^{\prime},{t}^{\prime})$ is seen to be of order $O(\epsilon )$. The cited KAM theorem can thus be applied directly to this rescaled system, provided that $\epsilon >0$ is chosen to be small enough.

We thus find a set of positive measure of invariant tori of *K*, in particular, the function *K* takes values in $(-\delta /2,\delta /2)$ on each of them, provided ε is small enough, and we deduce from [6, Theorem 2.1] that when $r>9$, these invariant tori are accumulated by periodic orbits of *K*. In view of Remark 2.2 and by Fubini’s theorem, the persisted invariant 2-tori form a set of positive measure in $\{K=0\}$ accumulated by periodic orbits, provided that ε is small enough. By assumption, $\tau \to \mathrm{\infty}$ when $\epsilon \to 0$.
∎

The existence of these two-dimensional KAM tori in the three-dimensional $\{K=0\}$ entails that these periodic orbits are “stable” in the sense that there are no large changes of the action variables τ and *I*, which translates into the energy and the amplitude of (1.1), respectively.

In [5], the existence of families of generalized periodic solutions is shown by an application of the Poincaré–Birkhoff theorem. In our case, these periodic solutions, which correspond to periodic points of the Poincaré map lying between invariant curves (which are themselves intersections of invariant 2-tori with the constructed Poincaré section), are stable à la Lagrange in the same sense that there are no large changes of the energy and the amplitude. Moreover, these invariant curves bound the positive but finite $|d\tau \wedge dt|$-measure in *D* preserved by $\mathcal{\mathcal{S}}$, which allows us to apply the Poincaré recurrence theorem (see [1, Section 2.6]) to confirm that $|d\tau \wedge dt|$-almost all points in the bounded regions (what is called Birkhoff’s region of instability) are recurrent. Therefore, the dynamics confined to these bounded regions is Poisson stable.

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