\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2009(2009), No. 54, pp. 1--17.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2009 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2009/54\hfil The first step normalization] {The first step normalization for hamiltonian systems with two degrees of freedom over \\ orbit cylinders} \author[G. D\'avila-Rasc\'on, Y. Vorobiev\hfil EJDE-2009/54\hfilneg] {Guillermo D\'avila-Rasc\'on, Yuri Vorobiev} % in alphabetical order \address{Guillermo D\'avila-Rasc\'on \newline Departamento de Matem\'aticas\\ Universidad de Sonora, Rosales y Blvd. Luis Encinas\\ Hermosillo, Sonora, 83000, Mexico} \email{davila@gauss.mat.uson.mx} \address{Yuri Vorobiev \newline Departamento de Matem\'aticas\\ Universidad de Sonora, Rosales y Blvd. Luis Encinas\\ Hermosillo, Sonora, 83000, Mexico} \email{yurimv@guaymas.uson.mx} \thanks{Submitted July 22, 2008. Published April 17, 2009.} \subjclass[2000]{37J05, 37J35, 37J40} \keywords{Perturbed Hamiltonian system; nearly integrable system; \hfill\break\indent orbit cylinder; Liouville tori; quasi-periodic motion; action-angle variables; \hfill\break\indent near identity transformation; time-1 flow} \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{corollary}[theorem]{Corollary} \newtheorem{example}[theorem]{Example} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \thanks{Partially supported by grant 55463 from CONACYT} \begin{abstract} The near integrability property is studied for a class of perturbed Hamiltonian systems with two degrees of freedom on a phase space whose symplectic form depends non-uniformly on a small parameter. \end{abstract} \maketitle \section{Introduction} The recognition of a Hamiltonian system as a nearly integrable system is the first step of the normalization procedures in the framework of KAM theory or averaging methods. The question on the near integrability arises in the study of Hamiltonian dynamics near an invariant non-zero dimensional submanifold, typically, a periodic trajectory or a quasi-periodic torus (see, for example \cite{Bruno1, Bruno2, BroHuiSevr, BelDobr, DavFloVor, KarVor}). In the present paper, we discuss the first step normalization for Hamiltonian systems with 2-degrees of freedom in the following setting which generalizes the case of a 2-submanifold of periodic trajectories (an orbit cylinder) \cite{Bruno1, Bruno2, DavFloVor, Vorob3}. Let $M = (\mathbb{R}^{1} \times \mathbb{S}^{1}) \times \mathbb{R}^{2}$ be the product manifold equipped with symplectic form non-uniformly depending on a parameter $\varepsilon > 0$, \begin{equation} \Omega^{\varepsilon} = \mathrm{d}s \wedge \mathrm{d}\varphi + \varepsilon \, \mathrm{d}p \wedge \mathrm{d}q, \label{nonunsymst} \end{equation} where $(s,\varphi \mod 2 \pi) \in \mathbb{R}^{1} \times \mathbb{S}^{1}$ and $(p,q) \in \mathbb{R}^{2}$. Consider a Hamiltonian system on $(M,\Omega^{\varepsilon})$ given by a smooth Hamiltonian of the form \begin{equation} H_{\varepsilon} = f(s) + \varepsilon \, F(s,\varphi,p,q). \label{Hameps} \end{equation} The corresponding equations of motion read \begin{gather} \dot{s} = - \varepsilon \, \frac{\partial F}{\partial \varphi}, \label{HamSys1}\\ \dot{\varphi} = \omega_{1}(s) + \varepsilon \, \frac{\partial F}{\partial s}, \label{HamSys2}\\ \dot{p} = - \frac{\partial F}{\partial q}, \label{HamSys3}\\ \dot{q} = \frac{\partial F}{\partial p}, \label{HamSys4} \end{gather} where $\omega_{1}(s) = {\partial f(s)}/{\partial s} > 0$. In general, this system is not completely integrable and our point is to study (\ref{HamSys1})-(\ref{HamSys4}) for small $\varepsilon$, in the context of the Hamiltonian perturbation theory. We remark that an alternative setting can be given on the standard phase space $(\mathbb{R}^{4}, \mathrm{d}p_{1} \wedge \mathrm{d}q_{1} + \mathrm{d} p_{2} \wedge \mathrm{d}q_{2})$ by considering the following class of Hamiltonian systems on $\mathbb{R}^{4}$: \begin{equation} \mathcal{H} = \mathcal{H}_{0}(p_{1},q_{1}) + \varepsilon \, \mathcal{H}_{1}(p_{1},q_{1},{p_{2}}/{\varepsilon^{\mu}}, {q_{2}}/{\varepsilon^{1-\mu}}), \label{HamR4} \end{equation} where $\mu \in [0,1]$. If an open domain $\mathcal{C} \subset \mathbb{R}^{2}$ is foliated by periodic trajectories of $\mathcal{H}_{0}$, then as $\varepsilon \to 0$, the behavior of $\mathcal{H}$ in a region $\{(p_{1},q_{1}) \in \mathcal{C}$, $p_{2} \sim \varepsilon^{\mu}$, $q_{2} \sim \varepsilon^{1-\mu} \}$ is described by a system like (\ref{HamSys1})-(\ref{HamSys4}). In particular, when $\mu ={1}/{2}$ and $\mathcal{H}_{1}$ is quadratic in $p_{2}$ and $q_{2}$, the Hamiltonian system $\mathcal{H}$ is independent of $\varepsilon$ and has an orbit cylinder $\mathcal{C}\times\{0\}$. Let $X_{\varepsilon} = X_{H_{\varepsilon}}$ be the Hamiltonian vector field of (\ref{HamSys1})-(\ref{HamSys4}) viewed as a perturbed dynamical system on $M$. Then, $X_{\varepsilon} = X_{0} + \varepsilon W$, where \begin{equation} X_{0} = \omega_{1}(s) \frac{\partial}{\partial\varphi} + \frac{\partial F}{\partial p} \frac{\partial}{\partial q} - \frac{\partial F}{\partial q}\frac{\partial}{\partial p}, \label{VfldX0} \end{equation} corresponds to the unperturbed dynamics and \begin{equation} W = - \frac{\partial F}{\partial\varphi} \frac{\partial}{\partial s} + \frac{\partial F}{\partial s} \frac{\partial}{\partial\varphi}, \label{VfldW} \end{equation} is a perturbation vector field. We assume that unperturbed system $X_{0}$ admits an additional integral of motion besides the trivial one, $s$. This means that the family of time-dependent Hamiltonian systems on $\mathbb{R}^{2}$ associated to $X_{0}$ is completely integrable. Nevertheless, in the context of near integrability of the perturbed system $X_{\varepsilon}$, the main difficulty is that the vector field $X_{0}$ (as an autonomous system) is not Hamiltonian relative to the original symplectic structure $\Omega^{\varepsilon}$. This effect comes from the singular dependence of $\Omega^{\varepsilon}$ on the parameter $\varepsilon$ ($\Omega^{\varepsilon}$ becomes degenerate at $\varepsilon = 0$). The idea is to search for a symplectic mapping $\Upsilon_{\varepsilon}$ (smoothly depending on $\varepsilon$) from $M$ to a canonical model phase space $N$ such that the transformed perturbed system $(\Upsilon_{\varepsilon})_{\ast} H_{\varepsilon}$ is $\varepsilon^{2}$-close to a completely integrable Hamiltonian system on $N$. The existence of such a \emph{twisting map} can be explained by the following observation \cite{DavVor, DavRas}: the unperturbed system $X_{0}$ is, in fact, Hamiltonian in a ``deformed'' non-canonical symplectic structure (see also \cite{DavFloVor, Vorob3}). Here, we show that the normalizing transformation is represented as the composition $\Upsilon_{\varepsilon} = \Psi_{\varepsilon} \circ \mathcal{T}$ of an $\varepsilon$-independent mapping $\mathcal{T}$ and a near identity transformation $\Psi_{\varepsilon}$. The mapping $\mathcal{T}$ transforms $X_{0}$ to a system with parallel dynamics but it is is not symplectic. The mapping $\Psi_{\varepsilon}$ is defined as the time-1 flow of a non-autonomous system and gives a near identity isomorphism between the transformed symplectic form $\mathcal{T}_{\ast}\Omega^{\varepsilon}$ and the canonical symplectic structure on $N$. Here we apply a parameter-dependent version of the Moser homotopy method \cite{GuilLerSter}. In general, the transformation $\Psi_{\varepsilon}$ is not infinitesimal, except for some particular cases, for example, when $F$ in (\ref{Hameps}) is independent of $s$ (see \cite{Bruno1, Bruno2, BelDobr}). In the linear case, when $F$ is quadratic in $p$, $q$, the unperturbed vector field $X_{0}$ describes the linearized dynamics at the orbit cylinder and the existence of an additional integral of motion is provided by the stability property of $X_{0}$. The mapping $\mathcal{T}$ corresponds to the reducibility transformation in the sense of the Floquet theory for linear periodic Hamiltonian systems \cite{Yakub}. \section{Main Results} \label{sec:mainres} On the phase space $M =(\mathbb{R}^{1} \times \mathbb{S}^{1}) \times \mathbb{R}^{2}$, consider the dynamical system of the unperturbed vector field $X_{0}$: \begin{gather} \dot{s} = 0, \label{HamSysS1}\\ \dot{\varphi} = \omega_{1}(s), \label{HamSysS2}\\ \dot{p} = - \frac{\partial F}{\partial q}(s,\varphi,p,q), \label{HamSysS3}\\ \dot{q} = \frac{\partial F}{\partial p} (s,\varphi,p,q). \label{HamSysS4} \end{gather} Denote by $\pi : M \to \mathbb{R}^{1} \times \mathbb{S}^{1}$ the canonical projection onto the first factor and consider $M$ as the total space of the trivial symplectic vector bundle $\pi$ over $\mathbb{R}^{1} \times \mathbb{S}^{1}$ with fiberwise symplectic structure $\mathrm{d}p \wedge \mathrm{d}q$. The base is trivially foliated by the periodic orbits of subsystem (\ref{HamSysS1}),(\ref{HamSysS2}) which is viewed as a Hamiltonian system with one degree of freedom. Geometrically, system (\ref{HamSysS1})-(\ref{HamSysS4}) belongs to the class of projectable systems, that is, the trajectories of $X_{0}$ project under $\pi$ to the periodic orbits of (\ref{HamSysS1}),(\ref{HamSysS2}). To each function $G$ on $M$, one can associate the \emph{vertical Hamiltonian vector field} \begin{equation} V_{G} = \frac{\partial G}{\partial p} \frac{\partial}{\partial q} - \frac{\partial G}{\partial q} \frac{\partial}{\partial p}. \label{VerVfld} \end{equation} It is clear that the trajectory of $V_{G}$ passing through a point $m \in M$ belongs to the fiber over $\pi(m)$. In other words, we can think of $V_{G}$ as a family of autonomous Hamiltonian system on $\mathbb{R}^{2}$ whose Hamiltonian $G_{s,\varphi}$ depends parametrically on $(s, \varphi) \in \mathbb{R}^{1} \times \mathbb{S}^{1}$. We assume that the following integrability hypothesis holds: \begin{itemize} \item[(IH)] There exists an open domain $\mathcal{M} \subset M$ and a smooth integral of motion $G : \mathcal{M} \to\mathbb{R} $ of $X_{0}$, \begin{equation} L_{X_{0}} G \equiv \omega_{1} \frac{\partial G}{\partial\varphi} + \frac{\partial F}{\partial p} \frac{\partial G}{\partial q} - \frac{\partial F}{\partial q} \frac{\partial G}{\partial p} = 0, \label{HypInt} \end{equation} such that $\mathcal{M}$ is foliated by the periodic trajectories of $V_{G}$, $\mathcal{M} \cap \pi^{-1}(b)$ is connected for every $b=(s,\varphi)$ and \begin{equation} \pi(\mathcal{M}) = \Delta \times \mathbb{S}^{1}, \label{piM} \end{equation} where $\Delta \subset \mathbb{R}$ is an open interval. \end{itemize} Consider the second product manifold \begin{equation} N = (\mathbb{R}^{1} \times \mathbb{S}^{1}) \times (\mathbb{R}^{1} \times \mathbb{S}^{1}) = \bigl\{ (s_{1}, \varphi_{1} \mod 2 \pi, s_{2}, \varphi_{2} \mod 2\pi ) \bigr\}, \label{SpaceN} \end{equation} with natural projection $\nu : N \to \mathbb{R}^{1} \times \mathbb{S}^{1}$ onto the first factor. Then, $N$ is the total space of the trivial symplectic bundle with fiberwise symplectic structure $\mathrm{d}s_{2} \wedge \mathrm{d}\varphi_{2}$. Therefore, we have two trivial symplectic bundles $\pi$ and $\nu$ over one and the same base $\mathbb{R}^{1} \times \mathbb{S}^{1}$. We say that a subset $\mathcal{N} \subset N$ is a \emph{simple toroidal domain} if $\nu(\mathcal{N}) = \Delta \times \mathbb{S}^{1}$ and $\iota(\mathcal{N)} = D_{\mathcal{N}} \times \mathbb{T}^{2}$, where $D = D_{\mathcal{N}} \subset \mathbb{R}^{2}$ is an open, connected and simply connected subset. Here $\iota : N \to \mathbb{R}^{2} \times \mathbb{T}^{2}$ denotes the canonical identification, $\iota(s_{1},\varphi_{1},s_{2}, \varphi_{2}) = (s_{1},s_{2},\varphi_{1},\varphi_{2})$. \begin{proposition}[Reducibility] \label{pro:reducib} There exist a simple toroidal domain $\mathcal{N} \subset N$ and a fibered diffeomorphism $\mathcal{T} : \mathcal{M} \to \mathcal{N}$ over $\Delta \times \mathbb{S}^{1}$, \begin{equation} \mathcal{T}(s,\varphi,p,q) = \bigl( s,\varphi,\mathcal{T}_{s,\varphi}(p,q) \bigr), \label{gaugeT1} \end{equation} which preserves the fiberwise symplectic structures, \begin{equation} (\mathcal{T}_{s,\varphi})_{\ast} \mathrm{d}p \wedge \mathrm{d}q = \mathrm{d}s_{2} \wedge \mathrm{d}\varphi _{2}, \label{gaugeT2} \end{equation} and such that the dynamical system of the push-forward $\mathcal{T}_{\ast} X_{0}$ takes the form \begin{gather} \dot{s_{1}} = 0, \label{PushSys1}\\ \dot{\varphi}_{1} = \omega_{1}(s_{1}), \label{PushSys2}\\ \dot{s_{2}} = 0, \label{PushSys3}\\ \dot{\varphi}_{2} = \omega_{2}(s_{1},s_{2}), \label{PushSys4} \end{gather} where $\omega_{2} = \omega_{2}(s_{1},s_{2})$ is a smooth function on $D_{\mathcal{N}}$. \end{proposition} As a consequence, we get that the domain $\mathcal{M}$ is trivially foliated by $2$-tori \begin{equation} \Lambda_{c_{1},c_{2}} = \mathcal{T}^{-1}(\mathbb{T}_{c_{1},c_{2}}^{2}), \label{LambTori} \end{equation} where $(c_{1},c_{2})$ runs over $D$ and $\mathbb{T}_{c_{1},c_{2}}^{2} = \{s_{1} = c_{1}, \; s_{2} = c_{2} \}$. Each torus $\Lambda_{c_{1},c_{2}}$ is the level set of the integrals of motion $s$ and $G$, carrying a quasi-periodic motion along the trajectories of $X_{0}$ with frequencies $\omega_{1}(c_{1})$ and $\omega_{2}(c_{1},c_{2})$. However, if ${\partial G}/{\partial \varphi} \neq 0$, then $\Lambda_{c_{1},c_{2}}$ are not Lagrangian tori with respect to the symplectic structure (\ref{nonunsymst}). An interpretation of $\Lambda_{c_{1},c_{2}}$ as Liouville tori is related with a Hamiltonian formulation for (\ref{HamSysS1})-(\ref{HamSysS4}) in a non-canonical symplectic structure on $M$ \cite{DavVor, DavRas}. In Section~\ref{sec:reducib}, we give a construction of $\mathcal{T}$ which is based on the Poincar\'e--Cartan invariant. In Section~\ref{sec:lincas}, we show that in the particular case when $X_{0}$ corresponds to the linearized dynamics around the orbit cylinder, the mapping $\mathcal{T}$ is just the Floquet--Lyapunov transformation \cite{FlorVor1, Yakub}. Fix a diffeomorphism $\mathcal{T}$ in Proposition~\ref{pro:reducib} and consider $N$ as a model phase space equipped with non-uniform canonical symplectic form \begin{equation} \widetilde{\Omega}^{\varepsilon} = \mathrm{d}s_{1} \wedge \mathrm{d} \varphi_{1} + \varepsilon \, \mathrm{d}s_{2} \wedge \mathrm{d}\varphi_{2}. \label{wtsymstr} \end{equation} The following observation says that the reducibility map is not symplectic. \begin{proposition} \label{pro:nonsym} The original symplectic structure $\Omega^{\varepsilon}$ \eqref{nonunsymst} is transformed under $\mathcal{T}$ to the following non-canonical symplectic form on $\mathcal{N}$, \begin{equation} \mathcal{T}_{\ast} \Omega^{\varepsilon} = \widetilde{\Omega}^{\varepsilon} - \varepsilon \, \mathrm{d}Q, \label{pushTOme} \end{equation} where \begin{equation} Q = Q_{1}(s_{1},\varphi_{1},s_{2},\varphi_{2}) \, \mathrm{d}s_{1} + Q_{2}(s_{1}, \varphi_{1}, s_{2}, \varphi_{2}) \, \mathrm{d}\varphi_{1}, \label{HorFrmQ} \end{equation} is a horizontal $1$-form. \end{proposition} The $2$-form (\ref{pushTOme}) gives a special deformation of $\widetilde{\Omega}^{\varepsilon}$ and belongs to the class of the so-called \emph{weak coupling of symplectic structures} \cite{GuilLerSter}. To complete the normalization procedure, we search for an isomorphism between $\mathcal{T}_{\ast} \Omega^{\varepsilon}$ and $\widetilde{\Omega}^{\varepsilon}$ in the class of near identity transformations. Given $\Delta$ and $\mathcal{M}$ in (IH), we say that another open domain $\mathcal{M}_{0}\subset M$, also satisfying the hypothesis (IH), is \emph{admissible} if (\ref{piM}) holds for a certain open interval $\Delta_{0}$ such that $\Delta_{0} \subset \Delta$, the closure $\mathcal{\overline{M}}_{0}$ is compact and $\mathcal{\overline{M}}_{0} \subset \mathcal{M}$. It is clear that an admissible domain always exists. Now, we formulate our main result. \begin{theorem} \label{teo:mainthm} Let $\mathcal{M}_{0} \subset \mathcal{M}$ be an admissible domain and $\mathcal{N}_{0} = \mathcal{T}(\mathcal{M}_{0})$. For sufficiently small $\varepsilon \geq 0$, there exists a diffeomorphism $\Psi_{\varepsilon} : \mathcal{N}_{0} \to \mathcal{N}$ onto its image, smoothly depending on $\varepsilon$, with $\Psi_{0} = \mathrm{id}$ and such that \begin{equation} \Upsilon_{\varepsilon} = \Psi_{\varepsilon} \circ \mathcal{T} , \label{sympmapY} \end{equation} is a symplectic map, \begin{equation} (\Upsilon_{\varepsilon})_{\ast} \Omega^{\varepsilon} = \widetilde{\Omega}^{\varepsilon}, \quad (\varepsilon > 0), \label{pushYpsi} \end{equation} transforming the original Hamiltonian system \eqref{HamSys1}-\eqref{HamSys4} into the normal form, \begin{equation} \widetilde{H}_{\varepsilon} := (\Upsilon_{\varepsilon})_{\ast} H_{\varepsilon} = f(s_{1}) + \varepsilon \, h(s_{1},s_{2}) + O(\varepsilon^{2}), \label{tildeHe} \end{equation} where \begin{equation} \frac{\partial h}{\partial s_{2}} (s_{1},s_{2}) = \omega_{2}(s_{1},s_{2}). \label{functh} \end{equation} \end{theorem} In Section~\ref{sec:nearIdT}, the near identity transformation $\Psi_{\varepsilon}$ is constructed by means of a parameter dependent version of the Moser homotopy method \cite{GuilLerSter}. The transformed perturbed Hamiltonian $\widetilde{H}_{\varepsilon}$ is $\varepsilon^{2}$-close to the Hamiltonian $\widetilde{H}_{\varepsilon}^{(0)} = f(s_{1}) + \varepsilon \, h(s_{1},s_{2})$ which defines a completely integrable Hamiltonian system on $(N, \widetilde{\Omega}^{\varepsilon})$. The invariant tori $\mathbb{T}_{c_{1},c_{2}}^{2}$ of system (\ref{PushSys1})-(\ref{PushSys4}) are now the Liouville tori of $\widetilde{H}_{\varepsilon}^{(0)}$ which carry the quasi-periodic motion with deformed frequencies $\omega_{1}(c_{1}) + \varepsilon \, {\partial h(c_{1},c_{2})}/{\partial s_{1}}$ and $\omega(c_{1},c_{2})$. If the frequencies satisfy some appropriate nondegeneracy condition (for example, in the sense of Kolmogorov or R\"ussmann), then one can apply the KAM type results \cite{BroHuiSevr} to state the persistence of quasi-periodic tori $\Upsilon_{\varepsilon}^{-1}(\mathbb{T}_{c_{1},c_{2}}^{2})$ for perturbed system (\ref{HamSys1})-(\ref{HamSys4}) as $\varepsilon \to 0$. \begin{example} \label{exa2.4} \rm On the phase space $(\mathbb{R}^{4} = \{ (y,x,p,q) \}, \Omega^{\varepsilon} = \mathrm{d} y \wedge \mathrm{d} x + \varepsilon \, \mathrm{d} p \wedge \mathrm{d} q )$, consider the perturbed Hamiltonian system \begin{equation} \label{ExHampert} H_{\varepsilon} = \frac{y^{2}}{2} + U_{0}(x) + \varepsilon \Big( \frac{p^{2}}{2} + \frac{U_{0}'(x)}{2 x} q^{2} + \frac{1}{x^{2}} U_{1}\left( \frac{q}{x} \right) \Big), \end{equation} where $U_{0}$, $U_{1}$ are arbitrary smooth functions. Then, the corresponding unperturbed system \begin{gather} \dot{y} = - U_{0}'(x), \label{exaeq1}\\ \dot{x} = y, \label{exaeq2}\\ \dot{p} = - \frac{U_{0}' (x)}{x} q - \frac{1}{x^{3}} U_{1}'\left( \frac{q}{x} \right), \label{exaeq3}\\ \dot{q} = p, \label{exaeq4} \end{gather} has the following integral of motion, \cite{Lewis, Haas}, \begin{equation} \label{ExGintmot} G = \frac{1}{2} \left( x p - y q \right)^{2} + U_{1}'\left( \frac{q}{x} \right). \end{equation} Suppose that the potentials $U_{0}$ and $U_{1}$ have local nondegenerate minima at some points $x_{0}$ and $z_{0}$, respectively. In this case, under passing to the standard action-angle variables $(y,x) \mapsto (s, \varphi)$ around $(0,x_{0})$ associated with \eqref{exaeq1}, \eqref{exaeq2}, system \eqref{exaeq1}-\eqref{exaeq4} takes the form (\ref{HamSysS1})-(\ref{HamSysS4}). Moreover, for a fixed $(y,x)$, the equilibrium $(y z_{0}, x z_{0})$ of $V_{G}$ is surrounded by periodic trajectories. Therefore, the integral of motion $G$ in (\ref{ExGintmot}) satisfies the condition (IH) and we can apply Theorem \ref{teo:mainthm}. \end{example} In the linear case, when $F$ in (\ref{Hameps}) is quadratic in $p$, $q$, the integrability hypothesis holds if the unperturbed system $X_{0}$ is \emph{strongly stable}. The linear version of Theorem~\ref{teo:mainthm} is discussed in Section~\ref{sec:lincas}. \section{Reducibility} \label{sec:reducib} Here we describe an algorithm for the construction of the reducibility transformation $\mathcal{T}$ in Proposition~\ref{pro:reducib}, assuming that hypothesis (IH) holds and the corresponding data $(\Delta, \mathcal{M}, G)$ are given. Consider the vector field $X_{0}$ of system (\ref{HamSysS1})-(\ref{HamSysS4}) which can be rewritten in the form \begin{equation} X_{0} = \omega_{1} \frac{\partial}{\partial\varphi} + V_{F}, \label{X0Vf} \end{equation} where $V_{F}$ is the vertical Hamiltonian vector field given by (\ref{VerVfld}). Then, $X_{0}$ has two integrals of motion, namely, $s$ and $G$. Let $J : \mathcal{M} \to \mathbb{R}^{2}$ be the corresponding ``momentum'' map, $J(s,\varphi, p, q) = \bigl(s, G(s,\varphi,p,q) \bigr)$. It follows from (IH) that $\big( \frac{\partial G}{\partial p},\frac{\partial G}{\partial q} \big) \neq 0$ on $\mathcal{M}$ and hence, $J$ is a surjective submersion onto its image. Moreover, $\mathcal{M}$ is foliated by the compact connected $2$-manifolds \begin{equation} \Lambda_{\xi} = J^{-1}(\xi), \quad \bigl( \xi \in J(\mathcal{M}) \bigr). \label{Momtmmap} \end{equation} We have also the following properties of vector fields $X_{0}$ and $V_{G}$: \begin{itemize} \item[(a)] $X_{0}$ and $V_{G}$ are linear independent on $\mathcal{M}$; \item[(b)] $V_{G}$ is tangent to each fiber $\Lambda_{\xi}$; \item[(c)] $X_{0}$ and $V_{G}$ commute, \begin{equation} [ X_{0},V_{G} ] = V_{L_{X_{0}}G} = 0. \label{commX0Vg} \end{equation} \end{itemize} It follows from here \cite{Arnld1, Kozl} that every level set $\Lambda_{\xi}$ is diffeomorphic to the $2$-torus and carries a quasi-periodic motion along the trajectories of $X_{0}$. In particular, $X_{0}$ is a complete vector field on $\mathcal{M}$. However, as we mentioned above, $X_{0}$ is not Hamiltonian relative to the original symplectic structure and we can not directly apply the Arnold-Liouville theorem on the action-angle variables to construct $\mathcal{T}$. Our argument is based on the Poincar\'e--Cartan invariant for time-dependent Hamiltonian systems. Denote by $\mathsf{Fl}_{X_{0}}^{t} : \mathcal{M} \to \mathcal{M}$ the flow of $X_{0}$. For a fixed $s \in \Delta$, one can associate to $X_{0}$ the time-dependent Hamiltonian system on $\mathbb{R}^{2}$ with Hamiltonian $F(s,\omega_{1}(s) \, t, p, q)$. Then, we have the following fact \cite{Arnld1,Kozl}: for any closed curve $\Gamma \subset \{ s \} \times \mathbb{S}^{1} \times \mathbb{R}^{2}$ transversal to $X_{0}$, the integral \begin{equation} \oint_{\Gamma_{t}} \Big[ p\, \mathrm{d}q - F(s,\varphi,p,q) \frac{\mathrm{d}\varphi}{\omega_{1}(s)} \Big] = \mathrm{const}, \label{PoinCarInv} \end{equation} that is, does not depend on $t$. Here $\Gamma_{t} = \mathsf{Fl}_{X_{0}}^{t}(\Gamma)$. For every $(s,\varphi) \in \Delta \times \mathbb{S}^{1}$, denote by $G_{s,\varphi} : \mathbb{R}^{2} \to \mathbb{R}$ the function given by $G_{s,\varphi}(p,q) = G(s,\varphi,p,q)$. Let $\pi^{-1}(s,\varphi) = \{ (s,\varphi) \} \times \mathbb{R}^{2}$ be the fiber of $\pi$ over $(s,\varphi)$. We have \begin{equation} \pi^{-1}(s,\varphi) \cap \mathcal{M} = \{ (s,\varphi) \} \times \mathcal{U}_{s,\varphi}, \label{fiberinvM} \end{equation} where $\mathcal{U}_{s,\varphi}$ is an open connected domain in $\mathbb{R}^{2}$. Observe that under varying $(s,\varphi)$, the set of values of $G_{s,\varphi}$ on $\mathcal{U}_{s,\varphi}$ is independent of $\varphi$. This follows from (\ref{commX0Vg}) and the property that the flow $\mathsf{Fl}_{X_{0}}^{t}$ preserves the fibers of $\pi$. Taking into account that $G$ can be renormalized by multiplication for any nonzero function of $s$, without lost of generality, we may assume that \begin{equation} G_{s,\varphi}(\mathcal{U}_{s,\varphi}) = (E_{1},E_{2}), \label{GsfiE1E2} \end{equation} for some constants $E_{1} < E_{2}$. Fix $(s,\varphi)$ and consider the Hamiltonian system on $\mathbb{R}^{2}$ associated with the restriction of the vertical field $V_{G}$ to the fiber over $(s,\varphi)$: \begin{gather} \frac{\mathrm{d}p}{\mathrm{d}t} = -\frac{\partial G_{s,\varphi}}{\partial q}, \label{HamsysG1}\\ \frac{\mathrm{d}q}{\mathrm{d}t} = \frac{\partial G_{s,\varphi}}{\partial p}. \label{HamsysG2} \end{gather} By the hypothesis, the level set \begin{equation} \gamma_{s,\varphi}(E) = \{ (p,q) \in \mathcal{U}_{s,\varphi} : G_{s,\varphi}(p,q) = E \}, \label{pathgamm} \end{equation} is a periodic trajectory of system (\ref{HamsysG1}), (\ref{HamsysG2}) for every $E \in (E_{1},E_{2})$. It is clear that $\mathcal{U}_{s,\varphi}$ is trivially foliated by $\gamma_{s,\varphi}(E)$ over $(E_{1},E_{2})$. Thus, one can introduce the standard action-angle variables on $\mathcal{U}_{s,\varphi}$ associated to this foliation. The point is to choose these coordinates to be smooth functions of the parameters $(s,\varphi)$. Let $g : \mathcal{M} \to (\Delta \times \mathbb{S}^{1}) \times (E_{1},E_{2})$ be the surjective submersion defined as $g = \pi \times G$. Then, the fibers \begin{equation} \Gamma_{s,\varphi}(E) := g^{-1}(s,\varphi,E) \equiv \{(s,\varphi) \} \times \gamma_{s,\varphi}(E), \label{GammasfiE} \end{equation} are just the periodic trajectories of $V_{G}$ in $\mathcal{M}$. It follows from (\ref{commX0Vg}) that the flow $\mathsf{Fl}_{X_{0}}^{t}$ is also a fiber preserving map with respect to $g$, \begin{equation} \mathsf{Fl}_{X_{0}}^{t}(\Gamma_{s,\varphi}(E)) = \Gamma_{s,\varphi + \omega_{1}(s)t}(E). \label{FlowX0} \end{equation} Putting $\Gamma = \Gamma_{s,\varphi}(E)$ into (\ref{PoinCarInv}) and using (\ref{FlowX0}), we get that the action along $\gamma_{s,\varphi}(E)$ is independent of $\varphi$, \begin{equation} a(s,E) = \frac{1}{2\pi} \oint_{\gamma_{s,\varphi}(E)} p \, \mathrm{d}q, \label{intgampdq} \end{equation} and defines a smooth function $a: \Delta \times ( E_{1},E_{2} ) \to \mathbb{R}$. The period of $\gamma_{s,\varphi}(E)$ is given by \begin{equation} T(s,E) = \frac{\partial a}{\partial E} (s,E). \label{periodT} \end{equation} Let us define \begin{equation} A := a\circ J, \label{ActionA} \end{equation} or, equivalently, $A(s, \varphi,p,q) = a(s, G(s, \varphi, p, q))$. It is clear that $A$ is an integral of motion of $X_{0}$ which represents a parameter-dependent action variable of (\ref{HamsysG1}), (\ref{HamsysG2}). The construction of the corresponding angle variable (smoothly depending on $s$ and $\varphi$) is related with the existence of a global section of the fibration $g$. \begin{lemma} \label{lem3.1} The domain $\mathcal{M}$ is trivially fibered by periodic trajectories $\Gamma_{s,\varphi}(E)$ of $V_{G}$, that is, there exists a smooth section $L : (\Delta \times \mathbb{S}^{1}) \times (E_{1},E_{2}) \to \mathcal{M}$ of $g$, $g \circ L = \mathrm{id}$. \end{lemma} \begin{proof} Let $g_{0} : \mathcal{M}_{0} \to \Delta \times(E_{1},E_{2})$ be the restriction of $g$ to the slice $\mathcal{M}_{0} = \mathcal{M}\cap \{ \varphi = 0 \}$ which is a contractible open subset. Then, there exists a smooth section $L_{0}$ of $g_{0}$, $L_{0}(s,E) \in \Gamma_{s,0}(E)$. To extend $L_{0}$ to the whole $\mathcal{M}$, let us consider a vector field on $\mathcal{M}$ of the form \[ Y = \frac{1}{\omega_{1}} X_{0} - \frac{\tau \circ J}{2\pi} V_{G}, \] Here $\tau$ is a smooth function on $\Delta \times (E_{1},E_{2})$ which is defined in the following way. Fix $(s,E)$ and consider the $2$-torus $\Lambda_{s,E} = J^{-1}(s,E)$. Then, $\Lambda_{s,E}$ is the disjoint union of the periodic trajectories of $V_{G}$, \[ \Lambda_{s,E} = \cup_{\varphi \in (0,2\pi]} \Gamma_{s,\varphi}(E). \] Pick a point $m = (s,\varphi,p,q) \in \Gamma_{s,\varphi}(E)$. It follows from (\ref{FlowX0}) that the trajectory of $X_{0}$ starting at $m$ stays on $\Lambda_{s,E}$ and meets again the trajectory $\Gamma_{s,\varphi}(E)$ at the point $\widetilde{m}$ after the time $t_{0} = {2\pi}/{\omega_{1}(s)}$, \[ \mathsf{F}l_{X_{0}}^{t_{0}}(m) = \widetilde{m} \in \Gamma_{s,\varphi}(E). \] Let $\tau$ be the time along the trajectory of $V_{G}$ from $m$ to $\widetilde{m}$, $\mathsf{Fl}_{V_{G}}^{\tau}(m) = \widetilde{m}$. One can show that $\tau$ does not depend on the choice of $m$ and $\tau = \tau(s,E)$ smoothly varies with $(s,E)$. Taking into account that $[Y,V_{G}] = 0$, we derive the following properties: (i) the flow of $Y$ is $2\pi$-periodic, and (ii) $\mathsf{Fl}_{Y}^{t}(\Gamma_{s,\varphi}(E)) = \Gamma_{s,\varphi + t}(E)$. Finally, we put \begin{equation} L(s,\varphi,E) = \mathsf{Fl}_{Y}^{\varphi}(L^{0}(s,E)). \label{sectL} \end{equation} \end{proof} We will suppose that a section $L$ in (\ref{sectL}) is given. Consider now the product manifold $N = (\mathbb{R}^{1} \times \mathbb{S}^{1}) \times (\mathbb{R}^{1} \times \mathbb{S}^{1})$ with coordinates $(s_{1},\varphi_{1} \mod 2\pi,s_{2}, \varphi_{2} \mod 2\pi)$. Denote by $E(s_{1},s_{2})$ the solution of the equation $s_{2} = a(s_{1},E)$. Here $(s_{1},s_{2})$ runs over the open domain \begin{equation} D = \cup_{s_{1} \in \Delta} \{ s_{1} \} \times D_{s_{1}} \; \subset \mathbb{R}^{2}, \label{DomD} \end{equation} where $D_{s_{1}} = \{ s_{2} = a(s_{1},E), \; E \in (E_{1},E_{2}) \}$. Define also \begin{equation} \mathcal{N} := \cup_{s_{1} \in \Delta} \{ s_{1} \} \times \mathbb{S}^{1} \times D_{s_{1}} \times \mathbb{S}^{1}. \label{DomN} \end{equation} It is clear that $\mathcal{N}$ is a simple toroidal domain in $N$. Let $Z = ({\mathcal{T} \circ J} / {2\pi}) V_{G}$ be the infinitesimal generator of the trivial $\mathbb{S}^{1}$-action on $\mathcal{M}$ whose orbits are just the periodic trajectories $\Gamma_{s,\varphi}(E)$. Using the section $L$ and the flow of $Z$, we define a mapping $\mathcal{R} : \mathcal{N} \to\mathcal{M}$ as follows: \begin{equation} \mathcal{R}(s_{1},\varphi_{1},s_{2},\varphi_{2}) := \mathsf{Fl}_{Z}^{\varphi_{2}} \bigl( L(s_{1},\varphi_{1},E(s_{1},s_{2})) \bigr). \label{Rtransf} \end{equation} It follows that $\mathcal{R}$ is a fiber preserving diffeomorphism covering the identity, with the inverse $\mathcal{R}^{-1}: \mathcal{M} \to \mathcal{N}$ given by \begin{gather} s_{1} \circ \mathcal{R}^{-1} = s, \quad \varphi_{1} \circ \mathcal{R} ^{-1} = \varphi, \label{Rtransf1}\\ s_{2} \circ \mathcal{R}^{-1} = A, \quad \varphi_{2} \circ \mathcal{R}^{-1} = \phi_{0}. \label{Rtransf2} \end{gather} Therefore, $A = A(s,\varphi,p,q)$ and $\phi_{0} = \phi_{0}(s,\varphi,p,q)$ in (\ref{Rtransf2}) are the standard action-angle coordinates of system (\ref{HamsysG1}), (\ref{HamsysG2}), parametrically depending on $s$ and $\varphi$. It follows that \begin{equation} \{ A, \phi_{0} \} := \frac{\partial A}{\partial p} \frac{\partial\phi_{0} }{\partial q} - \frac{\partial A}{\partial q} \frac{\partial \phi_{0} }{\partial p} = 1. \label{braktAfi} \end{equation} \begin{lemma} \label{lem3.2} Under the coordinate change $\mathcal{R}^{-1} : (s,\varphi,p,q) \mapsto (s_{1},\varphi_{1},s_{2},\varphi_{2})$, the equations of motion of $\mathcal{R}^{\ast}X_{0}$ take the form \begin{gather} \dot{s_{1}} = 0, \label{Systransf1}\\ \dot{\varphi}_{1} = \omega_{1}(s_{1}), \label{Systransf2}\\ \dot{s_{2}} = 0, \label{Systransf3}\\ \dot{\varphi}_{2} = \Theta(s_{1},\varphi_{1},s_{2}), \label{Systransf4} \end{gather} where $\Theta$ is $2\pi$-periodic function in $\varphi_{1}$. \end{lemma} \begin{proof} Equations (\ref{Systransf1})-(\ref{Systransf3}) follow directly from the definition of $\mathcal{R}$. The time evolution of $\varphi_{2} = \phi_{0} \circ \mathcal{R}$, according to (\ref{HamSysS1})-(\ref{HamSysS4}) is given by $\dot{\varphi}_{2} = \theta \circ \mathcal{R}$, where \begin{equation} \theta = \omega_{1}(s) \frac{\partial \phi_{0}}{\partial\varphi} + \{F, \phi_{0} \}. \label{thetafun} \end{equation} Observe that \begin{equation} \{\theta, A \} = 0. \label{brktthetaA} \end{equation} Indeed, combining relations (\ref{braktAfi}) and (\ref{thetafun}) with the Jacobi identity for the bracket $\{ , \}$, we derive \begin{align*} \{ \theta, A \} & = \omega_{1} \big\{ \frac{\partial \phi_{0}}{\partial \varphi} , A \big\} + \{ \{F, \phi_{0} \}, A\} \\ & = \omega_{1} \big\{ \frac{\partial \phi_{0}}{\partial \varphi}, A \big\} - \{ \{ \phi_{0}, A \}, F \} - \{ \{ A, F \},\phi_{0} \}\\ & = \omega_{1} \big\{ \frac{\partial \phi_{0}}{\partial \varphi}, A \big\} - \{\phi_{0}, \{ F, A\}\} \\ & = \omega_{1} \big\{ \frac{ \partial \phi_{0}}{\partial \varphi}, A \big\} + \omega_{1} \big\{ \phi_{0}, \frac{\partial A}{\partial \varphi} \big\} \\ &= \omega_{1} \frac{\partial}{\partial \varphi} \{ \phi_{0}, A \} = 0. \end{align*} Now, from (\ref{brktthetaA}) we have \[ \{ \theta,A \} \circ \mathcal{R} = \{ \theta \circ \mathcal{R}, A \circ \mathcal{R} \} = \{ \theta \circ \mathcal{R},s_{2} \} = - \frac{\partial }{\partial \varphi_{2}} (\theta \circ \mathcal{R}) = 0, \] and hence $\theta = \Theta(s,\varphi,A(s,\varphi,p,q))$, where $\Theta = \Theta(s_{1},\varphi_{1},s_{2})$ is a smooth function $2\pi $-periodic in $\varphi_{1}$. \end{proof} Next, given an arbitrary smooth function $\chi = \chi(s_{1},\varphi_{1},s_{2})$ which is $2\pi$-periodic in $\varphi_{1}$, one can correct the angle variable as follows: $\phi_{0} \mapsto \phi \equiv \phi_{0} + \chi \circ \mathcal{A}$. Here, $\mathcal{A}(s,\varphi,p,q) = (s,\varphi,A(s,\varphi,p,q))$. It is clear that such a transformation preserves bracket relation (\ref{braktAfi}). In order to eliminate the dependence on $\varphi_{1}$ in the right hand side of (\ref{Systransf4}), we have to put \begin{equation} \chi = \omega_{2}(s_{1},s_{2})\varphi_{1} - \int_{0}^{\varphi_{1}} \Theta (s_{1},\varphi_{1}',s_{2}) \, \mathrm{d}\varphi_{1}', \label{funcchi} \end{equation} where \begin{equation} \omega_{2}(s_{1},s_{2}) =\frac{1}{2\pi} \int_{0}^{2\pi} \Theta(s_{1},\varphi _{1}, s_{2}) \, \mathrm{d}\varphi_{1}, \label{freqw2} \end{equation} for $(s_{1},s_{2}) \in D$. Summarizing, we get that the transformation $\mathcal{T} : \mathcal{M} \to \mathcal{N}$ satisfying the assertions of Proposition~\ref{pro:reducib} is given by the formula \begin{equation} \mathcal{T}(s,\varphi,p,q) = \bigl( s, \varphi, A(s, \varphi,p,q), \phi(s,\varphi, p, q) \bigr), \label{Tsfipq} \end{equation} with \begin{equation} A = a\circ J, \quad \phi = \varphi_{2} \circ \mathcal{R}^{-1} + \chi\circ \mathcal{A}. \label{formulaA} \end{equation} \begin{remark} \label{em3.3} \rm According to the standard time-dependent Hamiltonian approach {\cite{Arnld1, Kozl}}, one can associate to $X_{0}$ the $s$-parameter family of completely integrable Hamiltonian systems $\widetilde{F}_{s} = \omega_{1}(s) \, \eta + F(s,\varphi,p,q)$ on the phase space $(\mathbb{R}^{1} \times \mathbb{S}^{1}) \times\mathbb{R}^{2}$ with canonical symplectic structure $\mathrm{d} \eta \wedge \mathrm{d}\varphi + \mathrm{d}p \wedge \mathrm{d}q$. Then, functions $A$ and $\phi$ in {(\ref{formulaA})} can be also derived form the action-angle variables associated to the trivial foliation by the Liouville $2$-tori $\{ \widetilde{F}_{s} = \mathrm{const}, G_{s} = \mathrm{const} \}$ (see, for example, \cite{Giach}). \end{remark} \section{Constructing a Near Identity Transformation} \label{sec:nearIdT} Fix a section $L$ in (\ref{sectL}) and consider the corresponding reducibility map $\mathcal{T} : \mathcal{M} \to \mathcal{N}$ given by (\ref{Tsfipq}). First we show that the symplectic structure $\Omega^{\varepsilon}$ is transformed under $\mathcal{T}$ by the rule (\ref{pushTOme}). Pick a $(s,\varphi) \in\mathbb{R}^{1} \times \mathbb{S}^{1}$ and consider the connected open domain $\mathcal{U}_{s,\varphi} \subset \mathbb{R}^{2}$ in (\ref{fiberinvM}). Let $\mathcal{T}_{s,\varphi} : \mathcal{U}_{s,\varphi} \to \mathbb{R}^{1} \times \mathbb{S}^{1}$ be a mapping defined by $\mathcal{T}_{s,\varphi}(p,q) = \mathcal{T}(s,\varphi,p,q)$. Introduce the following $1$-form on $\mathcal{U}_{s,\varphi}$, \[ \alpha_{s,\varphi} = \mathcal{T}_{s,\varphi}^{\ast}(s_{2} \, \mathrm{d} \varphi_{2}) - p \, \mathrm{d}q. \] Denote by $\mathrm{d}_{1}$ and $\mathrm{d}_{2}$ the partial exterior derivatives on $M$ along the factors $\mathbb{R}^{1} \times \mathbb{S}^{1}$ and $\mathbb{R}^{2}$, respectively. Then, $\mathrm{d}= \mathrm{d}_{1} + \mathrm{d}_{2}$ and $\mathrm{d}_{1} \circ \mathrm{d}_{2} + \mathrm{d}_{2} \circ \mathrm{d}_{1} = 0$. The following observation says that $\alpha_{s,\varphi}$ is exact on $\mathcal{U}_{s,\varphi} \subset \mathbb{R}^{2}$ and there exists a primitive which smoothly varies with $(s,\varphi)$. \begin{lemma} \label{lem4.1} There exists a smooth function $K = K(s,\varphi,p,q)$ on $\mathcal{M} \subset (\mathbb{R}^{1} \times\mathbb{S}^{1}) \times\mathbb{R}^{2}$ such that \begin{equation} \alpha_{s,\varphi} = - \mathrm{d}_{2} K, \label{primK} \end{equation} on $\mathcal{U}_{s,\varphi}$. \end{lemma} \begin{proof} For every $(s,\varphi)$, the open domain $\mathcal{U}_{s,\varphi}$ is trivially foliated by periodic orbits $\gamma_{s,\varphi}(E)$ over $(E_{1},E_{2})$ and it is isomorphic to the $1$-cylinder. Taking into account that \[ \mathcal{T}_{s,\varphi}^{\ast}(s_{2} \, \mathrm{d}\varphi_{2}) = A(s,\varphi,p,q) \big[ \frac{\partial\phi}{\partial p} \, \mathrm{d}p + \frac{\partial\phi}{\partial q} \, \mathrm{d}q \big], \] we get \begin{align*} \oint_{\Gamma_{s,\varphi}(E)} \alpha_{s,\varphi} & = \oint_{\Gamma_{s,\varphi}(E)} \mathcal{T}_{s,\varphi}^{\ast}(s_{2} \, \mathrm{d}\varphi_{2}) - \oint_{\Gamma_{s,\varphi}(E)} p \, \mathrm{d}q\\ & = \int_{0}^{2\pi} a(s,E) \, \mathrm{d}\phi - 2\pi a(s,E) = 0. \end{align*} This means that $\alpha_{s,\varphi}$ is exact on $\mathcal{U}_{s,\varphi}$. Using the section $L$ in (\ref{sectL}) and fixing $E_{0} \in (E_{1},E_{2})$, we define the primitive $- K_{s,\varphi}$ of $\alpha_{s,\varphi}$ smoothly depending on $s$ and $\varphi$ (as parameters) by \[ K_{s,\varphi}(p,q) = - \int_{(p^{0},q^{0})}^{(p,q)} \alpha_{s,\varphi}. \] Here, the integral is taken over a curve joining any point $(p,q) \in\mathcal{U}_{s,\varphi}$ and the point $(p^{0},q^{0}) = \bigl( p^{0}(s,\varphi,E_{0}), q^{0}(s,\varphi,E_{0}) \bigr)$ given by \[ L(s,\varphi, E_{0}) = \{ ( s,\varphi) \} \times \bigl\{ \bigl( p^{0}(s,\varphi,E_{0}), q^{0}(s,\varphi, E_{0}) \bigr) \bigr\}. \] \end{proof} We remark that in terms of $A$ and $\phi$, condition (\ref{primK}) is rewritten as follows \begin{equation} A \, \mathrm{d}_{2} \phi = p \, \mathrm{d}q - \mathrm{d}_{2} K, \label{Ad2fi} \end{equation} or, equivalently, \[ \frac{\partial K}{\partial p} = - A \, \frac{\partial\phi}{\partial p}, \quad \frac{\partial K}{\partial q} = - A \, \frac{\partial\phi}{\partial q} + p. \] \begin{lemma} The pull-back of the symplectic form $\widetilde{\Omega}^{\varepsilon}$ under the transformation $\mathcal{T}$ is \begin{equation} \mathcal{T}^{\ast} \widetilde{\Omega}^{\varepsilon} = \Omega^{\varepsilon} + \varepsilon \, \mathrm{d}P, \label{pubakTOme} \end{equation} where $P = P_{1} \, \mathrm{d}s + P_{2} \, \mathrm{d}\varphi$ is a horizontal $1$-form on $\mathcal{M}$ with coefficients \begin{gather} P_{1} = A \, \frac{\partial\phi}{\partial s} + \frac{\partial K}{\partial s}, \label{horPfrmcoef1}\\ P_{2} = A \, \frac{\partial\phi}{\partial\varphi} + \frac{\partial K}{\partial \varphi}. \label{horPfrmcoef2} \end{gather} \end{lemma} \begin{proof} By (\ref{horPfrmcoef1}), (\ref{horPfrmcoef2}) we have \begin{equation} P = A \, \mathrm{d}_{1} \phi + \mathrm{d}_{1} K. \label{pubkTds2dfi2} \end{equation} From (\ref{Ad2fi}) and (\ref{pubkTds2dfi2}) we derive that $P = A \, \mathrm{d}\phi - p \, \mathrm{d}q + \mathrm{d}K$, which implies (\ref{pubakTOme}). \end{proof} As a consequence of (\ref{pubakTOme}), we get formula (\ref{pushTOme}), where $Q = Q_{1} \, \mathrm{d}s_{1} + Q_{2} \, \mathrm{d} \varphi_{1}$ is a horizontal $1$-form on $\mathcal{N}$ with coefficients \begin{gather} Q_{1} = \Big( A \, \frac{\partial\phi}{\partial s} + \frac{\partial K}{\partial s} \Big) \circ \mathcal{T}^{-1}, \label{Qcoeff1}\\ Q_{2} = \Big( A \, \frac{\partial\phi}{\partial\varphi} + \frac{\partial K}{\partial \varphi} \Big) \circ \mathcal{T}^{-1}. \label{Qcoeff2} \end{gather} This proves Proposition~\ref{pro:nonsym}. Using (\ref{Qcoeff1}), (\ref{Qcoeff2}), one can show that $Q_{1}$ and $Q_{2}$ are related by the following ``zero curvature'' equation \cite{DavVor, Vorob3}: \begin{equation} \frac{\partial Q_{2}}{\partial s_{1}} - \frac{\partial Q_{1}}{\partial \varphi_{1}} + \{Q_{1},Q_{2}\} = 0. \label{zerocurveq} \end{equation} Let us denote $\widetilde{\Omega}_{Q}^{\varepsilon} = \widetilde{\Omega }^{\varepsilon} - \varepsilon \, \mathrm{d}Q$. It is clear that $\widetilde{\Omega}_{Q}^{\varepsilon}$ is nondegenerate on $\mathcal{N}$ for all $\varepsilon \neq 0$. Observe also that $\widetilde{\Omega}_{Q}^{\varepsilon}$ admits the following representation: \begin{equation} \widetilde{\Omega}_{Q}^{\varepsilon} = \mathrm{d}s_{1} \wedge \mathrm{d} \varphi_{1} + \frac{1}{\varepsilon} \, \Gamma^{1}\wedge \Gamma^{2}, \label{frmOmetilep} \end{equation} where \begin{gather} \Gamma^{1} = \varepsilon \big[ \mathrm{d}s_{2} + \frac{\partial Q_{1}}{\partial\varphi_{2}}\, \mathrm{d}s_{1} + \frac{\partial Q_{2}}{\partial\varphi_{2}} \, \mathrm{d}\varphi_{1} \big], \label{Gam1Gam1}\\ \Gamma^{2} = \varepsilon \big[ \mathrm{d}\varphi_{2} - \frac{\partial Q_{1}}{\partial s_{2}}\, \mathrm{d}s_{1} - \frac{\partial Q_{2}}{\partial s_{2}} \, \mathrm{d}\varphi_{1} \big]. \label{Gam2Gam2} \end{gather} From (\ref{zerocurveq}), (\ref{frmOmetilep}) we derive that the Poisson bracket on $\mathcal{N}$ corresponding to $\widetilde{\Omega}_{Q}^{\varepsilon}$ is given by the relations: \begin{gather} \{s_{1},\varphi_{1}\}_{\mathcal{N}} = 1, \label{poisbra1}\\ \{s_{1},s_{2}\}_{\mathcal{N}} = -\frac{\partial Q_{2}}{\partial\varphi_{2}}, \quad \{s_{1},\varphi_{2}\}_{\mathcal{N}} = \frac{\partial Q_{2}}{\partial s_{2}}, \label{poisbra2}\\ \{\varphi_{1}, s_{2}\}_{\mathcal{N}} = \frac{\partial Q_{1}}{\partial\varphi_{2}}, \quad \phantom{-} \{\varphi_{1},\varphi_{2}\}_{\mathcal{N}} = - \frac{\partial Q_{1}}{\partial s_{2}}, \label{poisbra3}\\ \{s_{2},\varphi_{2}\}_{\mathcal{N}} = \frac{1}{\varepsilon} + \Big( - \frac{\partial Q_{1}}{\partial\varphi_{2}} \frac{\partial Q_{2}}{\partial s_{2}} + \frac{\partial Q_{1}}{\partial s_{2}} \frac{\partial Q_{2}}{\partial \varphi_{2}} \Big). \label{poisbra4} \end{gather} Now we proceed to the construction of a near identity symplectomorphism $\Psi_{\varepsilon}$ and a proof of Theorem~\ref{teo:mainthm}. Consider the original perturbed system (\ref{HamSys1})-(\ref{HamSys4}). The push-forward of $H_{\varepsilon}$ by $\mathcal{T}$ gives a Hamiltonian system on $(\mathcal{N}, \widetilde{\Omega}_{Q}^{\varepsilon})$ with Hamiltonian $H_{\varepsilon} \circ \mathcal{T}^{-1} = f + \varepsilon \, F \circ \mathcal{T}^{-1}$. By Proposition~\ref{pro:reducib}, the corresponding dynamical system on $\mathcal{N}$ is of the form \begin{gather} \dot{s_{1}} = O(\varepsilon), \label{newsys1}\\ \dot{\varphi}_{1} = \omega_{1}(s_{1}) + O(\varepsilon), \label{newsys2}\\ \dot{s_{2}} = O(\varepsilon), \label{newsys3}\\ \dot{\varphi}_{2} = \omega_{2}(s_{1},s_{2}) + O(\varepsilon), \label{newsys4} \end{gather} where $\omega_{2}$ is given by (\ref{freqw2}). The function $K$ is uniquely determined by (\ref{primK}) up to adding an arbitrary function depending on $s_{1},\varphi_{1}$. This means that we have certain freedom in choosing the $1$-form $Q$. To fix $Q$, we use the following criterion. \begin{lemma} \label{lem4.3} One can choose $Q$ in \eqref{Qcoeff1}, \eqref{Qcoeff2} and a smooth function $h = h(s_{1}, s_{2})$ on $D$ such that \begin{equation} \frac{\partial h(s_{1},s_{2})}{\partial s_{2}} = \omega_{2}(s_{1},s_{2}), \label{partialh} \end{equation} and \begin{equation} F \circ \mathcal{T}^{-1} = - \omega_{1} \, Q_{2} + h. \label{FcomTinv} \end{equation} \end{lemma} \begin{proof} Suppose we are given some $Q$ in (\ref{Qcoeff1}), (\ref{Qcoeff2}) and $h$ satisfying (\ref{partialh}). Computing the components of the Hamiltonian vector field of $H_{\varepsilon} \circ \mathcal{T}^{-1}$ relative to bracket (\ref{poisbra1})-(\ref{poisbra4}) up to $O(\varepsilon)$ and comparing with the right hand side of (\ref{newsys1})-(\ref{newsys4}), we obtain the following relationship between $F$, $Q$ and $h$: \begin{equation} F \circ \mathcal{T}^{-1} = - \omega_{1} \, Q_{2} + h + \mu \label{FcomTinv2} \end{equation} where $\mu = \mu(s_{1},\varphi_{1})$ is a smooth function, $2\pi$-periodic in $\varphi_{1}$. Clearly, $Q$ and $h$ are uniquely determined up to the transformations \[ h \mapsto h + c_{1}, \quad Q \mapsto Q + \mathrm{d}_{1} c_{2}, \] for any smooth functions $c_{1} = c_{1}(s_{1})$ and $c_{2} = c_{2}(s_{1},\varphi_{1}) = c_{2}(s_{1},\varphi_{1} + 2\pi)$. To eliminate $\mu$ in (\ref{FcomTinv2}), we take \[ c_{1} = \frac{1}{2\pi} \int_{0}^{2\pi} \mu \, \mathrm{d}\varphi_{1}, \quad c_{2} = \frac{1}{\omega_{1}} \Big( \int_{0}^{\varphi_{1}} \mu \, \mathrm{d}\varphi_{1}' - \varphi_{1}c_{1} \Big). \] \end{proof} We shall assume that $h$ and $Q$ satisfying (\ref{partialh}) and (\ref{FcomTinv}) are given. Introduce the following curve of closed $2$-forms $[0,1] \ni \lambda \mapsto \sigma_{\lambda}^{\varepsilon}$ joining $\widetilde{\Omega}_{Q}^{\varepsilon}$ and $\widetilde{\Omega}^{\varepsilon}$: \begin{align*} \sigma_{\lambda}^{\varepsilon} & = ( 1 - \lambda) \, \widetilde{\Omega}_{Q}^{\varepsilon} + \lambda \, \widetilde{\Omega}^{\varepsilon}\\ & = \mathrm{d}s_{1} \wedge \mathrm{d}\varphi_{1} + \varepsilon \, \mathrm{d}s_{2} \wedge \mathrm{d}\varphi_{2} - \varepsilon \, (1 - \lambda) \, \mathrm{d} Q. \end{align*} Using (\ref{zerocurveq}), by straightforward computations we show that $\sigma_{\lambda}^{\varepsilon}$ has the representation \begin{equation} \sigma_{\lambda}^{\varepsilon} = m \, \mathrm{d}s_{1} \wedge \mathrm{d} \varphi_{1} + \frac{1}{\varepsilon} \, \Gamma_{\lambda}^{1} \wedge \Gamma_{\lambda}^{2}, \label{sigmalambe} \end{equation} where \[ m = 1 - \lambda \, (1-\lambda) \varepsilon \Big( \frac{\partial Q_{2}}{\partial s_{1}} - \frac{\partial Q_{1}}{\partial\varphi_{1}} \Big), \] and $1$-forms $\Gamma_{\lambda}^{1}$, $\Gamma_{\lambda}^{2}$ are defined by formulas (\ref{Gam1Gam1}), (\ref{Gam2Gam2}) under replacing $Q$ by $(1 - \lambda) \, Q$. Let $\mathcal{M}_{0} \subset \mathcal{M}$ be an admissible domain $\mathcal{N}_{0} = \mathcal{T}(\mathcal{M}_{0}) \subset \mathcal{N}$. Pick another open domain $\mathcal{W}$ in $\mathcal{N}$ such that $\mathcal{N}_{0} \subset \mathcal{W}$ and $\overline{\mathcal{W}}$ is compact. Then, the functions $Q_{1}$ and $Q_{2}$ are bounded on $\overline{\mathcal{W}}$ and \[ \delta_{0} = \frac{1}{4 \max_{\overline{\mathcal{W}}} | \frac{\partial Q_{2}}{\partial s_{1}} - \frac{\partial Q_{1}}{\partial\varphi_{1}} |} > 0. \] From here and (\ref{sigmalambe}), we derive the key property. \begin{lemma} \label{lem4.4} For any $\varepsilon \in (0,\delta_{0})$ and $\lambda \in [0,1]$ the $2$-form $\sigma_{\lambda}^{\varepsilon}$ is nondegenerate on $\mathcal{W}$. \end{lemma} Therefore, for every $\varepsilon \in (0,\delta_{0})$, we have the family $\{\sigma_{\lambda}^{\varepsilon} \}_{\lambda \in [0,1]}$ of symplectic structures on $\mathcal{W}$. According to a general scheme \cite{GuilLerSter}, an isomorphism between $\sigma_{\lambda}^{\varepsilon}$ and $\sigma_{0}^{\varepsilon}$ is given by the flow $\mathsf{Fl}_{Z_{\lambda}}^{\lambda}$ of a time-dependent vector field $Z_{\lambda}$ satisfying the homological equation \[ L_{Z_{\lambda}} \sigma_{\lambda}^{\varepsilon} - \frac{\mathrm{d}\sigma_{\lambda}^{\varepsilon}}{\mathrm{d}\lambda} = 0. \] In this case, $(\mathsf{Fl}_{Z_{\lambda}}^{\lambda})_{\ast} \sigma_{\lambda}^{\varepsilon} = \sigma_{0}^{\varepsilon}$. By standard arguments, finding $Z_{\lambda}$ is reduced to solving the algebraic equation \begin{equation} \mathbf{i}_{Z_{\lambda}} \sigma_{\lambda}^{\varepsilon} = Q. \label{intprsigQ} \end{equation} By (\ref{sigmalambe}), we derive that the dynamical system of the vector field $Z_{\lambda}$ satisfying (\ref{intprsigQ}), is of the form \begin{gather} \frac{\mathrm{d}s_{1}}{\mathrm{d}\lambda} = - \frac{\varepsilon}{m} \, Q_{2}, \label{solhomeq1}\\ \frac{\mathrm{d}\varphi_{1}}{\mathrm{d}\lambda} = \frac{\varepsilon}{m} \, Q_{1}, \label{solhomeq2}\\ \frac{\mathrm{d}s_{2}}{\mathrm{d}\lambda} = \frac{\varepsilon (1-\lambda)}{m} \big[ Q_{2} \frac{\partial Q_{1}}{\partial\varphi_{2}} - Q_{1} \frac{\partial Q_{2}}{\partial\varphi_{2}} \big], \label{solhomeq3}\\ \frac{\mathrm{d}\varphi_{2}}{\mathrm{d}\lambda} = \frac{\varepsilon (1-\lambda)}{m} \big[ Q_{1} \frac{\partial Q_{2}}{\partial s_{2}} - Q_{2}\frac{\partial Q_{1}}{\partial s_{2}} \big], \label{solhomeq4} \end{gather} Therefore, for every $\varepsilon \in [ 0,\delta_{0} )$ we have a time-dependent vector field $Z_{\lambda}$ on $\mathcal{W}$ which vanishes at $\varepsilon = 0$. From this property and the compactness of $\mathcal{N}_{0}$ it follows that there is a $\delta_{1}\in( 0,\delta_{0} )$ such that the flow $\mathsf{Fl}_{Z_{\lambda}}^{\lambda}$ on $\mathcal{N}_{0}$ is well-defined for all $\varepsilon \in [ 0,\delta_{1} )$ and $\lambda \in [ 0,1 ]$. We arrive at the following result. \begin{lemma} \label{lem4.5} For every \ $\varepsilon\in\lbrack0,\delta_{1})$, the time-1 flow $\Psi_{\varepsilon} = \mathsf{Fl}_{Z_{\lambda}}^{1} : \mathcal{N}_{0} \to N$ of system \eqref{solhomeq1}-\eqref{solhomeq4} is a near identity symplectomorphism between $\tilde{\Omega}_{Q}^{\varepsilon}$ and $\widetilde{\Omega}^{\varepsilon}$, \[ \Psi_{0} = \mathrm{id} \quad \text{and} \quad (\Psi_{\varepsilon})_{\ast} \widetilde{\Omega}_{Q}^{\varepsilon} = \widetilde{\Omega}^{\varepsilon}. \] \end{lemma} We remark that the inverse $\Psi_{\varepsilon}^{-1}$ is defined as the time-$1$ flow of the following non-autonomous system \begin{gather*} \frac{\mathrm{d}s_{1}}{\mathrm{d}\lambda} = \frac{\varepsilon}{m} Q_{2},\\ \frac{\mathrm{d}\varphi_{1}}{\mathrm{d}\lambda} = - \frac{\varepsilon}{m} Q_{1},\\ \frac{\mathrm{d}s_{2}}{\mathrm{d}\lambda} = \frac{\varepsilon \lambda}{m} \big[ Q_{1} \frac{\partial Q_{2}} {\partial\varphi_{2}} - Q_{2} \frac{\partial Q_{1}}{\partial\varphi_{2}} \big],\\ \frac{\mathrm{d}\varphi_{2}}{\mathrm{d}\lambda} = \frac{\varepsilon\lambda}{m} \big[ Q_{2} \frac{\partial Q_{1}}{\partial s_{2}} - Q_{1}\frac{\partial Q_{2}}{\partial s_{2}} \big], \end{gather*} which corresponds to the vector field $\widetilde{Z}_{\lambda} = - Z_{1 - \lambda}$. Finally, using (\ref{partialh}) and (\ref{FcomTinv}), for $\widetilde{H}_{\varepsilon} = H_{\varepsilon} \circ \mathcal{T}^{-1}$, we compute \begin{align*} \widetilde{H}_{\varepsilon} \circ \Psi_{\varepsilon}^{-1} & = \widetilde{H}_{\varepsilon} - L_{Z_{\lambda}} \widetilde{H}_{\varepsilon} + O(\varepsilon^{2})\\ & = f + \varepsilon \, (-\omega_{1}Q_{2} + h ) - \varepsilon \, L_{Z_{\lambda}}f + O(\varepsilon^{2})\\ & = f + \varepsilon \, h + O(\varepsilon^{2}). \end{align*} It follows that $\Upsilon_{\varepsilon} = \Psi_{\varepsilon}\circ T$ satisfies (\ref{pushYpsi}), (\ref{tildeHe}). This completes the proof of Theorem~\ref{teo:mainthm}. \begin{corollary} If $F$ in \eqref{Hameps} is independent of $s$, $F = F(\varphi,p,q)$, then one can choose $Q_{1} = 0$ and $Q_{2} = Q_{2}(\varphi_{1},s_{2},\varphi_{2})$. In this case, $\Psi_{\varepsilon}$ is an infinitesimal transformation of the form \[ \Psi_{\varepsilon}(s_{1},\varphi_{1},s_{2},\varphi_{2}) = \bigl( s_{1} - \varepsilon \, Q_{2}(\varphi_{1},s_{2}, \varphi_{2}),\varphi_{1}, s_{2}, \varphi_{2} \bigr). \] \end{corollary} Such type of transformations appear in the normalization of Hamiltonian systems near an individual periodic trajectory \cite{Bruno1, Bruno2, BelDobr}. \section{The Linear Case} \label{sec:lincas} As an illustration of above results, we consider the case when the function $F$ in (\ref{Hameps}) is quadratic in coordinates $p$ and $q$, \[ F = \frac{1}{2} \bigl( w_{1} p^{2} + 2\, w_{2}p q + w_{3} q^{2} \bigr). \] Here $w_{i} = w_{i}(s,\varphi)$ $(i=1,2,3)$ are smooth functions, $2\pi$-periodic in $\varphi$. Then, unperturbed system (\ref{HamSysS1})-(\ref{HamSysS4}) takes the form \begin{gather} \dot{s} = 0, \label{lincase1}\\ \dot{\varphi} = \omega_{1}(s), \label{lincase2}\\ \begin{pmatrix} \dot{p}\\ \dot{q} \end{pmatrix} = \mathbb{J} \, W(s,\varphi) \begin{pmatrix} p \\ q \end{pmatrix}, \label{lincase3} \end{gather} where \[ W(s,\varphi) = \begin{bmatrix} w_{1}(s,\varphi) & w_{2}(s,\varphi)\\ w_{2}(s,\varphi) & w_{3}(s,\varphi) \end{bmatrix}, \quad \mathbb{J} = \begin{bmatrix} 0 & -1\\ 1 & \phantom{-}0 \end{bmatrix}. \] Therefore, (\ref{lincase1})-(\ref{lincase3}) corresponds to a $s$-parameter family of linear time-periodic Hamiltonian systems on $\mathbb{R}^{2}$. Recall that a linear periodic Hamiltonian system is said to be \emph{stable} (in the sense of Lyapunov) if all solutions are bounded for $t \in (-\infty,\infty)$. Moreover, a stable linear $T$-periodic Hamiltonian system is called \emph{strongly stable} ( or \emph{parametrically stable}), if all sufficiently small linear $T$-periodic Hamiltonian perturbations of this system are stable as well \cite{FlorVor1, KarVor, Yakub}. We assume that system (\ref{lincase1})-(\ref{lincase3}) is \emph{strongly stable for every} $s \in \Delta = (\Delta_{1},\Delta_{2})$. Let $\mathbb{F}(s,\varphi)$ be the fundamental solution of the corresponding linear problem, \begin{gather*} \omega_{1}(s) \frac{\mathrm{d}\mathbb{F}}{\mathrm{d}\varphi} = \mathbb{J} \, W(s,\varphi) \, \mathbb{F}, \\ \mathbb{F}(s,0) = I. \end{gather*} Then, $\mathbb{F}$ is a $\mathrm{Sp}(1, \mathbb{R})$-valued smooth function in $s,\varphi$ with $\det \, \mathbb{F}(s,\varphi) = 1$. In terms of the monodromy matrix $\mathfrak{M}(s) = \mathbb{F}(s,2\pi)$, the strongly stability condition is formulated as follows \cite{FlorVor1, Yakub}: $ -2 < \mathrm{tr} \, \mathfrak{M}(s) < 2$, for $s \in \Delta$. This means that the spectrum of the the monodromy matrix $\mathfrak{M}(s)$ is simple and belongs to the unit circle in the complex plane, $\mathop{\rm Spec} \mathfrak{M}(s) = \{ \exp(\pm 2\pi \,i \,\beta(s)) \}$, where $\beta(s) > 0$ is the Floquet exponent. Let us associate to system (\ref{lincase1})-(\ref{lincase3}) the following Riccati equation for a $\mathbb{C}$-valued function $(s,\varphi) \mapsto \mathfrak{D}(s,\varphi)$ (depending on $s$ as a parameter): \begin{equation} \omega_{1} \frac{\partial\mathfrak{D}}{\partial\varphi} + w_{1}\mathfrak{D}^{2} + 2 w_{2}\mathfrak{D} + w_{3} = 0. \label{Ricatieq} \end{equation} \begin{proposition} {\cite{FlorVor1}} If system \eqref{lincase1}-\eqref{lincase3} is strongly stable, then: \begin{itemize} \item[(a)] There exists a unique smooth solution $\mathfrak{D}(s,\varphi) = \mathfrak{D}_{1}(s,\varphi) + i \mathfrak{D}_{2}(s,\varphi)$ of the Riccati equation \eqref{Ricatieq} satisfying the following properties \begin{gather} \mathfrak{D}_{2}(s,\varphi) > 0, \label{Ricatsol1}\\ \mathfrak{D}_{2}(s,\varphi +2\pi) = \mathfrak{D}_{2}(s,\varphi), \label{Ricatsol2} \end{gather} for all $s$, $\varphi$. The Floquet exponent $\beta = \beta(s)$ is smoothly varying with $s$ and has the representation \[ \beta(s) = \frac{1}{2\pi\omega_{1}(s)} \int_{0}^{2\pi} w_{1}(s,\varphi) \, \mathfrak{D}_{2}(s,\varphi) \, \mathrm{d}\varphi. \] \item[(b)] System \eqref{lincase1}-\eqref{lincase3} admits an integral of motion $G : (\Delta \times \mathbb{S}^{1}) \times \mathbb{R}^{2} \to \mathbb{R}$ given by \begin{equation} G(s,\varphi,p,q) = \frac{1}{2 \, \mathfrak{D}_{2}} \bigl[(p - \mathfrak{D}_{1} \, q)^{2} + (\mathfrak{D}_{2} \, q)^{2} \bigr]. \label{Rcatsol3} \end{equation} \end{itemize} \end{proposition} The dynamical system of the vertical Hamiltonian vector field $V_{G}$ is easily integrated by reducing it to the harmonic oscillator under the change of variables \[ p \mapsto \frac{p - \mathfrak{D}_{1} \, q}{\sqrt{\mathfrak{D}_{2}}}, \quad q \mapsto \sqrt{\mathfrak{D}_{2}} \, q \] For every $(s,\varphi)$ and $E > 0$, the level set $\gamma_{s,\varphi}(E)$ of $G_{s,\varphi}$ in $\mathbb{R}^{2}$ is an elliptic trajectory of $V_{G}$ with period $T = 2\pi$. The action along $\gamma_{s,\varphi}(E)$ is $a(s,E) = E$. Let $\mathcal{U}_{s,\varphi}$ be the open domain defined as the union of $\gamma_{s,\varphi}(E)$, where $E$ runs over $(0,\infty)$. Then, the hypothesis (IH) is satisfied for the domain $\mathcal{M} = \cup_{(s,\varphi) \in \Delta \times \mathbb{S}^{1}} \mathcal{U}_{s,\varphi}$ and $G$ (\ref{Rcatsol3}). Taking $L(s,\varphi,E) = \bigl( s,\varphi, \sqrt{2E \, \mathfrak{D}_{2}(s,\varphi)}, 0 \bigr)$, and applying the algorithm for constructing $A$ and $\phi$ in Section~\ref{sec:reducib}, we get that the inverse $\mathcal{T}_{s_{1},\varphi_{1}}^{-1} : (s_{2},\varphi_{2}) \mapsto(p,q)$ of the reducibility transformation $\mathcal{T}$ (\ref{Tsfipq}) is given by \[ p = \sqrt{2s_{2} \, \mathfrak{D}_{2}} \, \cos(\varphi_{2} + \chi) + \mathfrak{D}_{1} \sqrt{\frac{2s_{2}}{\mathfrak{D}_{2}}} \, \sin(\varphi_{2} + \chi), \quad q = \sqrt{\frac{2s_{2}}{\mathfrak{D}_{2}}} \, \sin(\varphi_{2}+\chi). \] Here $\mathfrak{D}_{1} = \mathfrak{D}_{1}(s_{1},\varphi_{1}), \mathfrak{D}_{2} = \mathfrak{D}_{2}(s_{1},\varphi_{1})$ and \begin{gather*} \chi(s_{1},\varphi_{1}) = \frac{1}{\omega_{1}(s_{1})} \Big[ \omega_{2}(s_{1}) \varphi_{1} - \int_{0}^{\varphi_{1}} w_{1}(s_{1},\varphi_{1}') \, \mathfrak{D}_{2}(s_{1},\varphi_{1}') \, \mathrm{d}\varphi_{1}' \Big], \\ \omega_{2}(s_{1}) = \omega_{1}(s_{1})\, \beta(s_{1}) = \frac{1}{2\pi} \int_{0}^{2\pi} w_{1}(s_{1},\varphi_{1}) \, \mathfrak{D}_{2}(s_{1},\varphi_{1})\, \mathrm{d}\varphi_{1}. \end{gather*} It follows that \begin{equation} \mathcal{T}(s,\varphi,p,q) = \Big( s, \varphi, G(s,\varphi,p,q), \mathrm{arctg} \big( \frac{\mathfrak{D}_{2}}{ ({p}/{q}) - \mathfrak{D}_{1} } \big) + \chi \Big), \label{solTlin} \end{equation} and $\mathcal{N} = \mathcal{T}(\mathcal{M}) = (\Delta\times S^{1}) \times (\mathbb{R}_{+}^{1} \times S^{1})$. It is easy to see that a function $K$ in (\ref{primK}), can be chosen in the form $K = ({p \, q}/{2}) - \mathfrak{D}_{1} \, q^{2}$. Substituting the formulas for $K$ and $\mathcal{T}$ into (\ref{Qcoeff1}), (\ref{Qcoeff2}) gives \begin{gather} Q_{1} = \frac{s_{2}}{2 \, \mathfrak{D}_{2}} \Big[ \sin 2(\varphi_{2} + \chi) \, \frac{\partial \mathfrak{D}_{2}}{\partial s_{1}} + (1 - \cos 2(\varphi_{2} + \chi)) \, \frac{\partial\mathfrak{D}_{1}}{\partial s_{1}} + \frac{\partial\chi}{\partial s_{1}} \Big], \label{SolcoefQ1} \\ Q_{2} = \frac{s_{2}}{2 \, \mathfrak{D}_{2}} \Big[ \sin 2(\varphi_{2} + \chi) \, \frac{\partial \mathfrak{D}_{2}}{\partial \varphi_{1}} + (1-\cos 2(\varphi_{2} + \chi)) \, \frac{\partial\mathfrak{D}_{1}}{\partial\varphi_{1}} + \frac{\partial \chi}{\partial \varphi_{1}} \Big]. \label{SolcoefQ2} \end{gather} Finally, we observe that if we take $h(s_{1},s_{2}) = s_{2} \, \omega_{2}(s_{1})$, then (\ref{FcomTinv}) holds. Fix some $E_{1}$ and $E_{2}$ such that $0 < E_{1} < E_{2} < \infty$, and consider the admissible domain $\mathcal{M}_{0}$ which is the union of the open elliptic rings $\mathcal{U}_{s,\varphi}^{0} = \cup_{E_{1} < E < E_{2}} \, \gamma_{s,\varphi}(E)$ in $\mathbb{R}^{2}$. Then, according to Theorem~\ref{teo:mainthm}, for small enough $\varepsilon$, the symplectomorphism $\Upsilon_{\varepsilon} = \Psi_{\varepsilon} \circ \mathcal{T} : \mathcal{M}_{0}\to \mathcal{N}$ transforms $H_{\varepsilon}$ into the normal form $f(s_{1}) + \varepsilon \, s_{2} \, \omega_{2}(s_{1}) + O(\varepsilon^{2})$. Here, $\mathcal{T}$ is the reducibility transformation in (\ref{solTlin}) and $\Psi_{\varepsilon}$ is the time-$1$ flow of system (\ref{solhomeq1})-(\ref{solhomeq4}) with $Q_{1}$ and $Q_{2}$ given by (\ref{SolcoefQ1}), (\ref{SolcoefQ2}). Notice that the reducibility map (\ref{solTlin}) can be also derived from the Floquet theory. Let $\mathcal{F}_{s,\varphi} : \mathbb{R}^{2} \to \mathbb{R}^{2}$ be the standard Floquet-Lyapunov transformation \cite{Yakub}, \[ \mathcal{F}_{s,\varphi} = \exp \big( \frac{\varphi}{\omega_{1}(s)} \mathcal{K}(s) \big) \circ \mathbb{F}^{-1}(s,\varphi). \] Here $\mathcal{K}(s) = \bigl[ {\omega_{1}(s)}/{2\pi} \bigr] \ln \mathfrak{M}(s)$ and a real branch of the logarithm of $\mathfrak{M}(s)$ exists because of the stability assumption. Then, one can show that $\mathcal{T} = \mathcal{S} \circ \mathcal{F}$, where $\mathcal{S} : \mathcal{M} \to \mathcal{N}$ is a symplectic map uniquely determined by $\mathcal{K}(s)$. We remark that for time-dependent harmonic oscillators, invariants like (\ref{Rcatsol3}) were studied in \cite{Lewis}. \subsection*{Acknowledgements} The authors wish to thank the anonymous referee for the useful comments and critical remarks. \begin{thebibliography}{99} \bibitem{Arnld1} {V. I. Arnold}, {\it Mathematical Methods of Classical Mechanics}, Graduate Texts in Math. 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