\magnification=1200 \baselineskip 12pt \def\picture #1 by #2 (#3){ \vbox to #2{ \hrule width #1 height 0pt depth 0pt \vfill \special{picture #3} } } \def\scaledpicture #1 by #2 (#3 scaled #4){{ \dimen0=#1 \dimen1=#2 \divide\dimen0 by 1000 \multiply\dimen0 by #4 \divide\dimen1 by 1000 \multiply\dimen1 by #4 \picture \dimen0 by \dimen1 (#3 scaled #4)} } %DEFINIZIONI \let\e\varepsilon \let\o\omega \let\dst\displaystyle \vglue2cm \bf \centerline{CONSTRUCTION OF STABLE PERIODIC ORBITS} \medskip \centerline{FOR THE SPIN--ORBIT PROBLEM OF} \medskip \centerline{CELESTIAL MECHANICS} \rm \vskip.2in \centerline{\bf Alessandra Celletti$^{(1)}$ and Luigi Chierchia$^{(2)}$} \vglue1cm (1) Dipartimento di Matematica Pura e Applicata, Universit\`a di L'Aquila, Via Vetoio - I--67010 L'Aquila (Italy) e--mail: alessandra.celletti@aquila.infn.it \vskip.2in (2) Dipartimento di Matematica, Universit\`a Roma Tre, Largo San Leonardo Murialdo 1, I--00146 Roma (Italy) e--mail: luigi@matrm3.mat.uniroma3.it \vglue2cm\noindent \bf ABSTRACT. \rm Birkhoff periodic orbits associated to spin--orbit resonances in Celestial Mechanics and in particular to the Moon--Earth and Mercury--Sun systems are considered. A general method (based on a quantitative version of the Implicit Function Theorem) for the construction of such orbits with particular attention to ``effective estimates" on the size of the perturbative parameters is presented and tested on the above mentioned systems. Lyapunov stability of the periodic orbits (for small values of the perturbative parameters) is proved by constructing KAM librational invariant surfaces trapping the periodic orbits. \vglue2cm\noindent \bf KEYWORDS: \rm Periodic orbits, Spin--orbit resonances, Stability. \vfill\eject \noindent \bf \S1. INTRODUCTION AND RESULTS\rm \vskip.1in\noindent The study of periodic orbits in Celestial Mechanics is strongly motivated by the abundance of ``resonant relations" existing in the solar system. In particular, in this paper, we are concerned with commensurabilities between the revolutional and the rotational period, i.e., with the so--called \sl spin--orbit \rm resonances (see, e.g., Celletti 1990, 1994, Goldreich and Peale 1966, 1970, Murdock 1978, Peale 1973, Wisdom 1987). As is well known, most of the evolved satellites of the solar system point always the same face toward the host planet (the most familiar example being, of course, that of our Moon). In such a case one speaks of 1:1 or ``synchronous" spin--orbit resonance. The only exception to 1:1 spin--orbit resonances is provided by the Mercury--Sun system, which moves in a 3:2 resonance (in fact, the ratio between the revolutional period of Mercury around the Sun and its period of rotation amounts to 3/2 within a very good approximation). \vskip.1in \noindent In \S2 we introduce a mathematical model describing an approximation of the spin--orbit problem. In particular we reduce such a problem to the study of a Hamiltonian equation of the form $$ \ddot x\ -\ \varepsilon\ f_x(x,t)\ =\ 0\ , \eqno(1.1) $$ where $x$ represents the \sl librational \rm angle, $\varepsilon$ is a positive ``perturbative" parameter measuring the equatorial oblateness of the satellite and $f=f(x,t)$ is a smooth ($x$-- and $t$--)periodic function, which depends also on the eccentricity of the satellite's orbit assumed to be Keplerian. A spin--orbit resonance of order $p:q$ is a Birkhoff periodic orbit with frequency $\omega={p\over q}$. We present a (general) method (\S3), based on a quantitative version of the classical Implicit Function Theorem (applied to a Poincar\'e map associated to eq. (1.1)), which allows to construct such periodic orbits. In particular we provide explicit approximations to the initial conditions associated to the periodic orbit and we give explicit ``effective" estimate on the equatorial oblateness parameter $\varepsilon$ ensuring the existence of the periodic orbit. Results for the 1:1, 3:2, 2:1 resonances in the Moon--Earth and the Mercury--Sun systems are discussed in \S4. In particular, we are able to prove the existence of a synchronous periodic orbit for the observed parameters of the Moon. Instead we cannot establish an analogous result for the 3:2 and 2:1 resonances: this suggests a greater robustness (and therefore a bigger probability of capture) of the 1:1 resonance compared with other resonances. \vskip.1in \noindent The Mercury--Sun case appears to be different: the existence of the three main resonances cannot be proved for ``realistic" values of the parameters and a less pronounced discrepancy (compared with the Moon--Earth case) is found between the 1:1 and 3:2 resonances. \vskip.1in \noindent A comparison with the observed data on the libration in longitude given in the Astronomical Almanac is also provided. \vskip.1in\noindent Finally we consider the stability of the periodic orbits constructed in \S4 and show that Lyapunov stability can be obtained by proving the existence of \sl librational \rm KAM invariant surfaces trapping the periodic orbits. In particular, in \S5, we show that the ``Siegel--Moser conditions" (Siegel and Moser 1971) for the existence of librational invariant surfaces are satisfied in our model--problem. Here, however, we do not pay attention about optimal estimates on the parameters: such estimates will be discussed in a future work. \vskip.1in\noindent Details on the results of \S4 and \S5 are provided, respectively, in Appendix A and B. \vskip.1in \noindent We close this introduction by mentioning that a further extension of this work might concern the computation of the actual ephemeris of the Moon: in fact one might use our approximate periodic orbit as a starting point to compute the effective lunar motion, using a strategy similar to that adopted by Hill (Hill, 1878). \vglue2cm \noindent \bf \S2. THE SPIN--ORBIT MODEL \rm \vskip.1in\noindent In this section we discuss briefly the so--called ``spin--orbit" model in Celestial Mechanics. \medskip\noindent Let $S$ be a triaxial ellipsoidal satellite moving around a central planet $P$. We denote by $T_{rev}$ and $T_{rot}$ the revolutional period of the satellite around $P$ and the rotational period about an internal spin--axis. A \sl $p:q$ spin--orbit resonance \rm occurs whenever $$ {{T_{rev}}\over {T_{rot}}}\ =\ {p\over q}\ ,\qquad\qquad {\rm for}\ p,q\in {\bf N},\ q\not=0\ . $$ In particular, when $p=q=1$ we speak of 1:1 or \sl synchronous \rm spin--orbit resonance; in this case, the satellite always points the same face to the host planet. As is well known, most of the \sl evolved \rm satellites or planets of the solar system (like, e.g., the Moon) are trapped in a 1:1 resonance (Astronomical Almanac 1990). The only exception is provided by Mercury which is observed in a nearly 3:2 resonance. We introduce a mathematical model describing the spin--orbit coupling, assuming that \noindent $i)$ the center of mass of the satellite moves on a Keplerian orbit around $P$ with semimajor axis $a$ and eccentricity $e$ (secular perturbations on the orbital parameters are neglected); \noindent $ii)$ the spin--axis is perpendicular to the orbit plane (i.e., we neglect the so--called ``obliquity"); \noindent $iii)$ the spin--axis coincides with the shortest physical axis (i.e., the axis whose moment of inertia is largest); \noindent $iv)$ dissipative effects as well as perturbations due to other planets or satellites are neglected. \medskip\noindent Let $A0$, $0<\theta<1$, $z_0\in {\bf R}^n$ and let $A$ be a compact set of ${\bf R}^p$. Let $F:(z,\alpha)\in {\overline B}_\rho(z_0)\times A\to F(z,\alpha)\in {\bf R}^n$ (${\overline B}_\rho(z_0)$ denoting the closed ball of radius $\rho$ and center $z_0$) be a continuous function with continuous and invertible Jacobian matrix ${\partial F\over \partial z}(z_0,\alpha)$, for any $\alpha\in A$. Denote by $M(\alpha)\equiv \Big({\partial F\over \partial z}(z_0,\alpha)\Big)^{-1}$ and by $m$ an upper bound on $\sup_A\|M\|$ ($\|\cdot\|$ denoting the standard ``operator norm" on matrices). If $$ {\rm (i)}\quad \sup_{{\overline B}_\rho(z_0)\times A} \Big\| I - M {\partial F\over \partial z}\Big\|\le \theta\ ,\qquad {\rm (ii)} \quad \sup_A |F(z_0,\alpha)|\le (1-\theta) {\rho\over m}\ , $$ then there exists a unique continuous function $\alpha\in A\to z(\alpha)\in {\overline B}_\rho(z_0)$ such that $F(z(\alpha),\alpha)\equiv 0$ for any $\alpha \in A$.} \medskip\noindent The proof is standard: conditions (i) and (ii) are immediately seen to guarantee that the map $u\in X\equiv C(A,{\overline B}_\rho(z_0))\to \Phi u$ defined by $$(\Phi u)(\alpha)\equiv u(\alpha) - M(\alpha) F(u(\alpha),\alpha)$$ is a contraction from $X$ (equipped with the supremum--norm) into itself (with contraction constant $\theta$). Thus, by the contraction mapping theorem, there is a unique fixed point in $X$ which corresponds to $z(\alpha)$ in the thesis of the Implicit Function Theorem. \medskip\noindent {\bf Remarks.} (i) If $F$ is $C^2$ in $z$ then, choosing $\theta=1/2$ and $$\rho\equiv 2 m \sup_A |F(z_0,\alpha)|\ ,$$ (and using the mean value theorem) one sees immediately that conditions (i) and (ii) above are enforced by the requirement $$ 4 m^2 \sup_A|F(z_0,\alpha)|\ \sup_{{\overline{B}}_\rho(z_0)\times A}\Big\| {\partial^2 F\over \partial z^2}\Big\| \le 1\ . \eqno{(3.4)}$$ This is the form we shall use below. The choice of $\theta=1/2$ is the one that ``optimizes" condition $(3.4)$. \noindent (ii) The above formulation is meaningful also in the case $A$ is a singleton, $A\equiv \{\alpha_0\}$. \noindent (iii) Notice that it is not required to have an initial solution of the equation $F=0$ but it is enough to have {\sl an approximate solution} $z_0$ (uniformly in the parameter $\alpha$). \medskip\noindent \medskip\noindent To {\sl construct} an initial approximation for $(3.3)$ we proceed as follows. Considering $\e$ as a small parameter we write explicitely the {\sl first $\e$--order} of the general solution of $(3.2)$ with initial data $(x,y)$: let $$\cases{\displaystyle x_1(t)\equiv x_1(t;y,x)=\int_0^t y_1(s)ds \cr \dst y_1(t)\equiv y_1(t;y,x)=\int_0^t f_x(x+ys,s)ds \ ,\cr} $$ and let $\xi(t)$ and $\eta(t)$ be the solution of $$\cases{ \dot\xi=\eta\ ,\qquad \xi(0)=0\ ,\cr \dst\dot\eta={1\over\varepsilon}[f_x(x+yt+\varepsilon x_1(t)+\varepsilon^2\xi(t),t) -f_x(x+yt,t)]\ ,\qquad \eta(0)=0\ . \cr} \eqno{(3.5)} $$ Then, as one readily verifies, $$\eqalign{ x(t)&\equiv x+yt+\varepsilon x_1(t)+\varepsilon^2\xi(t) \cr y(t)&\equiv y+\varepsilon y_1(t)+\varepsilon^2\eta(t) \cr} \eqno{(3.6)} $$ solve $(3.2)$ with initial data $x(0)=x$ and $y(0)=y$. Notice that $\xi$ and $\eta$ are well defined and bounded also in $\e$ at $\e=0$ (interprete the right--hand--side of the second equation in (3.5) as $f_{xx}(x+yt,t)$). \medskip\noindent The initial data $(x,y)$ has now to be fixed so as to meet $(3.3)$ (and will, of course, depend on $\e$). We shall take $$ x=x_0+ \e x_1\ ,\qquad y=y_0+\e y_1\ , \eqno{(3.7)} $$ with $x_i$ and $y_i$ {\sl independent} of $\e$. The choice of $x_i$ and $y_i$ will be made in the natural fashion: the problem $(3.3)$ may be formally solved expanding in power series of $\e$ and equating coefficients and $x_0$, $x_1$, $y_0$, $y_1$ will be taken as the first orders of such formal series. Keeping this in mind, one finds that $$ y_0={p\over q}\ , \eqno{(3.8)}$$ and that $x_0$ has to be a {\sl nondegenerate critical point of the (periodic) function} $$ \beta\to \int_0^{2\pi q} f_x(\beta+y_0 s,s)ds\ , $$ i.e., $x_0$ is such that $$ \int_0^{2\pi q} f_x(x_0+y_0 s,s)ds= 0\ ,\qquad \tau\equiv\int_0^{2\pi q} f_{xx}(x_0+y_0 s,s)ds\neq 0\ . \eqno{(3.9)}$$ In fact we shall see in Appendix B that the sign of $\tau$ determines the type of the solution: $\tau>0$ corresponds to {\sl hyperbolic} periodic solutions while $\tau<0$ to {\sl elliptic} ones. The next ``orders" $x_1$ and $y_1$ are given by $$ y_1=-{1\over {2\pi q}}\int_0^{2\pi q} \int_0^t f_x(x_0+y_0s,s)dsdt\ , \eqno{(3.10)}$$ $$ x_1=-{1\over {\int_0^{2\pi q} f_{xx}^0(t)}dt}\ [y_1\int_0^{2\pi q} tf_{xx}^0(t)dt+\int_0^{2\pi q} f_{xx}^0(t) x_1(t;y_0,x_0)dt]\ , \eqno{(3.11)}$$ where $f_{xx}^0(t)\equiv f_{xx}(x_0+y_0t,t)$. \medskip\noindent Having fixed such {\sl approximate} initial data, the function $\xi(t)$ and $\eta(t)$ in $(3.5)$ are determined and hence the whole solution $x(t)$ and $y(t)$ in $(3.2)$ is also uniquely determined and one can proceed to apply the Implicit Function Theorem performing the necessary ``a--priori estimates" on the solution $x(t)$, $y(t)$ (see Appendix A for detailed estimates). \medskip\noindent We remark that the word ``approximate" refers to the equation $(3.3)$ and not to equations $(3.2)$ of which $x(t)$ and $y(t)$ give an {\sl exact} solution with initial data $(3.7)$. We also notice that having choosen the first two nontrivial orders in $\e$ is of course rather arbitrary (for example one could take {\sl higher $\e$--order approximations}). \vglue2cm \noindent {\bf\S4. PERIODIC SOLUTIONS FOR THE SPIN--ORBIT PROBLEM} \medskip\noindent Here we apply the theory described in the previous section to the spin--orbit model discussed in \S2. \medskip\noindent We consider the two most significative examples of spin--orbit coupling, namely the Moon--Earth and Mercury--Sun systems. As everybody looking up in the sky knows, the Moon--Earth system lies in a 1:1 spin--orbit resonance. The Mercury--Sun system lies instead in a 3:2 resonance. Here, besides the 1:1 and 3:2 resonances, we shall consider also the 2:1 resonance since numerical computations show that the 2:1 is surrounded, in phase space, by a ``librational region" which appears to be larger than the ones associated to the remaining resonances. \medskip\noindent In applying the theory of \S3 {\sl we shall fix the value of the eccentricity equal to the astronomically observed one}. For the astronomical observed values of the parameters for the Moon--Earth and Mercury--Sun systems see Table 1. \medskip\noindent The mathematical results are listed in Table 2, which shows the maximum value of the perturbing parameter $\varepsilon$ for which we can establish the existence of a periodic orbit with frequency $p\over q$, associated to a $p:q$ spin--orbit resonance. We stress that the results are obtained for the \sl true \rm values of the eccentricities, i.e. $e=0.0549$ for the Moon and $e=0.2056$ for Mercury. \medskip\noindent Table 2 shows that the existence of a synchronous periodic orbit close to the actual motion of the Moon can be proved for values of the perturbing parameter bigger than the corresponding physical value. Instead a stable 3:2 or 2:1 periodic orbit for the Moon \sl cannot \rm be established for values of the parameter consistent with the observations. This remark suggests that the most likely ending state for the Moon is toward the synchronous resonance, validating previous results (Goldreich and Peale 1966, Henrard 1985) on probablities of capture into a resonance. Less evident is the situation for Mercury, for which the discrepancy between the theoretical results on resonance's stability is less pronounced. \vskip.1in\noindent In Figure 2 we compare our theoretical results with the actual motion of the Moon as it can be found in the Astronomical Almanac, 1990. More precisely, we plot the value of the $x$--coordinate over a year every lunar month ($\sim 27^d$). These values are obtained by integrating (with a leap--frog method) the equation of motion $(2.3)$ with initial data ${\hat x}=x_0+\varepsilon x_1$, ${\hat y}=y_0+\varepsilon y_1$, where $x_0$, $x_1$, $y_0$, $y_1$ can be explicitely computed through eq.s $(3.8)$--$(3.11)$ of \S3. Since $({\hat x},{\hat y})$ provide a good approximation of the theoretical location of the 1:1 periodic orbit, the successive $x$--values are almost the same every lunar month. We compare these results with the \sl libration in longitude \rm provided by the 1990 Astronomical Almanac. The small oscillations of the observed data are due to the physical librations of the Moon. We remark that the difference between our theoretical periodic orbit and the astronomical variation amounts to some thousandths of degree. \vskip.2in\noindent In Figure 3 we compare the values obtained plotting over one lunar month the theoretical solution $$ x(t)\ =\ x_*+y_* t+\varepsilon\ x_1(t)\ , $$ where $x_1(t)=\sum_{j=1}^7 c_j\ ({{\sin((2 y_*-j)t+2 x_*)-\sin(2 x_*)}\over {(2 y_*-j)^2}} -{\cos(2 x_*)t\over{(2 y_*-j)}})$ (here $x_*$ and $y_*$ are the values $x$ and $y$ given in (3.7)). More precisely, the value of the angle $x(t)$ was decreased by the revolutional mean motion and compared to the average over one solar year of the librational values provided by the 1990 Astronomical Almanac. \medskip\noindent As is well known, periodic orbits are often used as the starting points for computing the effective motion of solar system objects. For example, concerning the Moon, Hill (Hill 1878) found an exact special orbit and computed neighboring trajectories. More precisely, the idea is to solve the variational equations around a suitable periodic orbit and to recover the actual ephemeris within a good precision. We suggest that a similar method might be implemented, using as a starting point the orbits constructed above, to derive a semi--analytical theory of the Moon's physical librations. \vglue2cm \noindent \bf \S5. KAM STABILITY FOR THE ELLIPTIC CASE\rm \vskip.1in\noindent Lyapunov stability of the periodic orbits constructed in \S3 can be obtained by proving the existence of surrounding \sl librational \rm invariant surfaces. More precisely, according to (Siegel and Moser 1971), we consider the 2--dimensional area--preserving Poincar\'e map associated to $(2.3)$, having the origin as a fixed point. We reduce ourselves to study the stability of the origin, which implies the stability of the periodic orbit for the system $(2.3)$. As an example, we study the synchronous periodic orbit having period $T=2\pi$. \noindent We outline the sketch of the proof, referring to the Appendix B for further details. Following (Siegel and Moser 1971), we reduce the Poincar\'e map for (3.2) to the form $$\eqalign{ \eta_1'&= a\eta_1+b\eta_2+p(\eta_1,\eta_2)\cr \eta_2'&= c\eta_1+d\eta_2+q(\eta_1,\eta_2)\ ,\cr} \eqno{(5.1)} $$ where $S=\left(\matrix{a&b\cr c&d\cr}\right)$ is the matrix associated to the linear part and $p(\eta_1,\eta_2)$, $q(\eta_1,\eta_2)$ are higher order polynomials (with degree greater or equal than 3) in $\eta_1,\eta_2$. Since, in the elliptic case, the matrix $S$ has complex conjugated eigenvalues $(\lambda,{\overline\lambda})$, we proceed by reducing $(5.1)$ to a diagonal form through a symplectic coordinate change $(\eta_1,\eta_2)\rightarrow (\tilde x,\tilde y)$ $$\eqalign{ \tilde x'&= \lambda \tilde x+\tilde p(\tilde x,\tilde y)\cr \tilde y'&= {\overline\lambda} \tilde y+\tilde q(\tilde x,\tilde y)\ ,\cr} \eqno{(5.2)} $$ for some complex polynomials $\tilde p(\tilde x,\tilde y)$, $\tilde q(\tilde x,\tilde y)$. Next we perform a symplectic transformation which conjugates $(5.2)$ to the form $$\eqalign{ \xi'&=\xi\ e^{i(\gamma_0+\gamma_1\xi\eta+\gamma_2(\xi\eta)^2+...)}\cr \eta'&=\eta\ e^{-i(\gamma_0+\gamma_1\xi\eta+\gamma_2(\xi\eta)^2+...)}\ ,\cr} $$ where the coefficients $\gamma_j$ depend on $\varepsilon$ and $e$. According to Siegel and Moser the existence of an invariant curve around the origin is guaranteed by the condition that at least one of the coefficients $\gamma_j$, for $j\geq 1$, is non--zero. In particular, an explicit computation shows that the leading order in $\e$ is given by $$ \gamma_1\ =\ {{(3+\varepsilon T^2)T}\over {2(3-4\varepsilon T^2+\varepsilon^2 T^4)}} \ \sqrt{2\varepsilon T^2-{2\over 3}\varepsilon^2 T^4}\ , $$ which implies that the origin of the Poincar\'e map is a stable fixed point, providing the stability of the above periodic orbits of the differential system $(2.3)$. \vfill\eject \noindent \bf APPENDIX A: \rm \vskip.2in\noindent In this appendix we perform the main estimates in order to prove the results of \S4, using the Implicit Function Theorem of \S3. We denote by $x_*=x\equiv x_0+\varepsilon x_1$, $y_*=y\equiv y_0+\varepsilon y_1$ as in $(3.7)$ and $z_0\equiv (x_*,y_*)$. In particular we provide explicit estimates to check condition $(3.4)$, i.e. $$ 4\ \|M\|^2\ \|{{\partial^2F}\over{\partial(x,y)^2}}\|_\rho\ \|F(x_*,y_*)\|\ \leq\ 1\ , $$ with $$ \rho\ =\ 2\ \|M\|\ \|F(x_*,y_*)\|\ ; $$ here the parameter vector $\alpha$ is replaced by $\varepsilon$ varying in the interval $A\equiv [0,\varepsilon_0]$. For simplicity we have denoted $\|\cdot\|\equiv \sup_A|\cdot|$, $\|\cdot\|_\rho\equiv\sup_{{\overline B}_\rho(z_0)\times A}\|\cdot\|$. From $(3.3)$, $(3.6)$ the function $F(x,y)=(F_1(x,y),F_2(x,y))$ is explicitely given by $$\eqalign{ F_1(x,y)&\equiv yT+\varepsilon_0\int_0^T\int_0^t f_x(x+ys,s)dsdt+\varepsilon_0^2\int_0^T\eta(t;y,x)dt-2\pi p\cr F_2(x,y)&\equiv \int_0^T f_x(x+yt+\varepsilon_0 x_1(t;y,x)+\varepsilon^2\xi(t;y,x),t)dt\ .\cr} $$ \vglue1cm \noindent Let $f_x(x,t)$ be as in $(2.3)-(3.2)$; the norm of $F(x_*,y_*)$ is obtained as $$\eqalign{ \|F(x_*, y_*;\varepsilon)\|&\equiv\sup\{\| F_1(x_*, y_*;\varepsilon)\|, \|F_2(x_*, y_*;\varepsilon)\|\}\cr \|F_1(x_*, y_*;\varepsilon)\|&\leq\varepsilon^2\Big[ \|f_{xx}\|\ \Big[ |x_1|+|y_1|{T\over 3}\Big]\ {{T^2}\over 2}+\|\eta\|T\Big]\cr \|F_2(x_*, y_*;\varepsilon)\|&\leq{\varepsilon_0^2}\Big[ \|f_{xx}\|T\|\xi\|+{1\over 2} \|f_{xxx}\|T\|\xi\|^2\varepsilon_0^2\Big]\ ,\cr} $$ (having choosen in ${\bf R}^2$ the sup--norm) where $$ \|x_1(t;x_*,y_*)\|\leq \sum_{j=1}^7|c_j|\ |{{\sin((2y_*-j)T+2x_*)-\sin(2 x_*)}\over {(2y_*-j)^2}} -{\cos(2 x_*)T\over{(2 y_*-j)}}| \eqno{(A.1)} $$ and $$\eqalign{ \|\xi\|&\leq{{T^2}\over 2}\ {{\|f_{xx}\|\|x_1(t;x_*,y_*)\|}\over {1-\varepsilon_0{{T^2}\over 2}\|f_{xx}\|}}\cr \|\eta\|&\leq T\|f_{xx}\|\ \Big(\|x_1(t;x_*,y_*)\|+\varepsilon_0\|\xi\|\Big)\ . \cr} \eqno{(A.2)} $$ \vskip.2in \noindent $\bullet$ Estimate of the norm of $M$: let $$ M\equiv \Big({{\partial F(x_*,y_*;\varepsilon)}\over {\partial(x,y)}}\Big)^{-1}\ . $$ Let us explicit the derivatives as follows. If we denote by $$\eqalign{ G_1(x,y)&\equiv Ty-2\pi p+\varepsilon\int_0^T y_1(t;y,x)dt\cr G_2(x,y)&\equiv \int_0^T f_x(x+yt+\varepsilon x_1(t;y,x),t)dt\ , \cr} $$ then, omitting the arguments of the functions, we obtain: $$\eqalign{ {{\partial F_1}\over{\partial x}}&={{\partial G_1}\over{\partial x}} +\varepsilon^2\int_0^T\ \eta_x\ dt\cr {{\partial F_1}\over{\partial y}}&={{\partial G_1}\over{\partial y}} +\varepsilon^2\int_0^T\ \eta_y\ dt\cr {{\partial F_2}\over{\partial x}}&={{\partial G_2}\over{\partial x}} +\varepsilon^2\int_0^T\ f_{xx}\ \xi_x\ dt\cr {{\partial F_2}\over{\partial y}}&={{\partial G_2}\over{\partial y}} +\varepsilon^2\int_0^T\ f_{xx}\ \xi_y\ dt\ .\cr} $$ Denoting by $$ H_G\equiv \Big({{\partial G(x_*,y_*)}\over {\partial(x,y)}}\Big)\ , $$ we can write $M^{-1}\equiv H_G+\varepsilon^2\tilde H$, where $$ \tilde H\ \equiv\ \left(\matrix{\int_0^T\ \eta_x\ dt&\int_0^T\ \eta_y\ dt\cr \int_0^T\ f_{xx}\ \xi_x\ dt&\int_0^T\ f_{xx}\ \xi_y\ dt\cr}\right)\ . $$ Therefore an estimate on $M$ can be obtained through the above quantities as $$ \|M\|\ \leq\ {{\|H_G^{-1}\|}\over {1-\varepsilon_0^2\|H_G^{-1}\|\ \|\tilde H\|}}\ . $$ \vskip.1in \noindent Now we provide estimates on $H_G^{-1}$ and $\tilde H$. Let $$ H_G\equiv \Big({{\partial G(x_*,y_*)}\over {\partial(x,y)}}\Big)\equiv \left(\matrix{\alpha&\beta\cr \gamma&\delta\cr}\right)\ ; $$ then $$ \|H_G^{-1}\|=\|{1\over {\alpha\delta-\beta\gamma}}\|\ \cdot\ \sup\{\|\beta\|+\|\delta\|,\ \|\alpha\|+\|\gamma\|\}\ . $$ Denoting by $a=2x_*$ and by $b_j=2y_*-j$, one has $$\eqalign{ \alpha&=\varepsilon\|f_{xx}\|{{T^2}\over 2}\cr \beta&=T+\varepsilon \sum_{i=1}^7 c_j \ \{-{T\over {b_j^2}}\cos(a+b_jT)+{2\over{b_j^3}}\sin(a+b_jT)-{T\over{b_j^2}}\cos(a)-{2\over{b_j^3}} \sin(a)\}\cr \gamma&=\|f_{xx}\|T\ (1+\varepsilon\|f_{xx}\|{{T^2}\over 2})\cr \delta&=\|f_{xx}\|T\ ({T\over 2}+\varepsilon\|f_{xx}\|{{T^3}\over {24}})\ .\cr} $$ \vskip.1in \noindent Concerning the estimate on $\tilde H$ we have: $$ \|\tilde H\|\ \equiv\ \sup\{\|H_{11}\|+\|H_{12}\|,\ \|H_{21}\|+\|H_{22}\|\}\ , $$ and $$\eqalign{ \|H_{11}\|&\leq T\|\eta_x\|\cr \|H_{12}\|&\leq T\|\eta_y\|\cr \|H_{21}\|&\leq \|f_{xx}\| T\|\xi_x\|\cr \|H_{22}\|&\leq \|f_{xx}\| T\|\xi_y\|\ , \cr} $$ where $\|\eta_x\|$, $\|\eta_y\|$, $\|\xi_x\|$, $\|\xi_y\|$ may be estimated as follows: $$\eqalign{ \|\eta_x\|&\leq T\|f_{xx}\|\ \Big(\|{{\partial x_1}\over{\partial x}}\|+ \varepsilon_0\|\xi_x\|\Big)+T\|f_{xxx}\|\Big(\|x_1\|+\varepsilon_0\|\xi\|\Big)\Big(1+ \varepsilon_0 \|{{\partial x_1}\over{\partial x}}\|+\varepsilon_0^2\|\xi_x\|\Big)\cr \|\eta_y\|&\leq T\|f_{xx}\|\ \Big(\|{{\partial x_1}\over{\partial y}}\|+ \varepsilon_0\|\xi_y\|\Big)+T\|f_{xxx}\|\Big(\|x_1\|+\varepsilon_0\|\xi\|\Big) \Big({T\over2}+ \varepsilon_0 \|{{\partial x_1}\over{\partial y}}\|+\varepsilon_0^2\|\xi_y\|\Big)\cr \|\xi_x\|&\leq{{T^2\|f_{xx}\|\|{{\partial x_1}\over{\partial x}}\|+T^2\|f_{xxx}\|\ \Big(\|x_1\|+\varepsilon_0\|\xi\|\Big)\Big(1+\varepsilon_0\|{{\partial x_1}\over{\partial x}}\| \Big)}\over {2\Big(1-{{T^2}\over 2}\|f_{xx}\|\varepsilon_0-{{T^2}\over 2}\|f_{xxx}\| \varepsilon_0^2\big(\|x_1\|+\varepsilon_0\|\xi\|\big)\Big)}}\cr \|\xi_y\|&\leq{{T^2\|f_{xx}\|\|{{\partial x_1}\over{\partial y}}\|+T^2\|f_{xxx}\| \Big(\|x_1\|+\varepsilon_0\|\xi\|\Big)\Big({T\over 2}+\varepsilon_0\|{{\partial x_1}\over{\partial y}}\| \Big)}\over {2\Big(1-{{T^2}\over 2}\|f_{xx}\|\varepsilon_0-{{T^2}\over 2}\|f_{xxx}\| \varepsilon_0^2\big(\|x_1\|+\varepsilon_0\|\xi\|\big)\Big)}}\ .\cr} \eqno{(A.3)} $$ The estimates for the derivatives of $\|x_1(t;x,y)\|$ (computed at the point $(x_*,y_*)$) are derived similarly as in $(A.1)$. \vskip.2in \noindent $\bullet$ Estimate on $\|{{\partial^2 F}\over {\partial (x,y)}}\|$: \noindent In order to give the estimate on $\|{{\partial^2 F}\over {\partial (x,y)}}\|$ we need to provide the norms of the second derivatives of the functions $\xi$, $\eta$ (the estimates on $\xi$, $\eta$ and their first derivatives were already given in $(A.2)$, $(A.3)$). We find: $$\eqalign{ \|\xi_{xx}\|&\leq \Big[1-{{T^2}\over 2}\|f_{xx}\|\varepsilon_0-{{T^2}\over 2}\|f_{xxx}\| \Big(\|x_1\|+\varepsilon_0\|\xi\|\Big)\varepsilon_0^2\Big]^{-1}\ \ {1\over 2}\Big[T^2\|f_{xx}\| \|{{\partial^2 x_1}\over{\partial x^2}}\|\cr &+2T^2\Big(\|{{\partial x_1}\over{\partial x}}\|+\varepsilon_0\|\xi_x\|\Big)\ \|f_{xxx}\|\ \Big(1+\varepsilon_0\|{{\partial x_1}\over{\partial x}}\|+\varepsilon_0^2\|\xi_x\|\Big)\cr &+T^2\|f_{xxxx}\| \Big(\|x_1\|+\varepsilon_0\|\xi\|\Big) \Big(1+\varepsilon_0\|{{\partial x_1}\over{\partial x}}\|+\varepsilon_0^2\|\xi_x\|\Big)^2\cr &+\varepsilon_0 T^2 \|{{\partial^2 x_1}\over{\partial x^2}}\|\|f_{xxx}\| \Big(\|x_1\|+\|\varepsilon_0\|\xi\|\Big)\Big]\cr \|\eta_{xx}\|&\leq T\|f_{xx}\|\Big(\|{{\partial^2 x_1}\over{\partial x^2}}\|+ \varepsilon_0\|\xi_{xx}\|\Big)+2T\Big(\|{{\partial x_1}\over{\partial x}}\|+ \varepsilon_0\|\xi_x\|\Big)\|f_{xxx}\|\ \cr &\Big(1+\varepsilon_0\|{{\partial x_1}\over{\partial x}}\|+\varepsilon_0^2\|\xi_x\|\Big) +T\|f_{xxx}\|\Big(\|x_1\|+\varepsilon_0\|\xi\|\Big)\cr &\Big(1+\varepsilon_0\|{{\partial x_1}\over{\partial x}}\|+\varepsilon_0^2\|\xi_x\|\Big)^2 +\Big(\|x_1\|+\varepsilon_0\|\xi\|\Big)\cr &\ T\|f_{xxx}\|\varepsilon_0\Big(\|{{\partial^2 x_1}\over {\partial x^2}}\|+\varepsilon_0\|\xi_{xx}\|\Big)\cr \|\xi_{yy}\|&\leq {T\over 2}\Big[1-{{T^2}\over 2}\|f_{xx}\|\varepsilon_0-{{T^2}\over 2}\|f_{xxx}\| \Big(\|x_1\|+\varepsilon_0\|\xi\|\Big)\varepsilon_0^2\Big]^{-1}\ \Big[T\|f_{xx}\| \|{{\partial^2 x_1}\over{\partial y^2}}\|\cr &+2T\Big(\|{{\partial x_1}\over{\partial y}}\|+\varepsilon_0\|\xi_y\|\Big)\ \|f_{xxx}\|\ \Big({T\over 3}+\varepsilon_0\|{{\partial x_1}\over{\partial y}}\|+\varepsilon_0^2\|\xi_y\|\Big)\cr &+\|f_{xxxx}\| \Big(\|x_1\|+\varepsilon_0\|\xi\|\Big)\cr &\Big({{T^3}\over 6}+T\varepsilon_0^2(\|{{\partial x_1}\over{\partial y}}\| +\varepsilon_0\|\xi_y\|)^2+\varepsilon_0 {{T^2}\over 3}(\|{{\partial x_1}\over{\partial y}}\|+\varepsilon_0\|\xi_y\|)\Big)\cr &+\varepsilon_0 T \|{{\partial^2 x_1}\over{\partial y^2}}\|\|f_{xxx}\| \Big(\|x_1\|+\|\varepsilon_0\|\xi\|\Big)\Big]\cr \|\eta_{yy}\|&\leq T\|f_{xx}\|\Big[\|{{\partial^2 x_1}\over{\partial y^2}}\|+ \varepsilon_0\|\xi_{yy}\|\Big]+2T\Big[\|{{\partial x_1}\over{\partial y}}\|+ \varepsilon_0\|\xi_y\|\Big]\|f_{xxx}\|\cr &\Big({T\over 2}+\varepsilon_0\|{{\partial x_1}\over{\partial y}}\|+\varepsilon_0^2\|\xi_y\|\Big) +\|f_{xxxx}\|\Big(\|x_1\|+\varepsilon_0\|\xi\|\Big)\Big({{T^3}\over 3}+ T\varepsilon_0^2(\|{{\partial x_1}\over{\partial y}}\|+\varepsilon_0\|\xi_y\|)^2\cr &+\varepsilon_0 T^2(\|{{\partial x_1}\over{\partial y}}\|+\varepsilon_0\|\xi_y\|)\Big)+\varepsilon_0 T\|f_{xxx}\|(\|x_1\|+\varepsilon_0\|\xi\|)\Big(\|{{\partial^2 x_1}\over {\partial y^2}}\|+\varepsilon_0\|\xi_{yy}\|\Big)\cr \|\xi_{yx}\|&\leq {1\over 2}\Big[1-{{T^2}\over 2}\|f_{xx}\|\varepsilon_0-{{T^2}\over 2}\|f_{xxx}\| (\|x_1\|+\varepsilon_0\|\xi\|)\varepsilon_0^2\Big]^{-1}\ \Big[T^2\|f_{xx}\| \|{{\partial^2 x_1}\over{\partial y\partial x}}\|\cr &+T^2\Big(\|{{\partial x_1}\over{\partial y}}\|+\varepsilon_0\|\xi_y\|\Big)\|f_{xxx}\| \Big(1+\varepsilon_0\|{{\partial x_1}\over{\partial x}}\|+\varepsilon_0^2\|\xi_x\|\Big)+\|f_{xxxx}\| \Big(\|x_1\|+\varepsilon_0\|\xi\|\Big)\cr &\Big(1+\varepsilon_0(\|{{\partial x_1}\over{\partial x}}\| +\varepsilon_0^2\|\xi_x\|\Big)^2 T^2\Big({T\over 3}+\varepsilon_0 \|{{\partial x_1}\over{\partial y}}\|+\varepsilon_0^2\|\xi_y\|\Big)\cr &+\varepsilon_0 T^2 \|{{\partial^2 x_1}\over{\partial y\partial x}}\|\|f_{xxx}\| \Big(\|x_1\|+\varepsilon_0\|\xi\|\Big)\cr &+\|f_{xxx}\|\Big(\|{{\partial x_1}\over{\partial x}}\|+\varepsilon_0\|\xi_x\|\Big) T^2\Big({T\over 3}+\varepsilon_0 \|{{\partial x_1}\over{\partial x}}\|+\varepsilon_0^2 \|\xi_y\|\Big)\Big]\cr} $$ $$\eqalign{ \|\eta_{yx}\|&\leq T\|f_{xx}\|\Big(\|{{\partial^2 x_1}\over{\partial y\partial x}}\|+ \varepsilon_0\|\xi_{yx}\|\Big)+T\Big(\|{{\partial x_1}\over{\partial y}}\|+ \varepsilon_0\|\xi_y\|\Big)\|f_{xxx}\|\cr &\Big(1+\varepsilon_0\|{{\partial x_1}\over{\partial x}}\|+\varepsilon_0^2\|\xi_x\|\Big) +\|f_{xxxx}\|\Big(\|x_1\|+\varepsilon_0\|\xi\|\Big)\Big(1+ \varepsilon_0\|{{\partial x_1}\over{\partial x}}\|\cr &+\varepsilon_0^2\|\xi_x\|\Big)\ T \Big({T\over 2}+\varepsilon_0\|{{\partial x_1}\over{\partial y}}\|+\varepsilon_0^2\|\xi_y\|\Big) +\varepsilon_0 T\|f_{xxx}\|\Big(\|x_1\|+\varepsilon_0\|\xi\|\Big)\cr &\Big(\|{{\partial^2 x_1}\over {\partial y\partial x}}\|+\varepsilon_0\|\xi_{yx}\|\Big) +\|f_{xxx}\|\Big(\|{{\partial x_1}\over{\partial x}}\|+\varepsilon_0\|\xi_x\|\Big)\cr &T\Big({T\over 2}+\varepsilon_0 \|{{\partial x_1}\over{\partial x}}\|+\varepsilon_0^2 \|\xi_y\|\Big)\ .\cr} $$ \vskip.1in \noindent Thus, $$ \|{{\partial^2 F}\over {\partial (x,y)}}\|\ =\ \sup\{ \|{{\partial^2 F_1}\over {\partial y^2}}\|+\|{{\partial^2 F_1}\over {\partial x^2}}\| +2\|{{\partial^2 F_1}\over {\partial y\partial x}}\|,\ \|{{\partial^2 F_2}\over {\partial y^2}}\|+\|{{\partial^2 F_2}\over {\partial x^2}}\| +2\|{{\partial^2 F_2}\over {\partial y\partial x}}\|\}\ , $$ and $$\eqalign{ \|{{\partial^2 F_1}\over {\partial y^2}}\|&+\|{{\partial^2 F_1}\over {\partial x^2}}\| +2\|{{\partial^2 F_1}\over {\partial y\partial x}}\|\leq \varepsilon_0\|f_{xxx}\|\Big[{{T^2}\over 2}+{{T^3}\over 3}+{{T^4}\over {12}}\Big]\cr &+\varepsilon_0^2\Big[\|\eta_{xx}\|+\|\eta_{yy}\|+2\|\eta_{xy}\|\Big]\cr \|{{\partial^2 F_2}\over {\partial y^2}}\|&+\|{{\partial^2 F_2}\over {\partial x^2}}\| +2\|{{\partial^2 F_2}\over {\partial y\partial x}}\|\leq \|f_{xxx}\|\Big[({{T^3}\over 3}+(\varepsilon_0\|{{\partial x_1}\over {\partial y}}\|+ \varepsilon_0^2\|\xi_y\|)T^2 +(\varepsilon_0\|{{\partial x_1}\over {\partial y}}\|\cr &+\varepsilon_0^2\|\xi_y\|)^2T) +T(1+\varepsilon_0\|{{\partial x_1}\over {\partial x}}\|+ \varepsilon_0^2\|\xi_x\|)^2+2({{T^2}\over 2}+(\varepsilon_0 \|{{\partial x_1}\over {\partial y}}\|\cr &+\varepsilon_0^2\|\xi_y\|)T)(1+ \varepsilon_0\|{{\partial x_1}\over {\partial x}}\|+ \varepsilon_0^2\|\xi_x\|)\Big]\cr &+\|f_{xx}\|T\Big[\varepsilon_0\|{{\partial^2 x_1}\over {\partial y^2}}\|+ \varepsilon_0^2\|\xi_{yy}\|+\varepsilon_0\|{{\partial^2 x_1}\over {\partial x^2}}\|+ \varepsilon_0^2\|\xi_{xx}\|\cr &+2\varepsilon_0\|{{\partial^2 x_1}\over {\partial y\partial x}}\|+ 2\varepsilon_0^2\|\xi_{yx}\|\Big]\ .\cr} $$ In the above formulae, the function $x_1(t;x,y)$ and its derivatives are estimated on the domain of radius $\rho$, rather then being computed on the point $(x_*,y_*)$. Let us remark that the function $x_1(t;x,y)$ involves a term which dominates due to the resonance relation; therefore, we evidentiate this term labelling it with the index $k$. In particular it is $k=2$ for the 1:1 resonance, $k=3$ for the 3:2, $k=4$ for the 2:1. More precisely, we write $x_1(t)\equiv x_1(t;x,y)$ as $$\eqalign{ x_1(t)=\sum_{j=1,j\not=k}^7 &c_j \Big[{{\sin((2y-j)t+2x)-\sin(2 x)}\over {(2y-j)^2}} -{\cos(2 x)t\over{(2 y-j)}}\Big]\cr &+c_k \Big[{{\sin((2y-k)t+2x)-\sin(2 x)}\over {(2y-k)^2}} -{\cos(2 x)t\over{(2 y-k)}}\Big]\ .\cr} $$ Denoting by $M\equiv T(2\rho+|2y_*-k|)$, $b\equiv 2(|x_*|+\rho)$, $N_l\equiv|2\rho-|2y_*-l||$ for $l=1,...,7$, we obtain the following estimates: $$\eqalign{ \|x_1(t)\|&\leq \sum_{j=1,j\not=k}^7 |c_j| \Big[{2\over {N_j^2}}+{T\over {N_j}}\Big] +|c_k|\ \Big[T^2{{\sinh(M)-M}\over {M^2}}+bT^2{{\cosh(M)-1}\over{M^2}}\Big]\cr \|{{\partial x_1(t)}\over {\partial x}}\|&\leq 2\sum_{j=1,j\not=k}^7 |c_j| \Big[{2\over {N_j^2}}+{T\over {N_j}}\Big] +2|c_k|\ \Big[T^2{{\cosh(M)-1}\over {M^2}}+bT^2{{\sinh(M)-M}\over{M^2}}\Big]\cr \|{{\partial^2 x_1(t)}\over {\partial x^2}}\|&\leq 4\sum_{j=1,j\not=k}^7 |c_j| \Big[{2\over {N_j^2}}+{T\over {N_j}}\Big] +4|c_k|\ \Big[T^2{{\sinh(M)-M}\over {M^2}}+bT^2{{\cosh(M)-1}\over{M^2}}\Big]\cr} $$ $$\eqalign{ \|{{\partial x_1(t)}\over {\partial y}}\|&\leq \sum_{j=1,j\not=k}^7 |c_j| \Big[{8\over {N_j^3}}+{T\over {N_j^2}}\Big] +2|c_k|\ \Big[2T^3{{\sinh(M)-M}\over {M^3}}+T^3{{\cosh(M)-1}\over{M^2}}\cr &+2bT^3{{\cosh(M)-1}\over{M^3}}+bT^3{{\sinh(M)}\over {M^2}}\Big]\cr \|{{\partial^2 x_1(t)}\over {\partial y\partial x}}\|&\leq \sum_{j=1,j\not=k}^7 |c_j| \Big[{{16}\over {N_j^3}}+{{8T}\over {N_j^2}}\Big] +4|c_k|\ \Big[2bT^3{{\sinh(M)-M}\over {M^3}}+bT^3{{\cosh(M)-1}\over{M^2}}\cr &+2T^3{{\cosh(M)-1}\over{M^3}}+T^3{{\sinh(M)}\over {M^2}}\Big]\cr \|{{\partial^2 x_1(t)}\over {\partial y^2}}\|&\leq 4\sum_{j=1,j\not=k}^7 |c_j| \Big[{{12}\over {N_j^4}}+{{6T}\over {N_j^3}}+{{T^2}\over {N_j^2}}\Big]+4|c_k|\ \Big[T^4\Big(6{{\sinh(M)-M}\over {M^4}}\cr &+4{{\cosh(M)-1}\over{M^3}} +{{\cosh(M)}\over{M^2}}\Big) +bT^4\Big(6{{\cosh(M)-1}\over{M^4}}\cr &+4{{\sinh(M)}\over {M^3}}+{{\cosh(M)}\over {M^2}}\Big)\Big]\ .\cr} $$ \vfill\eject \noindent \bf APPENDIX B: \rm \vskip.2in\noindent In this appendix, following (Siegel and Moser, 1971) we provide details about the existence of KAM librational invariant surfaces around the synchronous periodic orbit. We do not claim to obtain optimal estimates on the parameters ensuring the stability of the periodic orbit, but just to prove that there exist invariant surfaces around the periodic orbit for \sl suitably \rm small values of the parameters. Therefore we consider the simpler problem with zero eccentricity, i.e. $$\eqalign{ \dot x&=y\cr \dot y&=-\varepsilon\sin(2x-2t)\equiv\varepsilon g(x,t)\cr} \eqno{(B.1)} $$ and perform all computations to first order in $\varepsilon$. Under suitable coordinate transformations, we reduce $(B.1)$ to the form $$\eqalign{ \xi'&=\xi\ e^{i(\gamma_0+\gamma_1\xi\eta+\gamma_2(\xi\eta)^2+...)}\cr \eta'&=\eta\ e^{-i(\gamma_0+\gamma_1\xi\eta+\gamma_2(\xi\eta)^2+...)}\cr} \eqno{(B.2)} $$ and we show that the first coefficient $\gamma_1=\gamma_1|_{e=0}$ of the normal form is different from zero. Since this coefficient depends analytically on $\varepsilon$, $e$, we can conclude that the condition $\gamma_1(\varepsilon,e)\not=0$ is satisfied for sufficiently small values of the parameters $\varepsilon$, $e$. We postpone to a later work the problem of proving the existence of invariant surfaces for realistic values of the parameters. \vglue2cm \noindent \bf B.1 Linearization of $\bf (B.1)$ \rm \noindent Let $({\overline x},{\overline y})$ be initial conditions on the periodic orbit and let $P({\overline x},{\overline y})\equiv(x(T),y(T))=(x',y')$ be the Poincar\'e map at time $T=2\pi q$. By $(B.1)$ we can rewrite the Poincar\'e map as $$\eqalign{ x'&={\overline x}+yT+\varepsilon\int_0^T\int_0^s g(x(\tau;{\overline x},{\overline y}), \tau)d\tau ds\cr y'&={\overline y}+\varepsilon\int_0^T g(x(s;{\overline x},{\overline y}),s) ds\ ,\cr} \eqno{(B.3)} $$ which has $({\overline x},{\overline y})$ as a fixed point. We shift the fixed point to the origin by means of the canonical transformation $$\eqalign{ \eta_1&=x-{\overline x}\cr \eta_2&=y-{\overline y}\ ,\cr} $$ so that $(B.3)$ becomes $$\eqalign{ \eta_1'&=\eta_1+\eta_2T+{\overline y}T+\varepsilon\int_0^T\int_0^s g(x(\tau; \eta_1+{\overline x},\eta_2+{\overline y}),\tau)d\tau ds\cr \eta_2'&=\eta_2+\varepsilon\int_0^T g(x(s;\eta_1+{\overline x},\eta_2+{\overline y}), s) ds\ .\cr} \eqno{(B.4)} $$ We recall that $({\overline x},{\overline y})$ are power series in $\varepsilon$: $$\eqalign{ {\overline x}&=x_0+\varepsilon x_1+\varepsilon^2 x_2+...\cr {\overline y}&=y_0+\varepsilon y_1+\varepsilon^2 y_2+...\ ,\cr} $$ where, in particular, $x_0=0$, $y_0=1$. Since $$ x(\tau;{\overline x},{\overline y})={\overline x}+{\overline y}\tau+ \varepsilon\int_0^\tau\int_0^s g(x(t;{\overline x},{\overline y}),t)\ dt ds\ , $$ up to first order in $\varepsilon$ we have $$ x(\tau;{\overline x},{\overline y})=x_0+y_0 \tau+O(\varepsilon)\ . $$ Therefore, disregarding $O(\e^2)$, $(B.4)$ reduces to $$\eqalign{ \eta_1'&=\eta_1+\eta_2T+y_0T+\varepsilon y_1T+\varepsilon\int_0^T\int_0^s g(\eta_1+x_0+(\eta_2+y_0)\tau,\tau)d\tau ds\cr \eta_2'&=\eta_2+\varepsilon\int_0^T g(\eta_1+x_0+(\eta_2+y_0)s,s) ds\ .\cr} \eqno{(B.5)} $$ The development of the r.h.s. of $(B.5)$ in power series of $\eta_1$, $\eta_2$ is performed using the periodicity conditions $$\eqalign{ y_1T&= -\int_0^T\int_0^s g(x_0+y_0 \tau,\tau)\ d\tau ds\cr \int_0^Tg&(x_0+y_0s,s)\ ds=0\ . \cr} \eqno{(B.6)} $$ By means of $(B.6)$ we obtain $$\eqalign{ y_1T&+\int_0^T\int_0^s g(\eta_1+x_0+(\eta_2+y_0)\tau,\tau)d\tau ds\cr &=\int_0^T\int_0^s [g(\eta_1+x_0+(\eta_2+y_0)\tau,\tau)-g(x_0+y_0\tau,\tau)]d\tau ds\cr &=\int_0^T\int_0^s [g_x(x_0+y_0\tau,\tau)\ (\eta_1+\eta_2\tau)+{1\over 2} g_{xx}(x_0+y_0\tau,\tau)\ (\eta_1+\eta_2\tau)^2\cr &\ \ +{1\over 6} g_{xxx}(x_0+y_0\tau,\tau)\ (\eta_1+\eta_2\tau)^3]\ d\tau ds\cr &=-T^2\eta_1-{{T^3}\over 3}\eta_2+{2\over 3}T^2\eta_1^3+{2\over 3}T^3\eta_1^2\eta_2 +{1\over 3}T^4\eta_1\eta_2^2+{1\over {15}}T^5\eta_2^3\ . \cr} $$ In a similar way, we obtain $$\eqalign{ \int_0^T &g(\eta_1+x_0+(\eta_2+y_0)s,s) ds=\int_0^T [g(\eta_1+x_0+(\eta_2+y_0)s,s) -g(x_0+y_0s,s)]ds\cr &=\int_0^T [g_x(x_0+y_0s)\ (\eta_1+\eta_2\ s)+{1\over 2} g_{xx}(x_0+y_0s)\ (\eta_1+\eta_2\ s)^2\cr &\ \ +{1\over 6} g_{xxx}(x_0+y_0s)\ (\eta_1+\eta_2\ s)^3]ds\cr &=-2T\eta_1-T^2\eta_2+{4\over 3}T\eta_1^3+2T^2\eta_1^2\eta_2+{4\over 3}T^3\eta_1\eta_2^2 +{1\over 3}T^4\eta_2^3\ . \cr} $$ Therefore neglecting $O(\varepsilon^2)$ and polynomial terms of order higher than 3 in $\eta_1$, $\eta_2$, we rewrite $(B.5)$ as $$\eqalign{ \eta_1'&=a\eta_1+b\eta_2+p(\eta_1,\eta_2)\cr \eta_2'&=c\eta_1+d\eta_2+q(\eta_1,\eta_2)\ ,\cr} \eqno{(B.7)} $$ where $$\eqalign{ a&=1-\varepsilon T^2\qquad b=T-\varepsilon{{T^3}\over 3}\qquad c=-2T\varepsilon\qquad d=1-\varepsilon T^2\cr p(\eta_1,\eta_2)&=\ \varepsilon[{2\over 3}T^2\eta_1^3+{2\over 3}T^2\eta_1^2\eta_2 +{1\over 3}T^4\eta_1\eta_2^2+{1\over {15}}T^5\eta_2^3]\cr q(\eta_1,\eta_2)&=\ \varepsilon[{4\over 3}T\eta_1^3+2T^2\eta_1^2\eta_2+{4\over 3}T^3\eta_1 \eta_2^2+{1\over 3}\eta_2^3]\ .\cr} $$ Provided $\varepsilon<{3\over {T^2}}$, the eigenvalues $(\lambda,\mu)$ of the linear part are complex conjugated, $\mu={\overline\lambda}$, and precisely $\lambda=\lambda_1+i\lambda_2$, $\mu=\lambda_1-i\lambda_2$, with $$ \lambda_1=1-\varepsilon T^2\ ,\qquad\qquad \lambda_2=\sqrt{2\varepsilon T^2- {2\over 3}\varepsilon^2T^4}\ . $$ \vskip.2in \noindent \bf Remark: \rm Notice that the solution is of \sl elliptic \rm type if the eigenvalues of the linear part of eq. $(B.7)$ are complex conjugate. Such eigenvalues are determined as the solution of the secular equation $$ \lambda^2-(a+d)\lambda+ad-bc=0\ . $$ The eigenvalues are complex conjugate, if the discriminant is negative, i.e. $\Delta\equiv (a+d)^2-4< 0$, namely $-2