Corrigendum to: Oscillatory motions in restricted $N$-body problems [J. Differential Equations 265 (2018) 779-803]

TL;DR

This paper proves the existence of chaotic and oscillatory motions in the planar restricted N-body problem by analyzing the intersections of stable and unstable manifolds at infinity, using Melnikov theory.

Contribution

It introduces a novel scaling approach to analyze the Hamiltonian, enabling the demonstration of chaotic and oscillatory motions without restrictions on mass parameters.

Findings

Existence of transversal intersections of manifolds at infinity.

Chaotic motions guaranteed by Smale's horseshoe.

Oscillatory motions where the particle leaves and returns to bounded regions.

Abstract

We consider the planar restricted $N$-body problem where the $N-1$ primaries are assumed to be in a central configuration whereas the infinitesimal particle escapes to infinity in a parabolic orbit. We prove the existence of transversal intersections between the stable and unstable manifolds of the parabolic orbits at infinity which guarantee the existence of a Smale's horseshoe. This implies the occurrence of chaotic motions but also of oscillatory motions, that is, orbits for which the massless particle leaves every bounded region but it returns infinitely often to some fixed bounded region. Our achievement is based in an adequate scaling of the variables which allows us to write the Hamiltonian function as the Hamiltonian of the Kepler problem multiplied by a small quantity plus higher-order terms that depend on the chosen configuration. We compute the Melnikov function related to the first two non-null perturbative terms and characterize the cases where it has simple zeroes. Concretely, for some combinations of the configuration parameters, i.e. mass values and positions of the primaries, the Melnikov function vanishes, otherwise it has simple zeroes and the transversality condition is satisfied. The theory is illustrated for various cases of restricted $N$-body problems, including the circular restricted three-body problem. No restrictions on the mass parameters are assumed.

Figures

Equations

$$H(Q,P)=\frac{1}{2}{\left|P\right|}^2-Q^{T}JP-U(Q),$$

$$U(Q)=\sum _{k=1}^{N-1}\frac{m_k}{\left|Q-a_k\right|}.$$

$$a_k=-\sum _{\stackrel{j=1}{j\ne k}}^{N-1}\frac{m_j(a_j-a_k)}{|a_j-a_k{|}^3},k=1,\dots ,N-1\text{mbox}and\sum _{k=1}^{N-1}{m}_k{a}_k=0,$$

$$H(q,p,t)=\frac{1}{2}{\left|p\right|}^2-U(q,t),$$

$$U(q,t)=\sum _{k=1}^{N-1}\frac{m_k}{\left|e^{-tJ}q-a_k\right|}.$$

$$H_{\epsilon }(q,p,t)={\epsilon }^3\left(\frac{1}{2}|p{|}^2-\sum _{k=1}^{N-1}\frac{m_k}{\left|{\epsilon }^{-2}{e}^{-tJ}q-a_k\right|}\right).$$

$$H_{\epsilon }(q,p,t)={\epsilon }^3\left(\frac{1}{2}|p{|}^2-\frac{1}{|q|}\right)-\sum _{j=2}^{\infty }\frac{{\epsilon }^{2j+3}}{|q{|}^{j+1}}\sum _{k=1}^{N-1}{m}_k|a_k{|}^j{P}_j(\cos {\gamma }_k),$$

$$q=r{e}^{i\theta },p=R{e}^{i\theta }+\frac{\Theta }{r}i{e}^{i\theta },$$

$$\begin{array}[]{rcl}H_{\varepsilon}(r,\theta,R,\Theta,t)&=&\displaystyle\varepsilon^{3}\left(\frac{1}{2}\left(R^{2}+\frac{\Theta^{2}}{r^{2}}\right)-\frac{1}{r}\right)\\[4.30554pt] &&\displaystyle-\sum_{j=2}^{\infty}\frac{\varepsilon^{2j+3}}{r^{j+1}}\sum_{k=1}^{N-1}m_{k}|a_{k}|^{j}P_{j}\left(\cos\left(\alpha_{k}-(\theta-t)\right)\right).\end{array}$$

$$\dot{r}$$

$$\dot{R}$$

$$\dot{\theta }$$

$$\dot{\Theta }$$

$$r=x^{-2},R=-\sqrt{2}y,s=t-\theta ,$$

$$\begin{array}[]{rcl}\dot{x}&=&\mbox{$\frac{1}{\sqrt{2}}$}\varepsilon^{3}x^{3}y,\\[5.16663pt] \dot{y}&=&\varepsilon^{3}\left(\mbox{$\frac{1}{\sqrt{2}}$}x^{4}-\mbox{$\frac{1}{\sqrt{2}}$}\Theta^{2}\,x^{6}\right)\\ &&\displaystyle-\mbox{$\frac{1}{\sqrt{2}}$}\sum_{j=2}^{\infty}(j+1)\varepsilon^{2j+3}x^{2j+4}\sum_{k=1}^{N-1}m_{k}\left|a_{k}\right|^{j}P_{j}\left(\cos\left(\alpha_{k}+s\right)\right),\\[5.16663pt] \dot{s}&=&1-\varepsilon^{3}x^{4}\,\Theta,\\ \dot{\Theta}&=&\displaystyle\sum_{j=2}^{\infty}\varepsilon^{2j+3}x^{2j+2}\sum_{k=1}^{N-1}m_{k}\left|a_{k}\right|^{j}\sin(\alpha_{k}+s)Q_{j}\left(\cos\left(\alpha_{k}+s\right)\right).\end{array}$$

$$C=H_{\epsilon }(x,y,\theta ,\Theta ,t)-\Theta ,$$

$$\Theta =\frac{1\pm \sqrt{1+2{\epsilon }^3{x}^4(C+{\epsilon }^3(x^2-y^2))}}{{\epsilon }^3{x}^4}+O({\epsilon }^7).$$

$$\begin{split}\dot{x}&=\mbox{$\frac{1}{\sqrt{2}}$}\varepsilon^{3}x^{3}y,\\ \dot{y}&=\mbox{$\frac{1}{\sqrt{2}}$}\varepsilon^{3}x^{4}+\varepsilon^{3}x^{6}f(x,y,\varepsilon,s,C),\\ \dot{s}&=1+C\varepsilon^{3}x^{4}+\varepsilon^{6}x^{6}-\varepsilon^{6}x^{4}y^{2}+\varepsilon^{6}x^{8}g(x,y,\varepsilon,s,C).\end{split}$$

$$x(t)=0,y(t)=0,s(t)=s_0+t\text{mbox}mod2\pi .$$

$$\mathcal{P}:\left\{\begin{array}[]{ll}x\to&x+\sqrt{2}\>\pi\varepsilon^{3}x^{3}y+\varepsilon^{6}x^{7}y\,r_{1}(x,y,\varepsilon)\\[-4.30554pt] &\\ y\to&y+\sqrt{2}\>\pi\varepsilon^{3}x^{4}(1-C^{2}x^{2})+\varepsilon^{6}x^{6}r_{2}(x,y,\varepsilon)\end{array}\right.$$

$$\tilde{\mathcal{P}}:\left\{\begin{array}[]{ll}u\to&u+\varepsilon^{3}p_{1}(u,v)+\varepsilon^{3}s_{1}(u,v,\varepsilon)\\[-4.30554pt] &\\ v\to&v+\varepsilon^{3}p_{2}(u,v)+\varepsilon^{3}s_{2}(u,v,\varepsilon)\end{array}\right.$$

$$\displaystyle W^{s}_{\varepsilon}(0,0)=\big{\{}(x,y)\in B_{1}(\delta,\beta):\;\forall n\geq 0,\,\,\mathcal{P}^{n}(x,y)\in B_{1}(\delta,\beta),$$

$$\displaystyle\hskip 56.9055pt\;\lim_{n\to\infty}\mathcal{P}^{n}(x,y)=(0,0)\big{\}},$$

$$\displaystyle W^{u}_{\varepsilon}(0,0)=\big{\{}(x,y)\in B_{2}(\delta,\beta):\;\forall n\leq 0,\,\,\mathcal{P}^{n}(x,y)\in B_{2}(\delta,\beta),$$

$$\displaystyle\hskip 56.9055pt\;\lim_{n\to-\infty}\mathcal{P}^{n}(x,y)=(0,0)\big{\}},$$

$$W_{\epsilon }^s(\gamma )=\left\{{\phi }^t\left(x_{\epsilon },y_{\epsilon }\right):t\ge 0,\left(x_{\epsilon },y_{\epsilon }\right)\in W_{\epsilon }^s(0,0)\right\},$$

$$W_{\epsilon }^u(\gamma )=\left\{{\phi }^t\left(x_{\epsilon },y_{\epsilon }\right):t\le 0,\left(x_{\epsilon },y_{\epsilon }\right)\in W_{\epsilon }^u(0,0)\right\},$$

$$\begin{array}[]{ll}\dot{x}=\displaystyle\mbox{$\frac{1}{\sqrt{2}}$}\varepsilon^{3}x^{3}y,&\qquad\dot{y}=\displaystyle\varepsilon^{3}\left(\mbox{$\frac{1}{\sqrt{2}}$}x^{4}-\mbox{$\frac{1}{\sqrt{2}}$}\Theta^{2}x^{6}\right),\\ &\\ \dot{s}=1-\varepsilon^{3}x^{4}\Theta,&\qquad\dot{\Theta}=0.\end{array}$$

$$\frac{dx}{d\tau }=x^{\prime }=y,\frac{dy}{d\tau }=y^{\prime }=x-{\Theta }_0^2{x}^3.$$

$$x(\tau )=\frac{\sqrt{2}}{|{\Theta }_0|}\mathrm{sech}\tau ,y(\tau )=-\frac{\sqrt{2}}{|{\Theta }_0|}\tanh \tau \mathrm{sech}\tau ,$$

$$H_{D}=\mbox{$\frac{1}{2}$}y^{2}-\mbox{$\frac{1}{2}$}x^{2}+\mbox{$\frac{1}{4}$}x^{4}\Theta_{0}^{2}.$$

$$\begin{array}[]{rcl}\varphi_{s}(\tau,x_{\varepsilon}^{s},y_{\varepsilon}^{s})=\xi(\tau-\tau_{0})+\varepsilon^{4}\varphi_{s}^{1}(\tau,\tau_{0},\varepsilon)+...,\text{ for }\tau\geq\tau_{0},\\[6.45831pt] \varphi_{u}(\tau,x_{\varepsilon}^{u},y_{\varepsilon}^{u})=\xi(\tau-\tau_{0})+\varepsilon^{4}\varphi_{u}^{1}(\tau,\tau_{0},\varepsilon)+...,\text{ for }\tau\leq\tau_{0},\end{array}$$

$$H_D({\phi }_s({\tau }_0,x_{\epsilon }^s,y_{\epsilon }^s))-H_D({\phi }_u({\tau }_0,x_{\epsilon }^u,y_{\epsilon }^u))={\epsilon }^{\nu }{\mathcal{M}}_{\nu }({\tau }_0,\epsilon )+...,$$

$${\mathcal{M}}_{\nu }({\tau }_0,\epsilon )=$$

Full text

1. Introduction

In restricted $N$-body problems there exists a class of unbounded orbits called oscillatory, where the motion of the $N-1$ primaries remain bounded while the motion of the infinitesimal mass is unbounded, but nevertheless it leads back near the primaries an infinite number of times. This type of motion was hypothesized by Chazy [4] for the 3-body problem, Sitnikov [22] was the first to construct an oscillatory motion in a restricted 3-body problem, he also described an initial condition set for such solutions. Later on, using the theory of quasi-random dynamical systems, Alekseev [1] studied the Sitnikov problem showing the existence of oscillatory motion for small but positive infinitesimal mass. By the same time Melnikov [15] and Arnold [3] developed a method to study the formation of transversal intersections of stable and unstable manifolds. This technique, now known as the Melnikov method, replaces variational equations by the computation of certain integrals.

An orbit of an infinitesimal particle in a restricted $N$-body problem is parabolic if the infinitesimal particle escapes to infinity with zero limit radial velocity. In [14] McGehee introduced a suitable set of coordinates that brings the infinity into the origin. He also proved that the set of parabolic orbits is formed by two smooth manifolds, that can be regarded as the stable and unstable manifolds of an unstable periodic orbit, or a hyperbolic fixed point in a suitable Poincaré map at infinity. This result was used by Moser [19] to clarify the proof of the existence of oscillatory motion in the Sitnikov problem. The key mechanism is to show the existence of transversal homoclinic intersections of the parabolic manifolds, which leads to a Smale’s horseshoe map that guarantees the existence of symbolic dynamics, giving rise to the existence of oscillatory orbits as a consequence.

Llibre and Simó [11] followed Moser’s approach to prove the existence of oscillatory solutions in the planar circular restricted 3-body problem (RPC3BP). They achieved it by demonstrating the transversal intersection of the stable and unstable parabolic manifolds for a large Jacobi constant $C$ and a sufficiently small mass ratio $\mu$ between the primary bodies. Xia [23] treated the RPC3BP by the Melnikov method, where $\mu$ is used again as a perturbation parameter. He proved the transversality of the homoclinic manifolds for sufficiently small $\mu$, and $C$ close to $\pm\sqrt{2}$. After that, he used analytic continuation to extend the transversality to almost any value of $\mu$ with $C$ large enough.

Guardia et al. [7] demonstrated that oscillatory motions do occur in the RPC3BP for any $\mu\in(0,1/2]$ and a sufficiently large Jacobi constant. Their result was achieved through an asymptotic formula of the distance between the stable and unstable manifolds of infinity in a level set of the Jacobi constant, and making $C$ sufficiently large, they proved that these manifolds intersect transversally. More recently, Guardia et al. [6] followed the above method to prove the existence of oscillatory solutions in the planar elliptic restricted 3-body problem, for any mass ratio $\mu\in(0,1/2]$ and small eccentricities of the Keplerian ellipses (the primaries perform nearly circular orbits). This is done by constructing an infinite transition chain of fixed points of the Poincaré map and then providing a lambda lemma which gives the existence of an orbit which shadows the chain.

As far as the authors are aware, there are few studies in the literature related with oscillatory motions in restricted $N$-body problems for $N$ greater than or equal to four, see for instance [5].

Our goal is to analyze the planar restricted $N$-body problem where the $N-1$ primaries form a given central configuration and the infinitesimal particle goes to infinity in a parabolic orbit. More specifically, we show the existence of the stable and unstable manifolds of the parabolic orbits at infinity by means of a Poincaré map and establish the transversal intersections between them. This is done by the application of McGehee’s Theorem for the existence of manifolds associated to degenerate fixed points [14]. The transversal intersections allow us to prove the existence of a Smale’s horseshoe [8, 20] and the subsequent occurrence of chaotic motions. Besides the appearance of oscillatory motions is ensured. These motions correspond to orbits such that the infinitesimal particle leaves every bounded region but it returns infinitely often to some fixed bounded region [19].

The approach carried out here does not require any restriction in the mass parameters of the primaries. Instead of using a small parameter related to the masses we make a symplectic scaling of the variables so that the infinitesimal particle is placed far away from the center of mass of the primaries. This is the so called cometary problem [16, 17]. The scaling is performed through the introduction of a small parameter that measures the distance from the infinitesimal particle to the origin. Thence, the resulting system is given by the Kepler Hamiltonian multiplied by a power of the small parameter plus higher-order terms that are easily obtained through Legendre polynomials. The perturbation depends on the specific configuration of the primaries.

The transversal intersection between the manifolds is proved by applying the Melnikov’s method [9, 10, 8, 20] related to the first two non-null perturbative term. Indeed we compute a general Melnikov function that works for all the configurations, determining when it has simple zeroes. There are some cases where the calculation of higher-order terms of the Melnikov’s function are required. This concerns to two parameters which are given in terms of the masses and positions of the primaries.

In the context of oscillatory motions we generalize the analysis performed for the RPC3BP by considering any central configuration of the $N-1$ primaries. A key point in our analysis is that, instead of using a small parameter related to the masses we apply a symplectic scaling. This allows us to treat all the problems together so that we can check whether the Melnikov function for a specific configuration has simple zeros after replacing the coefficients of the configuration. In this manner we have been able to deal with a large variety of restricted problems in a systematic and straightforward way.

This paper has been structured as follows. In Section 2 we formulate the Hamiltonian of the restricted $N$-body problem where the primaries are in central configuration in a rotating and an inertial frame. We also define the cometary case, introducing an appropriate small parameter so that the problem is expressed as a 2-body Kepler Hamiltonian multiplied by a power of the small parameter plus a small perturbation. In Section 3, we introduce McGehee coordinates to study the behavior of the system in a vicinity of infinity as a Poincaré map near a homoclinic orbit of a degenerate periodic orbit with analytic stable and unstable manifolds. Section 4 is devoted to the unperturbed problem that can be expressed as a Duffing oscillator. The main results will be given in Section 5, where we establish the existence of transversal homoclinic intersections of stable and unstable manifolds for the perturbed Hamiltonian system. We analyze the Melnikov function up to perturbation orders 4 and 6, depending on the non vanishing order of the Melnikov function. This is the fundamental part in our study, since it gives a systematic way of concluding the transversality between the corresponding invariant manifolds which only depends on the mass parameters and the configuration of the primaries. Then, we use that the Melnikov function has simple zeroes to show the transverse homoclinic intersections in the perturbed problem. In general, the corresponding integrals are hard to calculate, however, we make use of the asymptotic estimates provided in [11, 12, 13]. In Section 6 we illustrate the theory developed in the previous sections in some restricted $N$-body problems. In the RPC3BP, we show that in general it is enough to calculate the terms of order 4 and 6 in $\varepsilon$. In addition to the above, other examples are considered, such as the restricted 4-body problem with primaries in equilateral (Lagrange) configuration, the 5-body problem with primaries in rhomboidal configuration, the collinear restricted $N$-body problem and some polygonal restricted $N$-body problems. Finally, the study of the Melnikov functions is addressed in the Appendix.

All the numeric and symbolic calculations have been performed with Mathematica. We have made the computations within 50 significant digits although we only show the first eight.

2. Problem statement

The planar restricted $N$-body problem is the study of the motion of an infinitesimal mass particle subject to the Newtonian gravitational attraction of $N-1$ bodies called the primaries. It is assumed that the masses of the primaries are so big in relation to the mass of the infinitesimal particle that the latter does not exert any significant influence in the primaries, hence the motion of the primaries becomes an ($N-1$)-body problem. Along this paper the primaries move in a central configuration rotating around their center of mass with a constant angular velocity. Without loss of generality we also set the total mass of the primaries to one and their angular velocity $\omega=1$.

The Hamiltonian that governs the motion of the infinitesimal particle in a rotating frame is

[TABLE]

where $Q,P\in\mathbb{R}^{2}$ are the position and momentum, $J=\left(\begin{smallmatrix}0&1\\ -1&0\end{smallmatrix}\right)$ and the potential is given by

[TABLE]

(We will at times identify $\mathbb{R}^{2}$ and $\mathbb{C}$, i.e., $(x,y)\in\mathbb{R}^{2}$ with $x+y\,i\in\mathbb{C}$.)The terms $a_{k}=a_{k1}+a_{k2}\,i$ and $m_{k}$ correspond to the position and mass respectively, of the $k$-th primary, $k=1,\dots,N-1$. Since they are in central configuration the following equations are satisfied:

[TABLE]

with $m_{1}+\cdots+m_{N-1}=1$. We observe that the Hamiltonian $H$ represents an autonomous system with two degrees of freedom. This quantity is preserved through the changes of coordinates in spite of the fact that some characteristics, like the time independence of the flow or the Hamiltonian structure, are lost. More details related to the restricted $N$-body problem can be seen in [16, 17].

In order to work in an inertial frame we define the change of coordinates $(Q,P)=e^{-tJ}(q,p)$. The Hamiltonian accounting for the motion of the infinitesimal particle in inertial coordinates yields

[TABLE]

where the potential is given by

[TABLE]

Now the Hamiltonian is time dependent.

We are interested in the study of the motion of the infinitesimal particle near parabolic orbits, that is when the particle escapes to infinity with zero limit velocity. Thus, it is convenient to scale the Hamiltonian by introducing a small positive parameter $\varepsilon$ through the change $q\to\varepsilon^{-2}q$, $p\to\varepsilon p$. This is a symplectic transformation with multiplier $\varepsilon$. Hence $H\to\varepsilon H$. This problem now is the cometary regime of the restricted $N$-body problem, see [16, 17, 18].

Thus, Hamiltonian (4) becomes

[TABLE]

The expansion of the potential in (6) in terms of Legendre polynomials yields

[TABLE]

where $P_{j}$ is the $j$-th term of the Legendre polynomial, and $\gamma_{k}$ is the angle between the $k$-th primary’s position and $e^{-tJ}q$. Let us note that the zero term of the sum is $-1/|q|$ and the next term is zero because we have placed the center of mass at the origin. Hence the problem is a Kepler problem (multiplied by $\varepsilon^{3}$) plus a perturbation term of order $\varepsilon^{7}$. The series is convergent in the region of the configuration space where $\varepsilon^{2}|a_{k}|<|q|$.

Now we make the symplectic polar change of variables

[TABLE]

where $r$ is the distance of the infinitesimal particle to the origin, $\theta$ is the argument of latitude, $R$ stands for the radial velocity and $\Theta$ for the angular momentum. For our analysis we restrict $\Theta$ to be non-zero and bounded, which is equivalent to consider the angular momentum before the scaling being a big quantity as long as $\varepsilon$ is small.

Let $\alpha_{k}$ be the angle between the position of $k$-th primary and the horizontal axis of the inertial frame, thus $\gamma_{k}=\alpha_{k}-(\theta-t)$. Then, Hamiltonian (7) can be written as

[TABLE]

We observe that the argument of the Legendre polynomial $P_{j}$ depends on $a_{k}$ through the relations $\cos(\alpha_{k})=a_{k1}/|a_{k}|$ and $\sin(\alpha_{k})=a_{k2}/|a_{k}|$.

The associated Hamiltonian system is given by

[TABLE]

where $Q_{j}(w)=dP_{j}(w)/dw$.

3. Application of McGehee’s Theorem

In order to study the motion of the infinitesimal mass near infinity, we make the change of coordinates introduced by McGehee [14]:

[TABLE]

where $s\in\mathbb{S}^{1}$. In these new coordinates the equations (2) become

[TABLE]

Similarly to the circular restricted 3-body problem, the above system can be smoothly extended to $x=0$; in addition, it has a Jacobi-like first integral given by equating Hamiltonian $H$ in (1) to a constant. Let $C$ be an arbitrary value of it. Thus

[TABLE]

where $H_{\varepsilon}(x,y,\theta,\Theta,t)$ is given in (8) after replacing $r$ and $R$ for their values in terms of $x$ and $y$. In order to verify the hypotheses of McGehee’s Theorem [14], we observe that $\Theta$ can be written in terms of $x$, $y$, $s$ and the value of the Jacobi-like integral $C$ as follows:

[TABLE]

We take the negative sign in (12) because we are interested in small values of $x$. Therefore, the variable $\Theta$ depends smoothly on $x\geq 0$, $y\in\mathbb{R}$, $s\in\mathbb{S}^{1}$. In fact $\Theta=-C$ when $x=y=0$, hence the equation referring to $\dot{\Theta}$ can be dropped in system (10). Expanding (12) in power series around $x=0$, replacing its value in (10), the resulting system becomes:

[TABLE]

Observe that the functions $f(x,y,\varepsilon,s,C)$ and $g(x,y,\varepsilon,s,C)$ are real analytic.

Now, we are going to exploit the fact that the solution $\gamma(t)$ of system (13) with $x=0$ and $y=0$ is given by

[TABLE]

This solution does not depend on the Jacobi-like first integral $C$ and is a $2\pi$-periodic solution in $t$. Let $\Sigma=\{\left(x,y,s\right):s=s_{0}\}$ be a transversal section to this periodic orbit. This set is parametrized by the coordinates $x$ and $y$. For the point $\kappa_{0}=\left(0,0,s_{0}\right)\in\gamma\cap\Sigma$, the Implicit Function Theorem shows that there exists an open set $V\subset\Sigma$ containing $\kappa_{0}$ and a smooth function $\sigma:V\to\mathbb{R}$, the return time, such that the trajectories starting in $V$ come back to $\Sigma$ in a time $\sigma$ close to $2\pi$.

Let $\varphi^{t}(x_{0},y_{0})=(x(t,x_{0},y_{0}),y(t,x_{0},y_{0}),s(t,x_{0},y_{0}))$ be the flow solution of (13) with initial condition $(x(0),y(0),s(0))=(x_{0},y_{0},s_{0}\,\mbox{mod}\,2\pi)$. This solution depends on $\varepsilon$, but it is usually omitted.

The Poincaré map $\mathcal{P}:V\subset\Sigma\to\Sigma$ of the periodic orbit $\gamma$ (or of the fixed point $(0,0,s_{0})$) is given by $\mathcal{P}(x,y)=\varphi^{\sigma(x,y)}(x,y)$ where

[TABLE]

and $r_{1}$ and $r_{2}$ are real analytic functions.

A straightforward computation shows that if we make the transformation $x=u+v$, $y=v-u$, the map $\mathcal{P}$ takes the form

[TABLE]

where $p_{1}(u,v)=-\sqrt{2}\pi u(u+v)^{3}$, $p_{2}(u,v)=\sqrt{2}\pi v(u+v)^{3}$ and $s_{1}$ and $s_{2}$ are real analytic functions starting with homogeneous polynomials of degree 6 in $u$ and $v$. For $u>0$ we have $p_{1}(u,0)=-\sqrt{2}\pi u^{4}<0$, $p_{2}(u,0)=0$ and $\frac{\partial p_{2}}{\partial v}(u,0)=\sqrt{2}\pi u^{3}>0$. If $\varepsilon\geq\varepsilon_{0}>0$ is small enough, these are the hypotheses of McGehee’s Theorem [14], which we proceed to state in our context now.

For $\delta=\delta(\varepsilon)>0$ and $\beta=\beta(\varepsilon)>0$, the sector centred on the line $y=-x$ is defined by $B_{1}(\delta,\beta)=\{(x,y):0\leq x\leq\delta,(-1-\beta)x\leq y\leq(-1+\beta)x\}$. Similarly, the sector centred on the line $y=x$ is $B_{2}(\delta,\beta)=\{(x,y):0\leq x\leq\delta,(1-\beta)x\leq y\leq(1+\beta)x\}$. The sets

[TABLE]

are called the stable and unstable manifolds of the fixed point $(0,0)$.

Theorem 3.1** (McGehee, [14]).**

For the map $\mathcal{P}:V\subset\Sigma\to\Sigma$ given by (15), there is $\delta=\delta(\varepsilon)>0$ and $\beta=\beta(\varepsilon)>0$ such that the manifolds $W^{s}_{\varepsilon}(0,0)\subset B_{1}(\delta,\beta)$ and $W^{u}_{\varepsilon}(0,0)\subset B_{2}(\delta,\beta)$ correspond to the graphs of two functions $\psi_{s}$, $\psi_{u}:[0,\delta]\to\mathbb{R}$ that are smooth, real analytic in $(0,\delta]$, and $\psi_{s}(0)=\psi_{u}(0)=0$, $\psi_{s}^{\prime}(0)=-1$, $\psi_{u}^{\prime}(0)=1$. In addition to that, they vary uniform and smoothly for $\varepsilon\geq\varepsilon_{0}>0$ small enough.

The sets $W^{s}_{\varepsilon}(0,0)$ and $W^{u}_{\varepsilon}(0,0)$ are curves, that is, one-dimensional manifolds. From here it follows that

[TABLE]

are smooth manifolds of dimension two. They are formed by the orbits that escape to infinity ($x=0$) with zero velocity ($y=0$). These are called the parabolic manifolds.

4. The main term of the Hamiltonian

The main part in Hamiltonian (8) corresponds to the Kepler problem, and the related equations (10) take the form

[TABLE]

At this point it is convenient to introduce a new time through $d\tau/dt=\varepsilon^{3}x^{3}/\sqrt{2}$. Fixing $\Theta=\Theta_{0}$ non zero, the equations for $x$ and $y$ become

[TABLE]

Thus (17) is a $\Theta_{0}$-parametrized Duffing equation. The origin $(0,0)$ is a hyperbolic saddle point and its stable and unstable manifolds form a homoclinic orbit $\xi(\tau)$ given by

[TABLE]

connecting the fixed point with itself as shown in Fig. 1.

The Keplerian Hamiltonian associated to (17) is rewritten as the Duffing Hamiltonian:

[TABLE]

5. Perturbed Problem

In this section we compare equations (13) and (16). We observe that the Poincaré map of the last one has $(0,0)$ as a fixed point with a homoclinic orbit parametrized by (18). The fixed point is preserved by the Poincaré map of (13), however the homoclinic orbit is broken into the curves $W^{s}_{\varepsilon}(0,0)$ and $W^{u}_{\varepsilon}(0,0)$. Indeed, the smooth dependence on $\varepsilon$ implies that $W^{s}_{\varepsilon}(\gamma)$ and $W^{u}_{\varepsilon}(\gamma)$ are parametrized by orbits of the form

[TABLE]

with $\tau_{0}=\tau(s_{0})$, the functions $\varphi_{s}^{1}$ and $\varphi_{u}^{1}$ are determined by the first variational equation of the $\varepsilon^{4}$-perturbation of (17) along the orbit $\xi(\tau)$, that is, the corrections to $\xi(\tau-\tau_{0})$ corresponding to the perturbation of the Duffing Hamiltonian of order $\varepsilon^{4}$, the next terms, i.e. $\varphi_{s}^{2}$, $\varphi_{u}^{2}$, correspond to the perturbation of order $\varepsilon^{7}$, etc. Besides, $\varphi_{s}(\tau,x_{\varepsilon}^{s},y_{\varepsilon}^{s})$, $\varphi_{u}(\tau,x_{\varepsilon}^{u},y_{\varepsilon}^{u})$ are taken so that $\varphi_{s}(\tau_{0},x_{\varepsilon}^{s},y_{\varepsilon}^{s}),\varphi_{u}(\tau_{0},x_{\varepsilon}^{u},y_{\varepsilon}^{u})\in\Sigma$, guaranteeing the existence of a parametrization of the manifolds as above, see [23].

Our purpose now is to determine the speed of breaking up of $W^{s}_{\varepsilon}(0,0)$ and $W^{u}_{\varepsilon}(0,0)$ (equivalently the breaking up of $W^{s}_{\varepsilon}(\gamma)$ and $W^{u}_{\varepsilon}(\gamma)$) under the perturbation. We point out that the normal direction to the level sets of $H_{D}$ is the only one to be taken into account.

To measure the rate of separation between $W^{s}_{\varepsilon}(\gamma)$ and $W^{u}_{\varepsilon}(\gamma)$ with respect to $\varepsilon$, we take

[TABLE]

where

[TABLE]

is the first term of the Melnikov function, $x$, $y$ are evaluated in the unperturbed homoclinic and the derivative of $H_{D}$ with respect to $\tau$ is considered up to the perturbation of order $\varepsilon^{\nu}$ in (17) such that the function ${\mathcal{M}}_{\nu}$ is not identically zero. The ellipsis in the formula measuring the splitting of the manifolds should not be underestimated. More specifically we decompose

[TABLE]

where

[TABLE]

is the so called Melnikov function and ${\mathcal{M}}_{2l}$ contain the main terms of the total variation of $H_{D}$; their computation is similar to that of ${\mathcal{M}}_{\nu}$ above. In other words, the series contains the whole Hamiltonian $H_{\varepsilon}$ with $x$, $y$ replaced by their values in (18) while ${\mathcal{R}}$ corresponds to the contribution to the separation provided by the higher order terms of the orbits on the stable and unstable manifolds given in (20). After changing from $\tau_{0}$ to $s_{0}$ the infinite series can be interpreted as a Fourier series in $s_{0}$ whose coefficients are given as asymptotic expressions in terms of $\varepsilon$. In the Appendix we will provide the leading terms of the series as well as the corresponding estimates for the higher orders. Regarding $\mathcal{R}$ we shall prove that it is smaller than the dominant terms of ${\mathcal{M}}$ at least in regions of the phase space containing portions of the stable and unstable manifolds of $\gamma$ big enough where transversality can be checked, as it will be detailed in the Appendix.

The consequence of the existence of transversal homoclinic manifolds is the appearance of a Smale’s horseshoe, and thence the occurrence of chaotic motion of the infinitesimal particle. This includes the existence of oscillatory motion, see [19, 8].

In order to apply the results in the previous paragraphs we need to make some arrangements to Hamiltonian (8). Up to terms of order $\varepsilon^{9}$, it reads as

[TABLE]

where $c_{1}$, $c_{2}$ and $c_{3}$ are the following constant terms, that depend only on $a_{k}$ and $m_{k}$:

[TABLE]

and the $d_{j}$ are given by

[TABLE]

Higher order terms appearing in $O(\varepsilon^{11})$ should also be taken into consideration as we shall realize in the proof of the main result of this section.

The next step consists in applying McGehee’s transformation to the equations of motion associated to (5). After replacing $t-\theta$ by $s$, one gets

[TABLE]

which corresponds to Hamiltonian equations (10).

With respect to the new time $\tau$, Eq. (25) gets transformed into

[TABLE]

To obtain the Melnikov function we use the corresponding integral of the main term given by $H_{D}$ in (19), and compute the total derivative, assuming that $\Theta_{0}$ in (19) is considered as $\Theta$, arriving at

[TABLE]

where $x^{\prime}$, $y^{\prime}$, $\Theta^{\prime}$ are given in (26). In this expression $x$ and $y$ are replaced by their explicit values obtained for the unperturbed problem and that were given in (18).

The next step consists in solving the differential equation for $s^{\prime}$ in (26) using the fact that $\Theta$ is assumed to be constant, i.e. $\Theta=\Theta_{0}$, and $x(\tau)$ is taken from the solution (18). The corresponding equation becomes

[TABLE]

We set $s(0)=s_{0}$ and the solution yields

[TABLE]

The upper signs in $s^{\prime}$, $s$ and the formulae that follow are used when $\Theta_{0}>0$ and the lower ones when $\Theta_{0}<0$.

The leading term of the resulting expression of the total derivative of $H_{D}$ with respect to $\tau$ becomes

[TABLE]

where

[TABLE]

and

[TABLE]

Proceeding as in the previous paragraphs, the total derivative of $H_{D}$ with respect to $\tau$ corresponding to the terms of order six in $\varepsilon$ is

[TABLE]

where

[TABLE]

and

[TABLE]

We observe that the total derivative of the Duffing Hamiltonian has become

[TABLE]

At this point we need to compute the integrals of $M_{4}$ and $M_{6}$ for $\tau$ between $-\infty$ and $\infty$. After performing the change of variable $z=\sinh\tau$ (note that $z^{\prime}=\cosh\tau>0$, hence the change is well-defined), simplifying the resulting expressions and dropping the odd terms with respect to $z$ as their respective integrals are zero, we arrive at

[TABLE]

where $\tilde{M}_{4}(z)=M_{4}(\tau(z))/\sqrt{z^{2}+1}$ and

[TABLE]

We have passed from $\tau_{0}$ to $s_{0}$ as the resulting expressions are much easier expressed in terms of them. From now the Melnikov function $\mathcal{M}$ will be written in terms of $s_{0}$.

In order to obtain the integral of $M_{6}$ with respect to $\tau$ we apply as before the change $z=\sinh\tau$, discard those terms with zero integral and simplify, ending up with

[TABLE]

with $\tilde{M}_{6}(z)=M_{6}(\tau(z))/\sqrt{z^{2}+1}$ and

[TABLE]

Higher order terms, say $M_{8}$, $M_{10}$, etc. are obtained in a similar way as $M_{4}$ and $M_{6}$. An important feature is that $\varepsilon^{4}M_{4}$ is the dominant term, the second one is $\varepsilon^{6}M_{6}$ and so on. However, the ordering of the functions ${\mathcal{M}}_{2k}$ is different, due to the contribution of $\varepsilon$ inside the arguments of the trigonometric functions. Indeed, this ordering depends now on the harmonics $\cos ks_{0}$, $\sin ks_{0}$, in such a way that the dominant terms are those corresponding to $\cos s_{0}$, $\sin s_{0}$, then those of $\cos 2s_{0}$, $\sin 2s_{0}$ and so on as we shall see in the Appendix. As a consequence, the terms in ${\mathcal{M}}_{6}$ containing $\cos s_{0}$, $\sin s_{0}$ represent the leading terms in the approximation in $\varepsilon$ of the Melnikov function of the restricted $N$-body problem (6) related to parabolic motions of the infinitesimal particle near infinity.

It is straightforward to check that the relevant factor of the terms related to $\cos s_{0}$, $\sin s_{0}$ always arise through expressions of the form $\varepsilon^{2(2l+1)}{\mathcal{M}}_{2(2l+1)}$, $l\geq 2$ and it is

[TABLE]

We state the following result.

Theorem 5.1**.**

There exists $\varepsilon_{0}>0$ such that for any $\varepsilon$ with $\varepsilon_{0}\leq\varepsilon\ll 1$ the stable and unstable manifolds of the periodic orbit $\gamma$ related to Hamiltonian (6) intersect transversally if one of the following situations is given: i) $d_{1}$ or $d_{2}$ do not vanish; ii) $d_{1}=d_{2}=0$ and there exists $l\geq 2$ such that $d_{1}^{(l)}$ or $d_{2}^{(l)}$ do not vanish; iii) all the previous terms are zero and $c_{2}$ or $c_{3}$ do not vanish; iv) all the preceding terms are zero and there is a non-null constant term accompanying $\cos ks_{0}$ or $\sin ks_{0}$ for some $k\geq 2$.

Proof.

In the Appendix, using an argument of Sanders [21] we prove that ${\mathcal{R}}$ can be maintained small enough in regions of the phase space that include parts of the stable and unstable manifolds of $\gamma$ such that transversality is satisfied. More precisely, we can control the size of ${\mathcal{R}}$ for all $|\tau|\geq K>0$ and $K$ a constant independent of $\varepsilon$, keeping it smaller than the dominant terms of $\mathcal{M}$. Then we prove that the leading terms of the Melnikov function are those factorizing $\cos s_{0}$, $\sin s_{0}$, then those factorizing $\cos 2s_{0}$, $\sin 2s_{0}$ and so on. Therefore, the most influential term in the Melnikov function is the one appearing in $\varepsilon^{6}{\mathcal{M}}_{6}$, that is $(d_{2}\cos s_{0}-d_{1}\sin s_{0}){\mathcal{F}}_{6,1}(\Theta_{0},\varepsilon)$. As ${\mathcal{F}}_{6,1}$ does not vanish for $\Theta_{0}\neq 0$ and $\varepsilon\geq\varepsilon_{0}$ small enough, we focus on the analysis of the possible zeros of the factor $f(s_{0})=d_{2}\cos s_{0}-d_{1}\sin s_{0}$. Multiple roots of $f(s_{0})=0$ occur only when $d_{1}=d_{2}=0$. In this case we consider the next most important term of the Melnikov function, that is, $\varepsilon^{10}{\mathcal{M}}_{10}$. Its corresponding constant terms are $d_{1}^{(2)}$, $d_{2}^{(2)}$ and they always appear in the form $d_{2}^{(2)}\cos s_{0}-d_{1}^{(2)}\sin s_{0}$, and similarly for $l>2$. Thus, it is enough that there is a non-null coefficient $d_{j}^{(l)}$, $j=1,2$, $l\geq 2$, to establish the transversality condition. When all terms related to $\cos s_{0}$, $\sin s_{0}$ vanish, we take into account the next main terms, that are those of $\varepsilon^{4}{\mathcal{M}}_{4}$. They appear in the form $c_{3}\cos 2s_{0}-c_{2}\sin 2s_{0}$, thus it is enough that $c_{2}$ or $c_{3}$ do not vanish to get transversality. When $c_{2}=c_{3}=0$ we need to consider the next terms related to $\cos 2s_{0}$, $\sin 2s_{0}$, and these terms appear in $\varepsilon^{8}{\mathcal{M}}_{8}$. We continue the procedure until we identify a non-null constant term accompanying $\cos ks_{0}$ or $\sin ks_{0}$, concluding the transversality of the manifolds in that case. ∎

In Section 6, more specifically when we will deal with the circular restricted 3-body problem and with the polygonal restricted $N$-body problem, we shall see how higher-order terms of the Melnikov function are needed to establish the transversality of the stable and unstable manifolds of the parabolic periodic orbits.

6. Applications

6.1. Restricted circular 3-body problem

We study the planar circular restricted 3-body problem. In this example the position and mass parameters can be chosen as

[TABLE]

where $0<\mu\leq\frac{1}{2}$.

With these values, the perturbation parameters appearing in (24) read as:

[TABLE]

Then, according to Theorem 5.1, when $\mu<1/2$ for $\varepsilon>0$ small enough the stable and unstable manifolds of the periodic orbit $\gamma$ related to Hamiltonian (4) intersect transversally.

For $\mu=1/2$ we have to consider higher-order terms, starting with the harmonics $\cos s_{0}$, $\sin s_{0}$. In this case we note that the parameters $d_{1}^{(l)}$, $d_{2}^{(l)}$ vanish for all $l\geq 2$, then we calculate $c_{2}$ and $c_{3}$, arriving at

[TABLE]

Thus, Theorem 5.1 applies and the stable and unstable manifolds of the periodic orbit $\gamma$ intersect transversally.

With the approach described above we have completed the analysis of the parabolic orbits for the circular restricted 3-body problem for any $\mu\in(0,1/2]$. This problem is also treated in [7].

6.2. Equilateral restricted 4-body problem

In this example the three massive particles form an equilateral triangle, therefore a central configuration, thus (3) is satisfied. The parameter values are:

[TABLE]

where $m_{1},m_{2}>0$ and $m_{1}+m_{2}<1$.

Then, the coefficients $d_{i}$ of (24) take the following values:

[TABLE]

Therefore the transversal intersection of the manifolds is established as Theorem 5.1 applies, except for the case $m_{1}=m_{2}=1/3$, which is the only case for which $d_{1}=d_{2}=0$ in the allowed region of the mass parameters.

When $m_{1}=m_{2}=1/3$ we check that the coefficients $d_{1}^{(l)}$, $d_{2}^{(l)}$ vanish for all $l\geq 2$. Moreover, $c_{2}=c_{3}=0$ and realize that all terms accompanying $\cos 2s_{0}$, $\sin 2s_{0}$ vanish as well. Thus, we consider the leading terms of $\cos 3s_{0}$, $\sin 3s_{0}$ and get

[TABLE]

Thence Theorem 5.1 is applied, accomplishing the intersection of the manifolds of the parabolic orbits.

The case $m_{1}=m_{2}=1/3$ is also included as a particular situation of the polygonal restricted $N$-body problem that will be addressed in Subsection 6.5.

6.3. Restricted rhomboidal 5-body problem

We consider the case where masses $m_{1}$ to $m_{4}$ are equal by pairs and form a convex polygon, a rhombus, see [2] and references therein. The parameters that define the problem are as follows:

[TABLE]

where $0<\mu<1/2$ and $x,y>0$. To get a central configuration the parameters $\mu$, $x$, $y$ must be related. For this purpose we impose that Eqs. (3) are satisfied. It is convenient to introduce two parameters $a,b>0$ such that

[TABLE]

then a central configuration occurs provided $\mu$ is taken as

[TABLE]

To ensure that $\mu\in(0,1/2)$ we must restrict $a$, $b$ so that $0<b<\sqrt{3}a<3b$.

In terms of $x$, $y$, the perturbation parameters in (24) are identically zero. Moreover the coefficients $d_{1}^{(l)}$, $d_{2}^{(l)}$ also vanish for all $l\geq 2$, thus we need to obtain $c_{2}$, $c_{3}$. We get

[TABLE]

When $c_{2}$ is non-zero Theorem 5.1 applies and we get the transversality of the manifolds. For $c_{2}=0$ we should go to higher orders in $\varepsilon$. The coefficient $c_{2}$ vanishes in three cases: (i) If $a=b$, then $\mu=1/4$, which corresponds to the square configuration, i.e. the polygonal restricted 4-body problem, that will be treated in the next subsection; (ii) when $a=1.32018439...b$; and (iii) the reverse case to (ii), i.e. $a=0.75746994...b$. The last two values come as the only (two) real roots of the 14-degree homogeneous polynomial equation given by

[TABLE]

According to Theorem 5.1, the next terms that have to be checked for the three cases pointed out above are the coefficients of $\cos 2s_{0}$, $\sin 2s_{0}$ that appear in the function $\varepsilon^{8}{\mathcal{M}}_{8}$. Specifically, we have calculated $\mp 10\varepsilon^{8}\Theta_{0}^{-10}{\mathcal{F}}_{8,1}(\Theta_{0},\varepsilon)(c_{2}^{(2)}\sin 2s_{0}-c_{3}^{(2)}\cos 2s_{0})$ where ${\mathcal{F}}_{8,1}$ is a function of order $\varepsilon^{-5/2}\exp(-2\Theta_{0}^{3}/(3\varepsilon^{3}))$ when $\Theta_{0}>0$ and of order $\varepsilon^{-1}\exp(2\Theta_{0}^{3}/(3\varepsilon^{3}))$ when $\Theta_{0}<0$. Besides $c_{2}^{(2)}$, $c_{3}^{(2)}$ are coefficients depending on the masses and positions. We get $c_{3}^{(2)}=0$ for the three cases, $c_{2}^{(2)}=0.20447308...$ for case (ii) and $c_{2}^{(2)}=-0.20447308...$ for case (iii). Thus we can conclude the transversality of manifolds. However in case (i) one has $c_{2}^{(2)}=0$. This is the case $a=b$, that corresponds to a particular situation of the polygonal restricted $N$-body problem dealt with in Subsection 6.5.

6.4. Collinear restricted $N$-body problem

In the following we present two examples where the $N-1$ massive particles are collinear in a planar central configuration.

Note that in all collinear problems $a_{k2}=0$ for $k=1,\ldots,N-1$. Then, the parameter $c_{3}$ of (23) always vanishes.

6.4.1. Collinear restricted $8$-body problem

Let us consider a collinear configuration of seven bodies with equal masses that are placed in a symmetric configuration with respect to the origin on the horizontal axis. In order to calculate the positions we impose that the conditions (3) are satisfied and so, the seven bodies form a central configuration. Then, by means of the determination of the resultant of two polynomials we find that a collinear configuration occurs if the parameters are chosen as follows:

[TABLE]

The parameters $d_{1}$, $d_{2}$ of (24) vanish. Furthermore the coefficients associated to the higher order terms of the harmonics $\cos s_{0}$, $\sin s_{0}$, i.e. $d_{1}^{(l)}$, $d_{2}^{(l)}$ are all zero for all $l\geq 2$.

Then, the perturbation parameters appearing in (23) are:

[TABLE]

Thus the conditions of Theorem 5.1 are satisfied. Thence, for $\varepsilon$ small enough with $\varepsilon\geq\varepsilon_{0}>0$, the stable and unstable manifolds of the periodic orbit $\gamma$ related to Hamiltonian (4) intersect transversally.

6.4.2. Collinear restricted $11$-body problem

Here we consider ten particles in collinear configuration placed at equidistant positions. Then, we calculate the masses so as to obtain a central configuration. For that, we impose that Eqs. (3) are satisfied and solve the resulting system of linear equations to get

[TABLE]

Note that, although here we put an approximation of the values correct to eight decimal digits, we have obtained them using integer arithmetic.

As in the previous case, the parameters $d_{1}$, $d_{2}$ of (24) become zero while the coefficients $d_{1}^{(l)}$, $d_{2}^{(l)}$ also vanish. Then we need to obtain the leading terms of the harmonics $\cos 2s_{0}$, $\sin 2s_{0}$. We calculate the coefficients (23) and get their values also exactly, an approximation of them accurate up to eight decimal places being:

[TABLE]

Then, as $c_{2}\neq 0$, Theorem 5.1 is satisfied, concluding the transversality condition of the manifolds of $\gamma$.

6.5. Polygonal restricted $N$-body problem

In this case particles 1 to $N-1$ form a regular $(N-1)$-gon determined by the following constants:

[TABLE]

for $k=1$ to $N-1$ where $N\geq 4$.

We want to write down Hamiltonian (8) for the specific cases of the polygonal restricted problem. After some manipulations and simplifications that include an induction over the integer $N$, we notice that the Hamiltonian function can be written in terms of symplectic polar coordinates in a rather compact way by

[TABLE]

where

[TABLE]

and $\Gamma$ stands for the gamma function.

An important feature of (32) is that terms of order higher than $\varepsilon^{2N+1}$ can depend on $\cos(N-1)(t-\theta)$ but they are of smaller influence than the one of order $\varepsilon^{2N+1}$.

To obtain the Melnikov function we emphasize the convenience of developing $H_{\varepsilon}$ to order $2N+1$ because the first appearance of $\theta$ occurs at this order and the previous orders would yield zero.

At this point we apply the same steps as in the previous sections, arriving at the total derivative

[TABLE]

with

[TABLE]

Now, we observe that by the parity of the derivative with respect to $\tau$, the only term with no zero integral in the derivative is that factorized by $W_{N-1}$, thus we introduce

[TABLE]

We perform the change $z=\sinh\tau$ and define the Melnikov function as

[TABLE]

with

[TABLE]

In order to illustrate how the theory of this paper applies, we particularize the calculations for two specific cases, namely $N=7,8$.

When $N=7$ we get

[TABLE]

where

[TABLE]

with

[TABLE]

For $N=8$ the corresponding Melnikov function reads as

[TABLE]

where

[TABLE]

with

[TABLE]

From the expressions of both Melnikov functions it is clearly deduced that the equations ${\mathcal{M}}_{12}=0$ and ${\mathcal{M}}_{14}=0$ have simple roots provided, respectively, ${\mathcal{F}}_{12}$ and ${\mathcal{F}}_{14}$ do not vanish. However ${\mathcal{F}}_{12}$, ${\mathcal{F}}_{14}$ are of the same type as the functions ${\mathcal{F}}_{4}$, ${\mathcal{F}}_{6,1}$, ${\mathcal{F}}_{6,2}$ of Section 5 analyzed in the Appendix. Then they have a global maximum and a global minimum, their graphs cut the horizontal axis at a few points and tend asymptotically to zero as long as $\Theta_{0}$ tends to $\pm\infty$, see also Figure 2.

The asymptotic estimates of the Appendix hold provided $\varepsilon$ is small enough so that ${\mathcal{F}}_{12}$ and ${\mathcal{F}}_{14}$ do not vanish. Thus, for the polygonal restricted $7$- and $8$-body problems, the stable and unstable manifolds of the parabolic orbits $\gamma$ intersect transversally.

Regarding the case $N=4$, corresponding to the equilateral restricted 4-body problem of Subsection 6.2 when $m_{1}=m_{2}=m_{3}=1/3$, the Melnikov function obtained from (33) is a particular case of the function ${\mathcal{M}}_{6}$ in (30), thus we can achieve the transversality from the analysis of this subsection. When $N=5$, the square configuration studied in Subsection 6.3 is analyzed using the Melnikov function (33) which is of order eight in $\varepsilon$. Proceeding similarly to what we did for $N=7,8$ we conclude that ${\mathcal{M}}_{8}$ has simple roots provided, thus extending the analysis performed in Subsection 6.3 when $a=b$.

Examining (33) carefully, it is not difficult to infer that the integrals appearing in the Melnikov function ${\mathcal{M}}_{2N-2}$ are of the same type as ${\mathcal{F}}_{4}$, ${\mathcal{F}}_{6,1}$ and so on. We also take into account that the smallest harmonic appearing in the Melnikov function is $\sin(N-1)s_{0}$, thus we end up with an expression like

[TABLE]

with $K$ a non-null constant. Applying the estimates provided in the Appendix and taking $\varepsilon$ small enough, we conclude that $\varepsilon^{2N-2}{\mathcal{M}}_{2N-2}$ behaves like $\varepsilon^{-N-1/2}\exp(-(N-1)\Theta_{0}^{3}/(3\varepsilon^{3}))(1+O(\varepsilon))$ when $\Theta_{0}>0$ and $\varepsilon^{-N+1}\exp((N-1)\Theta_{0}^{3}/(3\varepsilon^{3}))(1+O(\varepsilon))$ for negative $\Theta_{0}$. Thus, we can apply Theorem 5.1, achieving the transversality of the manifolds of $\gamma$ in the polygonal restricted $N$-body problem for all $N\geq 4$.

7. Appendix: Qualitative study of functions ${\mathcal{F}}_{4}(\Theta_{0},\varepsilon)$, ${\mathcal{F}}_{6,1}(\Theta_{0},\varepsilon)$ and ${\mathcal{F}}_{6,2}(\Theta_{0},\varepsilon)$

The function $\mathcal{F}_{4}$ has been defined in (29) and its graph is given in Figure 3. Considered as a function in $z$, the integrand is smooth in $\mathbb{R}$ and bounded by above in the intervals in $(-\infty,-1]\cup[1,\infty)$ by the improperly integrable function $80/|z|^{7}$. Calling $\tilde{\Theta}_{0}=\Theta_{0}/\varepsilon$, the comparison test for improper integrals implies that ${\mathcal{F}}_{4}(\tilde{\Theta}_{0})$ is a well defined smooth function for all $\tilde{\Theta}_{0}\in\mathbb{R}$. In addition it has two zeroes, namely $\tilde{\Theta}_{0}^{{}^{\prime}}=0$ and $\tilde{\Theta}_{0}^{\ast}=0.61078210...$, and ${\mathcal{F}}_{4}(\tilde{\Theta}_{0})<0$ for $\tilde{\Theta}_{0}<\tilde{\Theta}_{0}^{\ast}$ (excepting at $\tilde{\Theta}_{0}^{{}^{\prime}}$), while ${\mathcal{F}}_{4}(\tilde{\Theta}_{0})>0$ for $\tilde{\Theta}_{0}>\tilde{\Theta}_{0}^{\ast}$. The function $\mathcal{F}_{4}$ takes a global maximum and a global minimum values. Furthermore

[TABLE]

The functions $\mathcal{F}_{6,1}$, $\mathcal{F}_{6,2}$ were introduced in (31) and present an analogous behavior to the function $\mathcal{F}_{4}$, as it can be seen in Figure 4. The corresponding improper integrals are absolutely convergent. Specifically, $\mathcal{F}_{6,1}=0$ has its unique root at $\tilde{\Theta}_{0}=0$ whereas the roots of $\mathcal{F}_{6,2}=0$ occur at $\tilde{\Theta}_{0}=0$, $\tilde{\Theta}_{0}=0.15745028...$, $\tilde{\Theta}_{0}=0.87685728...$. Besides, $\mathcal{F}_{6,1}>0$ when $\tilde{\Theta}_{0}<0$, $\mathcal{F}_{6,1}<0$ when $\tilde{\Theta}_{0}>0$ while $\mathcal{F}_{6,2}>0$ when $\tilde{\Theta}_{0}<0$ and $0.15745028...<\tilde{\Theta}_{0}<0.87685728...$, $\mathcal{F}_{6,2}<0$ when $0<\tilde{\Theta}_{0}<0.15745028...$ and $\tilde{\Theta}_{0}>0.87685728...$. As ${\mathcal{F}}_{4}$, the functions $\mathcal{F}_{6,1}$, $\mathcal{F}_{6,2}$ take a global maximum as well as a global minimum. Finally,

[TABLE]

We remark that similar integrals have been analyzed and can be found in [12, 13]. In fact, due to the highly oscillatory character of the integrals, an asymptotic analysis of the functions $\mathcal{F}_{4}$, $\mathcal{F}_{6,1}$, $\mathcal{F}_{6,2}$ and other related functions appearing in the Melnikov functions obtained in Section 5 has to be done.

Following [13] we introduce the improper integrals

[TABLE]

In [12] it is proved that $J_{k}$ can be written in terms of $I_{k}$ through

[TABLE]

whereas for $\delta>0$ big enough, the following estimates hold:

[TABLE]

The functions $\mathcal{F}_{4}$, $\mathcal{F}_{6,1}$, $\mathcal{F}_{6,2}$ as well as the rest of the functions appearing in ${\mathcal{M}}_{2k}$ with $k>3$ can be cast in terms of $I_{k}$ and $J_{k}$, after performing a partial fraction decomposition to each of the rational parts of them.

We start with $\Theta_{0}>0$ and take $\varepsilon$ small enough with $\varepsilon\geq\varepsilon_{0}>0$ to avoid the possible zeroes of the functions ${\mathcal{F}}_{4}$, ${\mathcal{F}}_{6,1}$, etc. Concerning ${\mathcal{M}}_{4}$ we apply the estimates (34) to $\mathcal{F}_{4}$, and after arranging the function conveniently, we conclude that

[TABLE]

For ${\mathcal{M}}_{6}$ we get

[TABLE]

and similar expressions are obtained for ${\mathcal{M}}_{8}$, ${\mathcal{M}}_{10}$, and so on.

From the above calculations it is clear that for $\varepsilon$ small enough the most important terms are those related to $\exp(-\Theta_{0}^{3}/(3\varepsilon^{3}))$, then those related to $\exp(-2\Theta_{0}^{3}/(3\varepsilon^{3}))$, next those with $\exp(-\Theta_{0}^{3}/\varepsilon^{3})$ and so on. Furthermore, we observe that the terms with asymptotic estimates having the factor $\exp(-\Theta_{0}^{3}/(3\varepsilon^{3}))$ correspond to the harmonics $\cos s_{0}$, $\sin s_{0}$, and in general, the asymptotic expressions with $\exp(-k\Theta_{0}^{3}/(3\varepsilon^{3}))$ are related to the harmonics $\cos ks_{0}$, $\sin ks_{0}$. In addition to this, for $k\geq 2$ the terms of $\cos ks_{0}$, $\sin ks_{0}$ are of order $\varepsilon^{-k-3/2}\exp(-k\Theta_{0}^{3}/(3\varepsilon^{3}))(1+O(\varepsilon))$ while for $k=1$ the terms of $\cos s_{0}$, $\sin s_{0}$ have the estimate $\varepsilon^{-3/2}\exp(-\Theta_{0}^{3}/(3\varepsilon^{3}))(1+O(\varepsilon))$. This in turn implies that the leading terms in the Melnikov function are the ones depending on the coefficients $d_{1}$, $d_{2}$, the next ones those with estimate $\varepsilon^{-1/2}\exp(-\Theta_{0}^{3}/(3\varepsilon^{3}))$ to which follows the rest of terms with harmonics $\cos s_{0}$, $\sin s_{0}$. Next we consider the main terms factorized by $\cos 2s_{0}$, $\sin 2s_{0}$, they are the ones depending on $c_{2}$, $c_{3}$. We continue with the higher order terms and so on. The Melnikov function becomes the formal Fourier series

[TABLE]

with

[TABLE]

for $k\geq 2$ and

[TABLE]

Regarding the estimate of ${\mathcal{R}}$ we use the ideas of Sanders [21] for the case of exponentially small estimates. Given a vector field of the form $\dot{x}=f_{0}(x)+\varepsilon f_{1}(x,t,\varepsilon)$ with $x\in D\subset\mathbb{R}^{2}$ and $\varepsilon$ a small parameter, he defines the Melnikov integral $\Delta_{\varepsilon}(t,x)=\varepsilon^{-1}f_{0}(x^{s,u}_{0}(t))\wedge(x_{\varepsilon}^{u}(t)-x_{\varepsilon}^{s}(t))$ where $\wedge$ is the wedge product in $\mathbb{R}^{2}$, $x^{s,u}_{0}$ refers to the parametrization of the stable and unstable manifolds of the unperturbed system and $x_{\varepsilon}^{u}$, $x_{\varepsilon}^{s}$ denote solutions on the unstable and stable manifolds, respectively. After some assumptions on the smoothness of the vector field, and a lemma regarding the relationships between the unperturbed and perturbed manifolds, Sanders arrives at an expression of the form

[TABLE]

where $\Delta_{0}$ stands for the usual Melnikov function and $\mu$ is the Lipschitz constant associated to $f_{0}$. Systems of the type $\dot{x}=\varepsilon f(x,t,\varepsilon)$, after applying averaging and rescaling time by $\tau=\varepsilon t$, are transformed into $y^{\prime}=dy/d\tau=f_{0}(y)+\varepsilon f_{1}(y,\tau/\varepsilon,\varepsilon)$, thus admitting the estimate $O(\varepsilon(1+\exp({-\mu|\tau|/\varepsilon}))^{2}\min\{1,\exp({-\mu|\tau|/\varepsilon})\})=O(\varepsilon\exp({-\mu|\tau|/\varepsilon}))$ for $\tau\neq 0$.

We adopt Sanders’ point of view in our setting as follows. First, we notice that the hypotheses on the Hamiltonian and the existence of the manifolds are fulfilled. Then we realize that $t$ and $\tau$ are related through the change of time, assuming that $x(\tau)=\sqrt{2}/\Theta_{0}\,{\rm sech}\,\tau+O(\varepsilon^{3})$ we get $t=\Theta_{0}^{3}\tau/(2\varepsilon^{3})+O(1)$. The Lipschitz constant of $H_{D}$ along the homoclinic $\xi(\tau)$ is calculated, obtaining $\mu=\sqrt{2}/\Theta_{0}$. Finally, after adjusting the factor $\varepsilon^{4}$ in the whole expression of $\Delta_{\varepsilon}$ so that we identify $\Delta_{0}$ with $\mathcal{M}$, we get an upper bound on $\mathcal{R}$ as $O(\exp({-\Theta_{0}^{2}|\tau|/(\sqrt{2}\varepsilon^{3})}))$. To control the size of $\mathcal{R}$ we compare the estimate with the dominant term of $\mathcal{M}$. When $d_{1}$ or $d_{2}$ are not zero, due to the presence of the factor $\varepsilon^{-3/2}$ in $\alpha_{1}$, $\beta_{1}$, it is enough that $\exp({-\Theta_{0}^{3}/(3\varepsilon^{3})})>\exp({-\Theta_{0}^{2}|\tau|/(\sqrt{2}\varepsilon^{3})})$ from where it is deduced that $|\tau|\geq\frac{\sqrt{2}}{3}\Theta_{0}$, which is true in big portions of the manifolds as $\Theta_{0}$ is of moderate size. Next, the transversality condition is verified in the part of phase space where this restriction on $\tau$ holds, but it implies that transversality is satisfied for every $\tau$ as this property is preserved through diffeomorphisms. When $d_{1}=d_{2}=0$ one compares $\exp({-k\Theta_{0}^{3}/(3\varepsilon^{3})})$ with $\exp({-\Theta_{0}^{2}|\tau|/(\sqrt{2}\varepsilon^{3})})$, starting with $k=2$.

When $\Theta_{0}<0$ we notice that to apply the estimates given above we should consider the integrals $I_{k}$, $J_{k}$ with $\delta<0$. Then we notice that $I_{k}(\delta)=I_{k}(-\delta)$, $J_{k}(\delta)=-J_{k}(-\delta)$ and $J_{k+2}(\delta)=\delta/(2(k+1))I_{k}(-\delta)$ and the estimates for $I_{k}$ given in (34) apply replacing $\delta$ by $-\delta$ in the expressions. Proceeding similarly to the case $\Theta_{0}>0$ we arrive at

[TABLE]

Reasoning as in the positive case we realize that the main term in the Melnikov function is that of $\varepsilon^{6}{\mathcal{M}}_{6}$ having coefficients $d_{1}$, $d_{2}$, then the rest of terms related to $\cos s_{0}$, $\sin s_{0}$, next the term associated to $\cos 2s_{0}$, $\sin 2s_{0}$ beginning with those whose coefficients are $c_{2}$, $c_{3}$, and so on. Moreover, when $k\geq 2$ the terms of $\cos ks_{0}$, $\sin ks_{0}$ behave like $\varepsilon^{-k}\exp(k\Theta_{0}^{3}/(3\varepsilon^{3}))(1+O(\varepsilon))$, thus we obtain analogous results to the case $\Theta$ positive although with estimates of different order in $\varepsilon$.

The estimate analysis of $\mathcal{R}$ is alike the procedure for $\Theta_{0}>0$.

Acknowledgements

We appreciate the comments of Prof. M. Guardia, P. Martín and T.M. Seara who pointed out a crucial error in the previous version of the paper and helped substantially in its improvement. The authors have received partial support from Project Grant Red de Cuerpos Académicos de Ecuaciones Diferenciales, Sistemas Dinámicos y Estabilización. PROMEP 2011-SEP, Mexico and from Projects 2014–59433–C2–1–P of the Ministry of Economy and Competitiveness of Spain and 2017–88137–C2–1–P of the Ministry of Economy, Industry and Competitiveness of Spain. The facilities provided by Cinvestav – IPN (Mexico) are also acknowledged.