Ejection-Collision orbits in the symmetric collinear four–body problem

In this paper, we consider the collinear symmetric four-body problem, where four masses m3 = α, m1 = 1, m2 = 1, and m4 = α, α > 0, are aligned in this order and move symmetrically about their center of mass. We introduce regularized variables to deal with binary collisions as well as McGehee coordinates to study the quadruple collision manifold for a negative value of the energy. The paper is mainly focused on orbits that eject from (or collide to) quadruple collision. The problem has two hyperbolic equilibrium points, located in the quadruple collision manifold. We use high order parametrizations of their stable/unstable manifolds to devise a numerical procedure to compute ejectioncollision orbits, for any value of α. Some results from the explorations done for α = 1 are presented. Furthermore, we prove the existence of ejection-direct escape orbits, which perform a unique type of binary collisions.


Introduction
The classical n-body problem studies the dynamics of n point masses interacting according to Newtonian gravity. In the symmetric collinear four-body problem, the bodies are symmetrically distributed about the centre of mass by pairs, each of those pairs have equal mass and the configuration of the four bod- 5 ies is collinear at every instant. It is a two degrees of freedom problem which is a sub-problem of the trapezoidal four-body problem that has three degrees of freedom, see the works of Lacomba and Simó ( [1,2]).
The four-body problem has attracted the attention of numerous astronomers since through it, the gravitational interaction of many stellar or exoplanetary 10 systems can be modelled, as the interaction of two binary star systems or the interplay of two planets with a binary star system. Many of the studies, as the influence between two binaries, have been carried out from the numerical point of view ( [3], [4] , [5], [6]). Also, the close interaction of systems of few stars give rise to the possibility of collisions between two or more stars in a cluster, as close 15 encounters and direct physical collisions between stars are frequent in globular clusters, [7]. These collisions are more frequent as a binary-binary system than as a system formed by a single star and a binary one. Other numerical studies have been conducted to understand the numerical scattering of the influence between binary-binary or single-binary systems, see [8]. 20 We focus on the particular case of the collinear model of a four body problem.
A solution of the symmetric collinear four-body problem, denoted by SC4BP, experiences a collision if two or more particles come together at a certain time.
At such a time the potential energy approaches infinity, the equations of motion become undefined and the solution has a singularity. The analytical and 25 numerical study of this problem requires the McGehee's blow up technique to regularize the singularity corresponding to total (quadruple) collision and the regularization of binary collisions (collisions between m 1 and m 2 ) and simultaneous binary collisions (m 1 and m 3 collide as well as m 2 and m 4 ), see for example [9] and [10]. This singularity due to total collision is blown up and in 30 its place is glued an invariant total collision manifold. In [2], Simó and Lacomba analyze the flow on the total collision manifold and they find a family of connection orbits between two quadruple collisions which arise as the parameter of masses is varied. The flow on this manifold provides relevant information for the flow close to quadruple collision. showing the existence of periodic, quasiperiodic, fast-scattering and chaotic-40 scattering orbits. Still in the case of equal masses, Sekiguchi and Tanikawa [12] study the SC4BP both analytically and numerically. In particular, they classify a great variety of orbits by means of symbol sequences and they obtain the initial conditions leading to escape using escape criteria established in the paper. 45 Focussing on periodic orbits, Bakker et al. [13] and Sweatman [14], using analytic-numerical methods, study the existence of Schubart-like orbits (that is, periodic solutions with exactly two binary collisions and one simultaneous binary collision per period) as well as their stability depending on the mass parameter. Later on, Ouyang and Yan [15] and Huang [16] analytically prove 50 the existence of such Schubart periodic orbits by applying topological methods and variational calculus, respectively.
We finally mention, for the symmetric collinear four-body problem, the papers by Alvarez et al. [9,17], where the authors provide some analytical results concerning singularities and regularization, and analytically study the quadru- The main goal in this paper is to amalgamate both theoretical and numerical tools to investigate, on one hand, orbits that eject from quadruple collision and have a fast escape to infinity, and, on the other hand, ejection-collision orbits (also denoted by ECO), that is orbits that eject from quadruple collisions and go 65 back to quadruple collision. The latter are regarded as heteroclinic connections between the two equilibrium points (that lie on the total collision manifold).
In order to numerically compute ejection-escape orbits and ejection-collision ones we need the construction of parametrizations, up to certain order, of the stable and unstable invariant manifolds, W s and W u , of the equilibrium points, 70 using the methodology explained in [19]. At this point we mention the work by Sekiguchi and Tanikawa [12] where due to the Poincaré section considered, the ejection-collision orbits are all mixed up and indistinguishable. In the present paper a different Poincaré section has been taken into account that allows to classify and distinguish different types of ejection-collision orbits. Following this 75 classification, In [20] Lacomba and Medina proved analytically the existence of certain ejection collision orbits for specific values of the mass parameter. In this paper, a numerical method is explained to compute ECO for any value of the mass parameter α and negative energy h. The results are presented for α = 1 and h = −1.

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The paper is organized as follows: in Section 2, we recall briefly some known results about the dynamics of the SC4BP, including the regularization of total collision using McGehee's coordinates [10], the regularization of binary collisions, the description of the flow on the quadruple collision manifold and the existence of two hyperbolic equilibrium points, E ± . We compute high or-85 der parametrizations of the associated stable and unstable invariant manifolds, W s,u (E ± ) and some error tests have been carried out to control the accuracy of the approximations. Section 3 is devoted to the orbits that eject from (or collide to) quadruple collision and directly escape to (come from) infinity describing a unique type of binary collisions. In Section 4 we present some properties of the ejection-collision orbits and devise a numerical method to compute them. We show the results for the case α = 1.

The symmetric collinear four body-problem
The aim of this section is to present a summary of the equations and the main properties of the symmetric collinear four-body problem, SC4BP. For more 95 details see, for example, [17,2] and the references therein. In particular, we focus on the main features for the computation of ejection/collision orbits (orbits that start/end at a quadruple collision) and ejection-collision orbits (ECO, orbits that start and end at a quadruple collision): the dynamics on the total collision manifold, the stable and unstable invariant manifolds of the equilibrium points 100 and the Poincaré section used.

Equations of the SC4BP
The symmetric collinear four-body problem consists of four point masses,   In this set of coordinates, the Hamiltonian of the problem is given by where p x = 2 dx dt , p y = 2 dy dt and the potential function is The phase space of the problem is Notice that the equations have three singularities, one at x = 0 and y = 0, another at y = √ αx = 0, and a third one corresponding to x = y = 0. They correspond to the following collision configurations: • Single binary collision: the bodies m 1 and m 2 collide, while the other two bodies remain bounded away from them. This type of collision corre-
• Double (simultaneous) binary collision: the bodies m 1 and m 3 collide, and by the symmetry of the problem, so do the other two bodies.

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• Quadruple collision: the four bodies collide. This collision corresponds to x = y = 0.
We fix a value of the energy in H = h, so the motion takes place in a 3dimensional manifold and, using (1), it is confined in the configuration space (x, y) to the Hill's region given by The function U (x, y) (given in (2)) is strictly positive for all (x, y) ∈ U, so for h ≥ 0 the Hill's region coincides with the configuration space U, whereas for h < 0 the Hill's region is limited by U (x, y) = −h. In Figure 2 we show 125 the Hill's region for a negative value of the energy h, and an orbit which ends at the quadruple collision and performs different binary collisions. We will use the representation of the orbits in the configuration space (x, y) inside the Hill's region R h through the paper. In the N -body problem bounded motions can only occur if h < 0 (see, for 130 example Chapter 4 in [21]). Therefore, to study the ejection-collision orbits we consider only negative values of the energy.
Furthermore, in order to study the dynamics close to the quadruple collision, it is necessary to describe the total collision manifold and the flow on it.
For this purpose, we apply the blow-up technique introduced by McGehee [10].

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Moreover, since the solutions of the SC4BP typically perform several binary collisions, we will also regularize the singularities (due to collisions) that appear in the system of the ordinary of differential equations (ODE). The suitable transformations of the blow up and the regularization in the symmetric collinear four-body problem have been made by Alvarez-Ramírez et al. [17]. For com-140 pleteness of the present work, and in order to understand the meaning of the regularized variables, we summarize the changes carried out.
• Introduce polar coordinates , where θ α = arctan( √ α ) corresponds to double binary collisions (DBC). The associated potential function is • Introduce the McGehee's coordinates (v, u) and a change in time through the relations • Remove simultaneously all binary collisions, considering the regularized potential which is a positive, real analytic function in [θ α , π/2], and the change of time and coordinates given by In coordinates (r, v, θ, w) and s as time variable, the equations of the SC4BP become dr ds = rv∆(θ), The energy relation H = h from (1) becomes The vector field defined by equations (6) is an analytic vector field on the The solutions of the ODE, also called orbits, will be denoted by Γ = {γ(s)} s∈R or simply by γ(s).
A straightforward computation shows that the set of equations (6) satisfies the symmetry Therefore, if Γ is a solution given by γ(s) = (r(s), v(s), θ(s), w(s)), then Γ defined as is also a solution. We recall that, from (7) and the definition of w (5), v = w = 0 means a binary collision (single or double) or a point on the zero velocity curve. 160

Total collision manifold, equilibrium points and invariant manifolds
Notice that the system (6) is well defined for r = 0, which corresponds to the total collision manifold C, given by which is a 2-dimensional manifold, topologically equivalent to a sphere minus four points, independent of the total energy h and invariant under the flow (6). The total collision manifold C belongs to the boundary of the manifold defined by a constant energy, h, for any value of h. Furthermore, the flow on C is gradient-like with respect the variable v, that is, dv/ds ≥ 0. See, for more details, [17,22,2].
The SC4BP has two equilibrium points where θ c is the only solution of V (θ) = 0 (the potential V is defined in (4)).
Furthermore, the SC4BP has a specific solution for which θ = θ c for all s ∈ R.
It is called the homothetic solution because the ratio y/x = tan θ c remains 170 constant (see [17]). It is seen as a segment in the configuration plane U and divides it into two regions: the region of the DBC for θ ∈ [θ α , θ c ), and the region of the SBC for θ ∈ (θ c , π/2]. See Figure 2.
Both equilibrium points are hyperbolic: the differential of the vector field evaluated at the equilibrium points has four different real eigenvalues, two pos-175 itive and two negative, for any value of α, see Lemma 1 in the Appendix. Also, we give explicit formulas for the eigenvalues and the corresponding eigenvectors in terms of α, θ c and the energy h.
Therefore, there exist the corresponding stable and unstable invariant manifolds W s (E ± ) and W u (E ± ). On the constant energy manifold their dimensions are the following: In particular, the invariant manifolds W u (E − ) and W s (E + ) are embedded in the total collision manifold C. In Figure 3, we plot their projection in the (θ, v) 180 plane for two values of the mass parameter α.
The dynamics on the total collision manifold C is the key to understanding the solutions of the SC4BP that go close to quadruple collision, in particular, the ejection-collision orbits. In Our purpose in this paper is to give a numerical general methodology that allows to compute the ejection-collision orbits for any value of α. Clearly, these orbits belong to the intersection of the invariant manifolds W u (E + ) and W s (E − ) in R 4 . Thus, in order to deal with them, we construct an approximation of their parametrizations. The approximations of order one are given by where σ i are the corresponding eigenvectors and ξ > 0 is a small fixed quantity, the distance from the initial conditions to the equilibrium point. See the Ap-pendix for more details and (24) for the specific expressions of the first order 200 approximation for each variable. In fact, using the symmetry (9), for any orbit take the approximation of order one, Ψ ± 1 (ξ, ϕ), to work with a good precision, for example of order 10 −12 (see below for details), it will be necessary to consider values of ξ less than 10 −6 . However, with such small values of ξ, in order to show the richness of the dynamics around the homothetic we need to consider values ε 10 −9 .
230 Therefore, if we want to consider initial conditions close to the slow direction, we need to start farther away from the equilibrium point, that is, with bigger values of ξ. Following [19] (specifically, Chapters 1 and 2), we derive the parametrization of the invariant manifold up to different orders Ψ + m , for m ≤ 8, and we have performed several numerical tests to control the quality of the approximation. Specifically, one can compute how big ξ can be in order to maintain a certain accuracy. To do that, for each order m and distance ξ, compute the error in the orbit e o (s, ξ, ϕ) (see Section 2.5 of [19]) for s ∈ [0, 1] and the maximum value over all the orbits The same procedure can be applied to the errors in the invariance equation and the errors in the energy (see [19] for details). For the purposes of this work, we have considered, for α = 1, a parametrization of order m = 8 and ξ of order 10 −2 . With these values, we will show that all the ejection-collision trajectories will be found for ϕ ∈ (1/2 − ε, 1/2 + ε), for ε of order 10 −4 . 240

Poincaré section
A common tool, in order to study the dynamics of a problem given by an autonomous system of differential equations, is the Poincaré map, which is defined on a surface of section. In [12], Sekiguchi and Tanikawa  least once. That is not true. In the proof the authors forget about the invariant manifolds W u (E + ) and W s (E − ). We will show in Section 3 that there exist orbits that start at a quadruple collision and escape directly without crossing {θ = θ c } (see Theorem 1) and in Section 4 we will show ejection-collision orbits that do not cross that section (by cross we always mean transversal intersection).

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Taking into account that, typically, all the orbits perform binary collisions (with the exception of the homothetic orbit), we consider the section which corresponds to both types of binary collisions: SBC and DBC.

Escape orbits
Notice that there are two different ways for the particles to escape to infinity.
In one case, the outer bodies escape, while the inner bodies perform consecutive then the orbit has an escape of type 1.
In both cases, the orbit escapes forwards or backwards in time depending on whether Υ is positive or negative, respectively.

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Proof. We follow exactly the same methodology as in [12] using our notation.
Consider the first case θ > θ c (in the configuration plane, the region of SBC, where x is bounded). We want to see that if the sufficient condition is satisfied, then lim s→∞ y(s) = ∞. From the system of equations associated to the Hamiltonian (1), the equation for the variable y writes It means that the force acting on the external bodies is bounded from below by the force of the two-body problem given by − f (1/δ) y 2 . Thus, a sufficient condition for escape motion is that the total energy of the two-body problem should be to be positive, that is yẏ 2 ≥ 2f (1/δ). In our variables (θ, v, u) it becomes sin θ(v sin θ + u cos θ) 2 ≥ 4 √ 2f (1/δ).
Now, if we rewrite (13) in terms of the variables (θ, v, w), the result stated in the item 1. of the proposition is followed.
The next is to consider the case with θ < θ c , which corresponds to the region DBC in the configuration plane. In this hypothesis, escape motion means that the four masses escape to infinity. Therefore, both x, y → ∞ and we need to introduce Jacobi variables: That is, essentially, Q 1 is the center of mass and Q 2 is the distance between m 2 and m 4 . In these variables, the escape through the DBC region corresponds to Q 1 → ∞ whereas Q 2 remains bounded. The equation of motion corresponding to Q 1 becomesQ is a positive monotone increasing function for z ∈ (0, 1/ √ α). Therefore, when θ < θ c , we have y/x < δ andQ , Again, a sufficient condition for escape motion is the total energy of the two-body problem must be positive. It means that As in the previous case, if we rewrite (14) in terms of the variables (θ, v, w), the result stated in the item 2. of the proposition is followed. Theorem 1. There exist ejection-direct escape orbits of type 1 and of type 2, that is, orbits starting (or ending) at the quadruple collision and escaping to (coming from) infinity with binary collisions only of one type.
Proof. By the symmetry of the problem, it is enough to prove that there exist 300 orbits on W u (E + ) that escape forwards in time performing binary collisions only of one type. We will prove the result for orbits that have only SBC. The proof to obtain orbits with only DBC is similar.
We will use the criteria given in Proposition 1. As we want to prove escape forwards in time of type one, we must see that Using the approximation of the parametrization of order 1 of the invariant manifold Ψ + 1 (ξ, ϕ) given in (11) and (24), and omitting the variable r, a point on W u (E + ) close to the equilibrium point can be written, for ξ small enough, where p 0 = (v c , θ c , 0) and p 1 are the terms of order 1 in ξ for values ϕ ∈ (1/4, 1/2), so that we ensure that θ > θ c and the initial condition is in the region of SBC.
Recall that the values ϕ = 1/4, 3/4 correspond to the fast direction on the invariant manifold, whereas ϕ = 1/2 corresponds to the homothetic and the 315 slow direction. Therefore, the farther an initial condition from the homothetic solution is, the higher the probability to escape directly to infinity.
We show numerically, for α = −1 and h = −1, that the orbits with initial conditions Ψ + 8 (ξ, ϕ) for ϕ = 1/4 + ε and ϕ = 3/4 − ε are the ones that escape directly. We consider the unstable manifold W u (E + ), ξ = 10 −2 and vary ϕ ∈ 320 (1/4, 3/4). Given an initial condition, we integrate the ODE (6) forwards in time and, at each step, we control the escape condition. If the condition is satisfied and the orbit has not crossed the section {θ = θ c }, we save the time s e at that point. In Figure 5

Ejection-collision orbits
In this Section we present some results about the ejection-collision orbits, as 330 well as the methodology to compute and classify them. Also the results obtained for α = 1 are presented. An ECO is a solution that starts and ends in a quadruple collision. Therefore, the orbit belongs to the intersection W u (E + ) ∩ W s (E − ). More concretely, so it is a heteroclinic connection between the two equilibrium points.

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Using the symmetry of the problem given by (8), we can prove the following statements: Proposition 2. Let Γ be a solution of the SC4BP.  Let B be the set of all possible sequences just taking into account the elements 1 and 2. Thus, we can define For example, the escape orbits shown in Figure 5 satisfy P (Γ) = (1, 1, . . .) or 345 P (Γ) = (2, 2, . . .), that is, they only exhibit the same type of binary collision.
Using the map P we have the following characterizations and properties.

Γ is a symmetric ECO if and only if P (Γ) is a symmetric sequence.
Proof. Suppose that an ECO has infinity binary collisions. Then, there exists    The dynamics on the total collision manifold C is the key to prove the existence of ECO. In [20], the authors use that information to show the existence of some ECO for specific values of α. In particular, they prove that for any value of the mass parameter α and for any natural number n there exists an ECO exhibiting only and exactly n SBC or n DBC. They also prove that for 375 α ∈ (α 3 , α 4 ) (see Section 2.2), an ECO with P (Γ) = (1, 2, 1, 2, 1) exists (in particular this is true for α = 1).
Our aim is to present a methodology to compute and classify the ECO for any fixed value of α. We will show and analyze the results obtained for the specific value α = 1.

Methodology
In order to look for heteroclinic connections between the two equilibrium points, the main idea is to analyze the successive intersections of the orbits of the invariant manifolds W s (E − ) and W u (E + ) with the section Σ c , defined in (12). Due to the symmetry of the problem, it is enough to deal with one of the invariant manifolds. In what follows, we consider initial conditions on W u (E + ) and the approximation of its parametrization, Ψ + m (ξ, ϕ), for a suitable m and for a fixed value of ξ. For simplicity we denote the parametrization simply by Ψ(ϕ). Then, each orbit γ(s) ∈ W u (E + ) is characterized by its initial condition given by where ϕ ∈ (1/4, 3/4). For each ϕ, we integrate forward in time to compute the first n intersections of γ(s) with Σ c . We define the map where P n (ϕ) = (p 1 , p 2 , . . . , p n ) codes the first n intersections with Σ c of the solution γ(s) with initial condition Ψ(ϕ).
We want to notice here that for a given ϕ, p 1 is the first binary collision after the initial condition. Depending on the the initial distance ξ considered, 385 there could exist a binary collision before the initial condition (backwards in time, towards the quadruple collision). This is specially true for values of ϕ near to 1/4 and 3/4. As we explained in Section 3, the solutions corresponding to values far from ϕ = 1/2 (the homothetic orbit) escape directly to infinity. So the ECO will be found for values ϕ ∈ (1/2 − ε, 1/2 + ε), for ε small, depending on α and ξ. For example, for α = 1 and ξ of order 10 −2 , ε 10 −4 . We will show that for such values of ϕ, and for the order of the ECO computed, the first binary collision takes place far away from the equilibrium point, so P n (ϕ) starts with the very first binary collision of the orbit.

An illustration of Proposition 5 is shown in
Using Proposition 5, to detect the existence of an ECO of order n, we vary ϕ ∈ (1/4, 3/4) and we compute P n+1 (ϕ) integrating the equations (6) of the SC4BP up to the (n + 1)-th crossing with Σ c . To detect a change in the type of the binary collision, it is enough to track the value of θ at the (n+1)-th crossing, 410 θ n+1 (ϕ): when it changes from π/2 to θ α , or the other way around, we are in the situation of Proposition 5. Although the discontinuities of the function θ n+1 (ϕ) show the existence of ECO, we propose to use, instead, the function F n+1 (ϕ) = r(θ − θ c ), where r and θ are the values of the orbit at the (n + 1)-th intersection with the section Σ c (or the (n + 1)-th binary collision). Clearly, 415 the function F n+1 is continuous and due to the fact that r > 0, it changes sign depending on whether θ is greater or smaller that θ c . Therefore, each solution of F n+1 (ϕ) = 0 corresponds to an ECO orbit of order j ≤ n. We track the sign of the function F n+1 and apply an iterative method to obtain the value of ϕ (up to a certain precision) that corresponds to the ECO. We have repeated 420 the explorations for α = 1 and different values of ξ = 0.001, 0.01, 0.05. In all cases we have obtained the same results (that is, the same ECO). In Figure 7 we show the values of θ n+1 (ϕ) and F n+1 (ϕ) for n = 4, 5, 6 and ϕ < 1/2, using α = 1, ξ = 0.05 and h = −1. For n = 4 the function F 5 shows four zeros, corresponding to four ECO of order n ≤ 4 (see Table 1 in next section), for 425 n = 5, the function F 6 shows six zeros, corresponding to the same ECO and two new ones, of order 5. And so on. We want to notice two important issues. On one hand, the bigger the order n, the smaller the interval I = (1/2 − ε, 1/2 + ε) where some of the zeros of F n+1 exist. However, fixed an order n, to look for the zeros of F n+1 with a good 430 accuracy, it is important to take a big ε, and a suitable ξ (as big as possible).
This implies that a high order approximation of the parametrization of the invariant manifold is needed. where p 1 n+1 = p 2 n+1 for small enough. Numerically we observe that this is true.

Results
We present here, for α = 1 and h = −1, the ECO computed up to order n ≤ 7 by looking for the zeros of the function F 8 (ϕ) as explained in the previous Section. The orbits obtained are summarized in Tables 1-4. Recall that, for any ECO of order n of type (p 1 , . . . , p n ), there exists also the symmetric one 440 (p n , . . . , p 1 ) that traces the same path in configuration space, so they are not included.
For n ≤ 4, only ECO of type (1, n) . . ., 1) or (2, n) . . ., 2) exist, so there are only two ECOs for each order, see Table 1. For n = 5 we find four different ECO, all of them symmetric, see Table 2. The first non-symmetric orbits are found 445 for n ≥ 6. For n = 6, there exist eight different ECOs, two symmetric and six non-symmetric, see Table 3. We plot three of the non-symmetric ones, the (2) (1,1)   Table 4.
We finally notice that in [20], Lacomba and Medina give a graph, for certain values of α, that allows to identify the possible sequences of binary collisions for an orbit passing near the total collision. This is in accordance with [12], 455 where the authors mention that not all the possible sequences are realizable. We reproduce (in our notation) that graph in Figure 8 for α = 1. For example, from the graph in Figure 8 it is clear, that no ECO orbits of types (2, k) . . ., 2, 1, 1) and (1, k) . . ., 1, 2, 2) can exist (since from the graph, we see that the partial sequences (2, 1, 1) or (1, 2, 2) cannot exist).

Discussion and conclusions
As stated in [12], of E + and E − and the manifolds of the infinity. But this is work for a future paper.
Although the methodology described applies for any value of α > 0, h < 0 and any given order n for the ECO, we present the results obtained for α = 1 and h = −1, and show ECO up to order n = 7. We also remark that different 485 results, concerning the type of ECO, may be expected depending on α. More precisely, for α = 1, we notice that if there exists an ECO of type (p 1 , ..., p n ), then we also find the ECO of type ( p 1 , . . . , p n ), with p j = 3 − p j . This seems to be related with the same topological behaviour, concerning SBC and DBC, of the invariant manifolds of E ± on the quadruple collision manifold (but no 490 analytical proof is known so far). However, as shown in Figure 3 for α = 2, the behaviour of such manifolds, and therefore the type of ECO obtained, varies with α.

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Proof. Notice that, from (18), the two inequalities of the second statement are equivalent, so it is enough to prove one of them. Moreover, it is not difficult to see that both statements are equivalent to where θ c is the unique solution of V (θ) = 0, and V is given in (4). We write V (θ) = √ 2 2 cos θ h(z) where z = tan θ > √ α and h(z) = 1 + α 5/2 z + 8α 3/2 z z 2 − α .
The condition V (θ c ) = 0 is equivalent to where z c = tan(θ c ). Introducing this relation into the expression for V (θ c ), we have that the conditions in (19) become and where z c is the solution of (20).
Therefore, we can write the approximations of order one of the parametrizations of the invariant manifolds. For example, using the normalized eigenvectors 520 σ i = σ i /||σ i ||, we have that for W u (E + ): From the second statement of Lemma 1 we have that the eigenvector associated to λ 1 gives the slow direction in both invariant manifolds, whereas the strong directions are given by the eigenvectors associated to λ 3 (for W u (E + )) and λ 4 (for W s (E − )).