Corrigendum to: Oscillatory motions in restricted N -body problems [J. Differential Equations 265 (2018) 779-803]
CCORRIGENDUM TO: OSCILLATORY MOTIONS INRESTRICTED N -BODY PROBLEMS [J.DIFFERENTIAL EQUATIONS 265 (2018) 779–803] M. ALVAREZ-RAM´IREZ*, A. GARC´IA*, J.F. PALACI ´AN**, P. YANGUAS**
Abstract.
We consider the planar restricted N -body problemwhere the N − N -body problems, includ-ing the circular restricted three-body problem. No restrictions onthe mass parameters are assumed. Introduction
In restricted N -body problems there exists a class of unboundedorbits called oscillatory , where the motion of the N − Key words and phrases.
Restricted N -body problem; central configurations;cometary case; symplectic scaling; invariant manifolds at infinity; McGehee’s coor-dinates; Melnikov function; transversality of manifolds; Smale’s horseshoe; oscilla-tory motions. a r X i v : . [ m a t h . D S ] M a y ALVAREZ, GARC´IA, PALACI ´AN, YANGUAS times. This type of motion was hypothesized by Chazy [4] for the 3-body problem, Sitnikov [22] was the first to construct an oscillatorymotion in a restricted 3-body problem, he also described an initial con-dition set for such solutions. Later on, using the theory of quasi-randomdynamical systems, Alekseev [1] studied the Sitnikov problem showingthe existence of oscillatory motion for small but positive infinitesimalmass. By the same time Melnikov [15] and Arnold [3] developed amethod to study the formation of transversal intersections of stableand unstable manifolds. This technique, now known as the Melnikovmethod, replaces variational equations by the computation of certainintegrals.An orbit of an infinitesimal particle in a restricted N -body problemis parabolic if the infinitesimal particle escapes to infinity with zerolimit radial velocity. In [14] McGehee introduced a suitable set of co-ordinates that brings the infinity into the origin. He also proved thatthe set of parabolic orbits is formed by two smooth manifolds, thatcan be regarded as the stable and unstable manifolds of an unstableperiodic orbit, or a hyperbolic fixed point in a suitable Poincar´e mapat infinity. This result was used by Moser [19] to clarify the proof ofthe existence of oscillatory motion in the Sitnikov problem. The keymechanism is to show the existence of transversal homoclinic intersec-tions of the parabolic manifolds, which leads to a Smale’s horseshoemap that guarantees the existence of symbolic dynamics, giving rise tothe existence of oscillatory orbits as a consequence.Llibre and Sim´o [11] followed Moser’s approach to prove the ex-istence of oscillatory solutions in the planar circular restricted 3-bodyproblem (RPC3BP). They achieved it by demonstrating the transversalintersection of the stable and unstable parabolic manifolds for a largeJacobi constant C and a sufficiently small mass ratio µ between the pri-mary bodies. Xia [23] treated the RPC3BP by the Melnikov method,where µ is used again as a perturbation parameter. He proved thetransversality of the homoclinic manifolds for sufficiently small µ , and C close to ±√
2. After that, he used analytic continuation to extendthe transversality to almost any value of µ with C large enough.Guardia et al. [7] demonstrated that oscillatory motions do occurin the RPC3BP for any µ ∈ (0 , /
2] and a sufficiently large Jacobiconstant. Their result was achieved through an asymptotic formulaof the distance between the stable and unstable manifolds of infinityin a level set of the Jacobi constant, and making C sufficiently large,they proved that these manifolds intersect transversally. More recently, ESTRICTED N -BODY PROBLEMS 3 Guardia et al. [6] followed the above method to prove the existenceof oscillatory solutions in the planar elliptic restricted 3-body problem,for any mass ratio µ ∈ (0 , /
2] and small eccentricities of the Keplerianellipses (the primaries perform nearly circular orbits). This is done byconstructing an infinite transition chain of fixed points of the Poincar´emap and then providing a lambda lemma which gives the existence ofan orbit which shadows the chain.As far as the authors are aware, there are few studies in the literaturerelated 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 wherethe N − cometary problem [16, 17].The scaling is performed through the introduction of a small parameterthat measures the distance from the infinitesimal particle to the origin.Thence, the resulting system is given by the Kepler Hamiltonian mul-tiplied by a power of the small parameter plus higher-order terms thatare easily obtained through Legendre polynomials. The perturbationdepends on the specific configuration of the primaries.The transversal intersection between the manifolds is proved by ap-plying the Melnikov’s method [9, 10, 8, 20] related to the first twonon-null perturbative term. Indeed we compute a general Melnikovfunction that works for all the configurations, determining when it hassimple zeroes. There are some cases where the calculation of higher-order terms of the Melnikov’s function are required. This concerns to ALVAREZ, GARC´IA, PALACI ´AN, YANGUAS two parameters which are given in terms of the masses and positionsof the primaries.In the context of oscillatory motions we generalize the analysis per-formed for the RPC3BP by considering any central configuration ofthe N − N -body problem where the primariesare in central configuration in a rotating and an inertial frame. We alsodefine the cometary case, introducing an appropriate small parameterso that the problem is expressed as a 2-body Kepler Hamiltonian mul-tiplied by a power of the small parameter plus a small perturbation.In Section 3, we introduce McGehee coordinates to study the behaviorof the system in a vicinity of infinity as a Poincar´e map near a ho-moclinic orbit of a degenerate periodic orbit with analytic stable andunstable manifolds. Section 4 is devoted to the unperturbed problemthat can be expressed as a Duffing oscillator. The main results willbe given in Section 5, where we establish the existence of transversalhomoclinic intersections of stable and unstable manifolds for the per-turbed Hamiltonian system. We analyze the Melnikov function up toperturbation orders 4 and 6, depending on the non vanishing order ofthe Melnikov function. This is the fundamental part in our study, sinceit gives a systematic way of concluding the transversality between thecorresponding invariant manifolds which only depends on the mass pa-rameters and the configuration of the primaries. Then, we use that theMelnikov function has simple zeroes to show the transverse homoclinicintersections in the perturbed problem. In general, the correspondingintegrals are hard to calculate, however, we make use of the asymptoticestimates provided in [11, 12, 13]. In Section 6 we illustrate the theorydeveloped in the previous sections in some restricted N -body problems.In the RPC3BP, we show that in general it is enough to calculate theterms of order 4 and 6 in ε . In addition to the above, other examplesare considered, such as the restricted 4-body problem with primariesin equilateral (Lagrange) configuration, the 5-body problem with pri-maries in rhomboidal configuration, the collinear restricted N -body ESTRICTED N -BODY PROBLEMS 5 problem and some polygonal restricted N -body problems. Finally, thestudy of the Melnikov functions is addressed in the Appendix.All the numeric and symbolic calculations have been performed withMathematica. We have made the computations within 50 significantdigits although we only show the first eight.2. Problem statement
The planar restricted N -body problem is the study of the motion ofan infinitesimal mass particle subject to the Newtonian gravitationalattraction of N − N − ω = 1.The Hamiltonian that governs the motion of the infinitesimal particlein a rotating frame is H ( Q, P ) = 12 | P | − Q T J P − U ( Q ) , (1)where Q, P ∈ R are the position and momentum, J = ( − ) and thepotential is given by U ( Q ) = N − (cid:88) k =1 m k | Q − a k | . (2)(We will at times identify R and C , i.e., ( x, y ) ∈ R with x + y i ∈ C .)The terms a k = a k + a k i and m k correspond to the position andmass respectively, of the k -th primary, k = 1 , . . . , N −
1. Since theyare in central configuration the following equations are satisfied: a k = − N − (cid:88) j =1 j (cid:54) = k m j ( a j − a k ) | a j − a k | , k = 1 , . . . , N − N − (cid:88) k =1 m k a k = 0 , (3)with m + · · · + m N − = 1. We observe that the Hamiltonian H repre-sents an autonomous system with two degrees of freedom. This quan-tity is preserved through the changes of coordinates in spite of the factthat some characteristics, like the time independence of the flow or the ALVAREZ, GARC´IA, PALACI ´AN, YANGUAS
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 coordi-nates ( Q, P ) = e − tJ ( q, p ). The Hamiltonian accounting for the motionof the infinitesimal particle in inertial coordinates yields H ( q, p, t ) = 12 | p | − U ( q, t ) , (4)where the potential is given by U ( q, t ) = N − (cid:88) k =1 m k | e − tJ q − a k | . (5)Now the Hamiltonian is time dependent.We are interested in the study of the motion of the infinitesimal par-ticle near parabolic orbits, that is when the particle escapes to infinitywith zero limit velocity. Thus, it is convenient to scale the Hamilton-ian by introducing a small positive parameter ε through the change q → ε − q , p → εp . This is a symplectic transformation with multiplier ε . Hence H → εH . This problem now is the cometary regime of therestricted N -body problem, see [16, 17, 18].Thus, Hamiltonian (4) becomes H ε ( q, p, t ) = ε (cid:32) | p | − N − (cid:88) k =1 m k | ε − e − tJ q − a k | (cid:33) . (6)The expansion of the potential in (6) in terms of Legendre polynomialsyields H ε ( q, p, t ) = ε (cid:18) | p | − | q | (cid:19) − ∞ (cid:88) j =2 ε j +3 | q | j +1 N − (cid:88) k =1 m k | a k | j P j (cos γ k ) , (7)where P j is the j -th term of the Legendre polynomial, and γ k is theangle between the k -th primary’s position and e − tJ q . Let us note thatthe zero term of the sum is − / | q | and the next term is zero becausewe have placed the center of mass at the origin. Hence the problem isa Kepler problem (multiplied by ε ) plus a perturbation term of order ε . The series is convergent in the region of the configuration spacewhere ε | a k | < | q | .Now we make the symplectic polar change of variables q = re iθ , p = Re iθ + Θ r ie iθ , ESTRICTED N -BODY PROBLEMS 7 where r is the distance of the infinitesimal particle to the origin, θ isthe argument of latitude, R stands for the radial velocity and Θ forthe angular momentum. For our analysis we restrict Θ to be non-zeroand bounded, which is equivalent to consider the angular momentumbefore the scaling being a big quantity as long as ε is small.Let α k be the angle between the position of k -th primary and thehorizontal axis of the inertial frame, thus γ k = α k − ( θ − t ). Then,Hamiltonian (7) can be written as H ε ( r, θ, R, Θ , t ) = ε (cid:18) (cid:18) R + Θ r (cid:19) − r (cid:19) − ∞ (cid:88) j =2 ε j +3 r j +1 N − (cid:88) k =1 m k | a k | j P j (cos ( α k − ( θ − t ))) . (8)We observe that the argument of the Legendre polynomial P j dependson a k through the relations cos( α k ) = a k / | a k | and sin( α k ) = a k / | a k | .The associated Hamiltonian system is given by˙ r = ε R, ˙ R = ε (cid:18) − r + Θ r (cid:19) − ∞ (cid:88) j =2 ( j + 1) ε j +3 r j +2 N − (cid:88) k =1 m k | a k | j P j (cos ( α k − ( θ − t ))) , (9)˙ θ = ε Θ r , ˙Θ = ∞ (cid:88) j =2 ε j +3 r j +1 N − (cid:88) k =1 m k | a k | j Q j (cos ( α k − ( θ − t ))) sin ( α k − ( θ − t )) , 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]: r = x − , R = −√ y, s = t − θ, ALVAREZ, GARC´IA, PALACI ´AN, YANGUAS where s ∈ S . In these new coordinates the equations (9) become˙ x = √ ε x y, ˙ y = ε (cid:16) √ x − √ Θ x (cid:17) − √ ∞ (cid:88) j =2 ( j + 1) ε j +3 x j +4 N − (cid:88) k =1 m k | a k | j P j (cos ( α k + s )) , ˙ s = 1 − ε x Θ , ˙Θ = ∞ (cid:88) j =2 ε j +3 x j +2 N − (cid:88) k =1 m k | a k | j sin( α k + s ) Q j (cos ( α k + s )) . (10)Similarly to the circular restricted 3-body problem, the above systemcan be smoothly extended to x = 0; in addition, it has a Jacobi-likefirst integral given by equating Hamiltonian H in (1) to a constant.Let C be an arbitrary value of it. Thus C = H ε ( x, y, θ, Θ , t ) − Θ , (11)where H ε ( x, y, θ, Θ , t ) is given in (8) after replacing r and R for theirvalues in terms of x and y . In order to verify the hypotheses of McGe-hee’s Theorem [14], we observe that Θ can be written in terms of x , y , s and the value of the Jacobi-like integral C as follows:Θ = 1 ± (cid:112) ε x ( C + ε ( x − y )) ε x + O ( ε ) . (12)We take the negative sign in (12) because we are interested in smallvalues of x . Therefore, the variable Θ depends smoothly on x ≥ y ∈ R , s ∈ S . In fact Θ = − C when x = y = 0, hence the equationreferring to ˙Θ can be dropped in system (10). Expanding (12) in powerseries around x = 0, replacing its value in (10), the resulting systembecomes: ˙ x = √ ε x y, ˙ y = √ ε x + ε x f ( x, y, ε, s, C ) , ˙ s = 1 + Cε x + ε x − ε x y + ε x g ( x, y, ε, s, C ) . (13)Observe that the functions f ( x, y, ε, s, C ) and g ( x, y, ε, s, C ) are realanalytic.Now, we are going to exploit the fact that the solution γ ( t ) of system(13) with x = 0 and y = 0 is given by x ( t ) = 0 , y ( t ) = 0 , s ( t ) = s + t mod 2 π. (14) ESTRICTED N -BODY PROBLEMS 9 This solution does not depend on the Jacobi-like first integral C and is a 2 π -periodic solution in t . Let Σ = { ( x, y, s ) : s = s } be atransversal section to this periodic orbit. This set is parametrized bythe coordinates x and y . For the point κ = (0 , , s ) ∈ γ ∩ Σ, theImplicit Function Theorem shows that there exists an open set V ⊂ Σcontaining κ and a smooth function σ : V → R , the return time, suchthat the trajectories starting in V come back to Σ in a time σ close to2 π .Let ϕ t ( x , y ) = ( x ( t, x , y ) , y ( t, x , y ) , s ( t, x , y )) be the flow solu-tion of (13) with initial condition ( x (0) , y (0) , s (0)) = ( x , y , s mod 2 π ).This solution depends on ε , but it is usually omitted.The Poincar´e map P : V ⊂ Σ → Σ of the periodic orbit γ (or of thefixed point (0 , , s )) is given by P ( x, y ) = ϕ σ ( x,y ) ( x, y ) where P : (cid:40) x → x + √ πε x y + ε x y r ( x, y, ε ) y → y + √ πε x (1 − C x ) + ε x r ( x, y, ε ) (15)and r and r are real analytic functions.A straightforward computation shows that if we make the transfor-mation x = u + v , y = v − u , the map P takes the form˜ P : (cid:40) u → u + ε p ( u, v ) + ε s ( u, v, ε ) v → v + ε p ( u, v ) + ε s ( u, v, ε )where p ( u, v ) = −√ πu ( u + v ) , p ( u, v ) = √ πv ( u + v ) and s and s are real analytic functions starting with homogeneous polynomialsof degree 6 in u and v . For u > p ( u,
0) = −√ πu < p ( u,
0) = 0 and ∂p ∂v ( u,
0) = √ πu >
0. If ε ≥ ε > δ = δ ( ε ) > β = β ( ε ) >
0, the sector centred on the line y = − x is defined by B ( δ, β ) = { ( x, y ) : 0 ≤ x ≤ δ, ( − − β ) x ≤ y ≤ ( − β ) x } . Similarly, the sector centred on the line y = x is B ( δ, β ) = { ( x, y ) : 0 ≤ x ≤ δ, (1 − β ) x ≤ y ≤ (1 + β ) x } . The sets W sε (0 ,
0) = (cid:8) ( x, y ) ∈ B ( δ, β ) : ∀ n ≥ , P n ( x, y ) ∈ B ( δ, β ) , lim n →∞ P n ( x, y ) = (0 , (cid:9) ,W uε (0 ,
0) = (cid:8) ( x, y ) ∈ B ( δ, β ) : ∀ n ≤ , P n ( x, y ) ∈ B ( δ, β ) , lim n →−∞ P n ( x, y ) = (0 , (cid:9) , are called the stable and unstable manifolds of the fixed point (0 , Theorem 3.1 (McGehee, [14]) . For the map P : V ⊂ Σ → Σ given by(15), there is δ = δ ( ε ) > and β = β ( ε ) > such that the manifolds W sε (0 , ⊂ B ( δ, β ) and W uε (0 , ⊂ B ( δ, β ) correspond to the graphsof two functions ψ s , ψ u : [0 , δ ] → R that are smooth, real analytic in (0 , δ ] , and ψ s (0) = ψ u (0) = 0 , ψ (cid:48) s (0) = − , ψ (cid:48) u (0) = 1 . In addition tothat, they vary uniform and smoothly for ε ≥ ε > small enough. The sets W sε (0 ,
0) and W uε (0 ,
0) are curves, that is, one-dimensionalmanifolds. From here it follows that W sε ( γ ) = (cid:8) ϕ t ( x ε , y ε ) : t ≥ , ( x ε , y ε ) ∈ W sε (0 , (cid:9) ,W uε ( γ ) = (cid:8) ϕ t ( x ε , y ε ) : t ≤ , ( x ε , y ε ) ∈ W uε (0 , (cid:9) , are smooth manifolds of dimension two. They are formed by the orbitsthat escape to infinity ( x = 0) with zero velocity ( y = 0). These arecalled the parabolic manifolds .4. The main term of the Hamiltonian
The main part in Hamiltonian (8) corresponds to the Kepler prob-lem, and the related equations (10) take the form˙ x = √ ε x y, ˙ y = ε (cid:16) √ x − √ Θ x (cid:17) , ˙ s = 1 − ε x Θ , ˙Θ = 0 . (16)At this point it is convenient to introduce a new time through dτ /dt = ε x / √
2. Fixing Θ = Θ non zero, the equations for x and y become d xd τ = x (cid:48) = y, d yd τ = y (cid:48) = x − Θ x . (17)Thus (17) is a Θ -parametrized Duffing equation. The origin (0 ,
0) isa hyperbolic saddle point and its stable and unstable manifolds form ahomoclinic orbit ξ ( τ ) given by x ( τ ) = √ | Θ | sech τ, y ( τ ) = − √ | Θ | tanh τ sech τ, (18)connecting the fixed point with itself as shown in Fig. 1.The Keplerian Hamiltonian associated to (17) is rewritten as theDuffing Hamiltonian: H D = y − x + x Θ . (19) ESTRICTED N -BODY PROBLEMS 11 xy xy Figure 1.
From left to right, we show the flow of theDuffing Hamiltonian and the homoclinic loop for differ-ent values of Θ .5. Perturbed Problem
In this section we compare equations (13) and (16). We observe thatthe Poincar´e map of the last one has (0 ,
0) as a fixed point with ahomoclinic orbit parametrized by (18). The fixed point is preserved bythe Poincar´e map of (13), however the homoclinic orbit is broken intothe curves W sε (0 ,
0) and W uε (0 , ε implies that W sε ( γ ) and W uε ( γ ) are parametrized by orbits of the form ϕ s ( τ, x sε , y sε ) = ξ ( τ − τ ) + ε ϕ s ( τ, τ , ε ) + ..., for τ ≥ τ ,ϕ u ( τ, x uε , y uε ) = ξ ( τ − τ ) + ε ϕ u ( τ, τ , ε ) + ..., for τ ≤ τ , (20)with τ = τ ( s ), the functions ϕ s and ϕ u are determined by the firstvariational equation of the ε -perturbation of (17) along the orbit ξ ( τ ),that is, the corrections to ξ ( τ − τ ) corresponding to the perturbationof the Duffing Hamiltonian of order ε , the next terms, i.e. ϕ s , ϕ u ,correspond to the perturbation of order ε , etc. Besides, ϕ s ( τ, x sε , y sε ), ϕ u ( τ, x uε , y uε ) are taken so that ϕ s ( τ , x sε , y sε ) , ϕ u ( τ , x uε , y uε ) ∈ Σ, guaran-teeing 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 sε (0 , W uε (0 ,
0) (equivalently the breaking up of W sε ( γ ) and W uε ( γ )) underthe perturbation. We point out that the normal direction to the levelsets of H D is the only one to be taken into account. To measure the rate of separation between W sε ( γ ) and W uε ( γ ) withrespect to ε , we take H D ( ϕ s ( τ , x sε , y sε )) − H D ( ϕ u ( τ , x uε , y uε )) = ε ν M ν ( τ , ε ) + ..., where M ν ( τ , ε ) = (cid:90) ∞−∞ dH D dτ ( ξ ( τ ) , τ + τ ) dτ is the first term of the Melnikov function, x , y are evaluated in theunperturbed homoclinic and the derivative of H D with respect to τ is considered up to the perturbation of order ε ν in (17) such that thefunction M ν is not identically zero. The ellipsis in the formula measur-ing the splitting of the manifolds should not be underestimated. Morespecifically we decompose H D ( ϕ s ( τ , x sε , y sε )) − H D ( ϕ u ( τ , x uε , y uε )) = M ( τ , ε ) + R ( τ , ε ) , (21)where M ( τ , ε ) = ∞ (cid:88) l =2 ε l M l ( τ , ε )is the so called Melnikov function and M l contain the main terms ofthe total variation of H D ; their computation is similar to that of M ν above. In other words, the series contains the whole Hamiltonian H ε with x , y replaced by their values in (18) while R corresponds to thecontribution to the separation provided by the higher order terms ofthe orbits on the stable and unstable manifolds given in (20). Afterchanging from τ to s the infinite series can be interpreted as a Fourierseries in s whose coefficients are given as asymptotic expressions interms of ε . In the Appendix we will provide the leading terms ofthe series as well as the corresponding estimates for the higher orders.Regarding R we shall prove that it is smaller than the dominant termsof M at least in regions of the phase space containing portions of thestable and unstable manifolds of γ big enough where transversality canbe checked, as it will be detailed in the Appendix.The consequence of the existence of transversal homoclinic manifoldsis the appearance of a Smale’s horseshoe, and thence the occurrence ofchaotic motion of the infinitesimal particle. This includes the existenceof oscillatory motion, see [19, 8].In order to apply the results in the previous paragraphs we need tomake some arrangements to Hamiltonian (8). Up to terms of order ε , ESTRICTED N -BODY PROBLEMS 13 it reads as H ε ( r, θ, R, Θ , t ) = ε H + ε H + ε H + O ( ε )= ε (cid:18) (cid:18) R + Θ r (cid:19) − r (cid:19) − ε r ( c + c cos 2( t − θ ) + c sin 2( t − θ )) (22) − ε r ( d cos( t − θ ) + d sin( t − θ ) + d cos 3( t − θ )+ d sin 3( t − θ ))+ O ( ε ) , where c , c and c are the following constant terms, that depend onlyon a k and m k : c = N − (cid:88) k =1 m k ( a k + a k ) , c = 3 N − (cid:88) k =1 m k ( a k − a k ) ,c = − N − (cid:88) k =1 m k a k a k , (23)and the d j are given by d = 3 N − (cid:88) k =1 m k a k ( a k + a k ) , d = − N − (cid:88) k =1 m k a k ( a k + a k ) ,d = 5 N − (cid:88) k =1 m k a k ( a k − a k ) , d = − N − (cid:88) k =1 m k a k (3 a k − a k ) . (24)Higher order terms appearing in O ( ε ) should also be taken into con-sideration as we shall realize in the proof of the main result of thissection.The next step consists in applying McGehee’s transformation to theequations of motion associated to (22). After replacing t − θ by s , one gets˙ x = √ ε x y, ˙ y = √ ε (1 − Θ x ) x + √ ε ( c + c cos 2 s + c sin 2 s ) x + √ ε ( d cos s + d sin s + d cos 3 s + d sin 3 s ) x + O ( ε ) , ˙ s = 1 − ε Θ x , ˙Θ = − ε ( c cos 2 s − c sin 2 s ) x + ε ( d sin s − d cos s + 3 d sin 3 s − d cos 3 s ) x + O ( ε ) , (25)which corresponds to Hamiltonian equations (10).With respect to the new time τ , Eq. (25) gets transformed into x (cid:48) = y,y (cid:48) = (1 − Θ x ) x + ε ( c + c cos 2 s + c sin 2 s ) x + ε ( d cos s + d sin s + d cos 3 s + d sin 3 s ) x + O ( ε ) ,s (cid:48) = √ ε − − Θ x ) x − , Θ (cid:48) = − √ ε ( c cos 2 s − c sin 2 s ) x + √ ε ( d sin s − d cos s + 3 d sin 3 s − d cos 3 s ) x + O ( ε ) . (26)To obtain the Melnikov function we use the corresponding integral ofthe main term given by H D in (19), and compute the total derivative,assuming that Θ in (19) is considered as Θ, arriving at dH D dτ = ∂H D ∂x x (cid:48) + ∂H D ∂y y (cid:48) + ∂H D ∂ Θ Θ (cid:48) , where x (cid:48) , y (cid:48) , Θ (cid:48) are given in (26). In this expression x and y are replacedby their explicit values obtained for the unperturbed problem and thatwere given in (18).The next step consists in solving the differential equation for s (cid:48) in(26) using the fact that Θ is assumed to be constant, i.e. Θ = Θ ,and x ( τ ) is taken from the solution (18). The corresponding equationbecomes s (cid:48) ( τ ) = ± ε − Θ cosh τ ∓ τ. ESTRICTED N -BODY PROBLEMS 15 We set s (0) = s and the solution yields s ( τ ) = s ∓ τ / ± ε − Θ (9 sinh τ + sinh 3 τ ) . (27)The upper signs in s (cid:48) , s and the formulae that follow are used whenΘ > < H D with respect to τ becomes ε M = ∓ ε sech τ tanh τ (cid:0) c cosh τ + A cos 2 s + B sin 2 s (cid:1) , where A = ( c c Θ + c s Θ )(35 −
28 cosh 2 τ + cosh 4 τ ) − c c Θ − c s Θ )(7 sinh τ − sinh 3 τ ) − csch τ (8( c c Θ + c s Θ )(7 sinh τ − sinh 3 τ )+( c c Θ − c s Θ )(35 −
28 cosh 2 τ + cosh 4 τ )) ,B = ( c c Θ − c s Θ )(35 −
28 cosh 2 τ + cosh 4 τ )+8( c c Θ + c s Θ )(7 sinh τ − sinh 3 τ ) − csch τ (( c c Θ + c s Θ )(35 −
28 cosh 2 τ + cosh 4 τ ) − c c Θ − c s Θ )(7 sinh τ − sinh 3 τ )) , and c Θ = cos (cid:16) Θ ε (9 sinh τ + sinh 3 τ ) (cid:17) ,s Θ = sin (cid:16) Θ ε (9 sinh τ + sinh 3 τ ) (cid:17) . Proceeding as in the previous paragraphs, the total derivative of H D with respect to τ corresponding to the terms of order six in ε is ε M = ∓ ε sech τ tanh τ ( C sin s + D cos s + E sin 3 s + F cos 3 s ) , where C = 16 (cid:0) ( d ¯ c Θ + d ¯ s Θ )(3 − cosh 2 τ ) − d ¯ c Θ − d ¯ s Θ ) sinh τ (cid:1) cosh τ +12 cosh τ (cid:0) d ¯ c Θ + d ¯ s Θ ) cosh τ − ( d ¯ c Θ − d ¯ s Θ )(cosh 2 τ −
3) coth τ (cid:1) ,D = 16 (cid:0) ( d ¯ c Θ − d ¯ s Θ )(3 − cosh 2 τ ) + 4( d ¯ c Θ + d ¯ s Θ ) sinh τ (cid:1) cosh τ +12 cosh τ (cid:0) d ¯ c Θ − d ¯ s Θ ) cosh τ +( d ¯ c Θ + d ¯ s Θ )(cosh 2 τ −
3) coth τ (cid:1) ,E = ( d ˜ c Θ + d ˜ s Θ )(462 −
495 cosh 2 τ + 66 cosh 4 τ − cosh 6 τ ) − d ˜ c Θ − d ˜ s Θ )(198 sinh τ −
55 sinh 3 τ + 3 sinh 5 τ )+ csch τ (cid:0) ( d ˜ c Θ − d ˜ s Θ )(462 −
495 cosh 2 τ + 66 cosh 4 τ − cosh 6 τ )+4( d ˜ c Θ + d ˜ s Θ )(198 sinh τ −
55 sinh 3 τ + 3 sinh 5 τ ) (cid:1) ,F = ( d ˜ c Θ − d ˜ s Θ )(462 −
495 cosh 2 τ + 66 cosh 4 τ − cosh 6 τ )+4( d ˜ c Θ + d ˜ s Θ )(198 sinh τ −
55 sinh 3 τ + 3 sinh 5 τ )+ csch τ (cid:0) ( d ˜ c Θ + d ˜ s Θ )( −
462 + 495 cosh 2 τ −
66 cosh 4 τ + cosh 6 τ )+4( d ˜ c Θ − d ˜ s Θ )(198 sinh τ −
55 sinh 3 τ + 3 sinh 5 τ ) (cid:1) , and ¯ c Θ = cos (cid:16) Θ ε (9 sinh τ + sinh 3 τ ) (cid:17) , ¯ s Θ = sin (cid:16) Θ ε (9 sinh τ + sinh 3 τ ) (cid:17) , ˜ c Θ = cos (cid:16) Θ ε (9 sinh τ + sinh 3 τ ) (cid:17) , ˜ s Θ = sin (cid:16) Θ ε (9 sinh τ + sinh 3 τ ) (cid:17) . We observe that the total derivative of the Duffing Hamiltonian hasbecome dH D dτ = ε M + ε M + O ( ε ) . At this point we need to compute the integrals of M and M for τ between −∞ and ∞ . After performing the change of variable z =sinh τ (note that z (cid:48) = cosh τ >
0, hence the change is well-defined),simplifying the resulting expressions and dropping the odd terms with
ESTRICTED N -BODY PROBLEMS 17 respect to z as their respective integrals are zero, we arrive at M ( s ; Θ , ε ) = (cid:90) ∞−∞ ˜ M dz = ± F (Θ , ε )( c sin 2 s − c cos 2 s ) , (28)where ˜ M ( z ) = M ( τ ( z )) / √ z + 1 and F (Θ , ε ) = (cid:90) ∞−∞ z + 1) (cid:16) z − z + 1) cos( Θ ε z ( z + 3))+ z (3 z − z + 11) sin( Θ ε z ( z + 3)) (cid:17) dz. (29)We have passed from τ to s as the resulting expressions are mucheasier expressed in terms of them. From now the Melnikov function M will be written in terms of s .In order to obtain the integral of M with respect to τ we apply asbefore the change z = sinh τ , discard those terms with zero integraland simplify, ending up with M ( s ; Θ , ε ) = (cid:90) ∞−∞ ˜ M dz = ± (cid:0) F , (Θ , ε )( d cos s − d sin s )+ F , (Θ , ε )( d cos 3 s − d sin 3 s ) (cid:1) , (30)with ˜ M ( z ) = M ( τ ( z )) / √ z + 1 and F , (Θ , ε ) = (cid:90) ∞−∞ z + 1) (cid:16) (9 z −
1) cos( Θ ε z ( z + 3))+2 z (2 z −
3) sin( Θ ε z ( z + 3)) (cid:17) dz, F , (Θ , ε ) = (cid:90) ∞−∞ z + 1) (cid:0) (27 z − z + 69 z − ×× cos( Θ ε z ( z + 3))+2 z (2 z − z + 60 z −
11) sin( Θ ε z ( z + 3)) (cid:17) dz. (31)Higher order terms, say M , M , etc. are obtained in a similar wayas M and M . An important feature is that ε M is the dominantterm, the second one is ε M and so on. However, the ordering of thefunctions M k is different, due to the contribution of ε inside the ar-guments of the trigonometric functions. Indeed, this ordering dependsnow on the harmonics cos ks , sin ks , in such a way that the dominantterms are those corresponding to cos s , sin s , then those of cos 2 s , sin 2 s and so on as we shall see in the Appendix. As a consequence,the terms in M containing cos s , sin s represent the leading termsin the approximation in ε of the Melnikov function of the restricted N -body problem (6) related to parabolic motions of the infinitesimalparticle near infinity.It is straightforward to check that the relevant factor of the termsrelated to cos s , sin s always arise through expressions of the form ε l +1) M l +1) , l ≥ d ( l )2 cos s − d ( l )1 sin s with (cid:40) d ( l )1 = (cid:80) N − j =1 m j a j ( a j + a j ) l ,d ( l )2 = − (cid:80) N − j =1 m j a j ( a j + a j ) l . We state the following result.
Theorem 5.1.
There exists ε > such that for any ε with ε ≤ ε (cid:28) the stable and unstable manifolds of the periodic orbit γ related toHamiltonian (6) intersect transversally if one of the following situationsis given: i) d or d do not vanish; ii) d = d = 0 and there exists l ≥ such that d ( l )1 or d ( l )2 do not vanish; iii) all the previous terms arezero and c or c do not vanish; iv) all the preceding terms are zero andthere is a non-null constant term accompanying cos ks or sin ks forsome k ≥ .Proof. In the Appendix, using an argument of Sanders [21] we provethat R can be maintained small enough in regions of the phase spacethat include parts of the stable and unstable manifolds of γ such thattransversality is satisfied. More precisely, we can control the size of R for all | τ | ≥ K > K a constant independent of ε , keepingit smaller than the dominant terms of M . Then we prove that theleading terms of the Melnikov function are those factorizing cos s ,sin s , then those factorizing cos 2 s , sin 2 s and so on. Therefore, themost influential term in the Melnikov function is the one appearing in ε M , that is ( d cos s − d sin s ) F , (Θ , ε ). As F , does not vanishfor Θ (cid:54) = 0 and ε ≥ ε small enough, we focus on the analysis of thepossible zeros of the factor f ( s ) = d cos s − d sin s . Multiple rootsof f ( s ) = 0 occur only when d = d = 0. In this case we consider thenext most important term of the Melnikov function, that is, ε M .Its corresponding constant terms are d (2)1 , d (2)2 and they always appearin the form d (2)2 cos s − d (2)1 sin s , and similarly for l >
2. Thus, itis enough that there is a non-null coefficient d ( l ) j , j = 1 , l ≥ s , sin s vanish, we take into account the next main terms, that ESTRICTED N -BODY PROBLEMS 19 are those of ε M . They appear in the form c cos 2 s − c sin 2 s , thusit is enough that c or c do not vanish to get transversality. When c = c = 0 we need to consider the next terms related to cos 2 s ,sin 2 s , and these terms appear in ε M . We continue the procedureuntil we identify a non-null constant term accompanying cos ks orsin ks , concluding the transversality of the manifolds in that case. (cid:3) In Section 6, more specifically when we will deal with the circularrestricted 3-body problem and with the polygonal restricted N -bodyproblem, we shall see how higher-order terms of the Melnikov functionare needed to establish the transversality of the stable and unstablemanifolds of the parabolic periodic orbits.6. Applications
Restricted circular 3-body problem.
We study the planarcircular restricted 3-body problem. In this example the position andmass parameters can be chosen as a = 1 − µ, a = 0 , a = − µ, a = 0 , m = µ, m = 1 − µ, where 0 < µ ≤ .With these values, the perturbation parameters appearing in (24)read as: d = 3 µ (1 − µ )(1 − µ ) , d = 0 . Then, according to Theorem 5.1, when µ < / ε > γ related toHamiltonian (4) intersect transversally.For µ = 1 / s , sin s . In this case we note that the parameters d ( l )1 , d ( l )2 vanish for all l ≥
2, then we calculate c and c , arriving at c = 34 , c = 0 . Thus, Theorem 5.1 applies and the stable and unstable manifolds ofthe periodic orbit γ intersect transversally.With the approach described above we have completed the analysisof the parabolic orbits for the circular restricted 3-body problem forany µ ∈ (0 , / Equilateral restricted 4-body problem.
In this example thethree massive particles form an equilateral triangle, therefore a centralconfiguration, thus (3) is satisfied. The parameter values are: a = (1 − m − m ) , a = √ (1 − m ) ,a = (2 − m − m ) , a = − √ m ,a = − ( m + 2 m ) , a = − √ m ,m = 1 − m − m , where m , m > m + m < d i of (24) take the following values: d = ( m + 2 m − m + 2 m + 2 m m − m − m ) ,d = − √ m (2 m + 2 m + 2 m m − m − m + 1) , Therefore the transversal intersection of the manifolds is establishedas Theorem 5.1 applies, except for the case m = m = 1 /
3, which isthe only case for which d = d = 0 in the allowed region of the massparameters.When m = m = 1 / d ( l )1 , d ( l )2 vanishfor all l ≥
2. Moreover, c = c = 0 and realize that all terms accom-panying cos 2 s , sin 2 s vanish as well. Thus, we consider the leadingterms of cos 3 s , sin 3 s and get d = 0 , d = 53 √ . Thence Theorem 5.1 is applied, accomplishing the intersection of themanifolds of the parabolic orbits.The case m = m = 1 / N -body problem that will be addressed inSubsection 6.5.6.3. Restricted rhomboidal 5-body problem.
We consider thecase where masses m to m are equal by pairs and form a convexpolygon, a rhombus, see [2] and references therein. The parametersthat define the problem are as follows: a = − x, a = 0 , a = 0 , a = y,a = x, a = 0 , a = 0 , a = − y,m = m = µ, m = m = − µ, ESTRICTED N -BODY PROBLEMS 21 where 0 < µ < / x, y >
0. To get a central configuration theparameters µ , x , y must be related. For this purpose we impose thatEqs. (3) are satisfied. It is convenient to introduce two parameters a, b > x = a √ a + b (cid:18) a b − ( a + b ) a b − ( a + b )( a + b ) / (cid:19) / ,y = b √ a + b (cid:18) a b − ( a + b ) a b − ( a + b )( a + b ) / (cid:19) / , then a central configuration occurs provided µ is taken as µ = a (cid:0) b − ( a + b ) / (cid:1) a b − ( a + b )( a + b ) / ) . To ensure that µ ∈ (0 , /
2) we must restrict a , b so that 0 < b < √ a < b .In terms of x , y , the perturbation parameters in (24) are identicallyzero. Moreover the coefficients d ( l )1 , d ( l )2 also vanish for all l ≥
2, thuswe need to obtain c , c . We get c = − y + 6 µ ( x + y ) ,c = 0 . When c is non-zero Theorem 5.1 applies and we get the transversalityof the manifolds. For c = 0 we should go to higher orders in ε . Thecoefficient c vanishes in three cases: (i) If a = b , then µ = 1 /
4, whichcorresponds to the square configuration, i.e. the polygonal restricted 4-body problem, that will be treated in the next subsection; (ii) when a =1 . ...b ; and (iii) the reverse case to (ii), i.e. a = 0 . ...b .The last two values come as the only (two) real roots of the 14-degreehomogeneous polynomial equation given by a + 2 a b + 6 a b + 10 a b + 17 a b + 22 a b − a b − a b − a b + 22 a b + 17 a b + 10 a b + 6 a b + 2 ab + b = 0 . According to Theorem 5.1, the next terms that have to be checkedfor the three cases pointed out above are the coefficients of cos 2 s ,sin 2 s that appear in the function ε M . Specifically, we have cal-culated ∓ ε Θ − F , (Θ , ε )( c (2)2 sin 2 s − c (2)3 cos 2 s ) where F , is afunction of order ε − / exp( − / (3 ε )) when Θ > ε − exp(2Θ / (3 ε )) when Θ <
0. Besides c (2)2 , c (2)3 are coefficients de-pending on the masses and positions. We get c (2)3 = 0 for the threecases, c (2)2 = 0 . ... for case (ii) and c (2)2 = − . ... for case (iii). Thus we can conclude the transversality of manifolds. How-ever in case (i) one has c (2)2 = 0. This is the case a = b , that correspondsto a particular situation of the polygonal restricted N -body problemdealt with in Subsection 6.5.6.4. Collinear restricted N -body problem. In the following wepresent two examples where the N − a k = 0 for k = 1 , . . . , N − c of (23) always vanishes.6.4.1. Collinear restricted -body problem. Let us consider a collinearconfiguration of seven bodies with equal masses that are placed in asymmetric configuration with respect to the origin on the horizontalaxis. 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 polynomi-als we find that a collinear configuration occurs if the parameters arechosen as follows: a = − . ..., a = − . ...,a = − . ..., a = 0 ,a = − a , a = − a , a = − a ,a k = 0 , m k = , for k = 1 , . . . , . The parameters d , d of (24) vanish. Furthermore the coefficientsassociated to the higher order terms of the harmonics cos s , sin s , i.e. d ( l )1 , d ( l )2 are all zero for all l ≥ c = 1 . ..., c = 0 . Thus the conditions of Theorem 5.1 are satisfied. Thence, for ε smallenough with ε ≥ ε >
0, the stable and unstable manifolds of theperiodic orbit γ related to Hamiltonian (4) intersect transversally.6.4.2. Collinear restricted -body problem. Here we consider ten parti-cles in collinear configuration placed at equidistant positions. Then, wecalculate the masses so as to obtain a central configuration. For that,we impose that Eqs. (3) are satisfied and solve the resulting system of
ESTRICTED N -BODY PROBLEMS 23 linear equations to get a = − . ..., a = − . ...,a = − . ..., a = − . ...,a = − . ...,a = − a , a = − a , a = − a , a = − a , a = − a ,a k = 0 for k = 1 , . . . , ,m = m = 0 . ..., m = m = 0 . ...,m = m = 0 . ..., m = m = 0 . ...,m = m = 0 . .... Note that, although here we put an approximation of the valuescorrect to eight decimal digits, we have obtained them using integerarithmetic.As in the previous case, the parameters d , d of (24) become zerowhile the coefficients d ( l )1 , d ( l )2 also vanish. Then we need to obtainthe leading terms of the harmonics cos 2 s , sin 2 s . We calculate thecoefficients (23) and get their values also exactly, an approximation ofthem accurate up to eight decimal places being: c = 1 . ..., c = 0 . Then, as c (cid:54) = 0, Theorem 5.1 is satisfied, concluding the transversalitycondition of the manifolds of γ .6.5. Polygonal restricted N -body problem. In this case particles1 to N − N − a k = Re(e πi k − N − ) , a k = Im(e πi k − N − ) , m k = 1 N − , for k = 1 to N − N ≥ N , we notice thatthe Hamiltonian function can be written in terms of symplectic polar coordinates in a rather compact way by H ε ( r, θ, R, Θ , t ) = ε (cid:18) (cid:18) R + Θ r (cid:19) − r (cid:19) − N − (cid:88) j =1 ε j +3 r j/ U j − ε N +1 r N (cid:16) V N − + W N − cos( N − t − θ ) (cid:17) + O ( ε N +2 ) , (32)where U j = (1 + ( − j + 2 cos( jπ/ j/ / π (Γ( j/ ,V N − = (1 + ( − N − )(Γ( N/ π (Γ( N/ / ,W N − = 2Γ( N − / √ π Γ( N ) , and Γ stands for the gamma function.An important feature of (32) is that terms of order higher than ε N +1 can depend on cos( N − t − θ ) but they are of smaller influence thanthe one of order ε N +1 .To obtain the Melnikov function we emphasize the convenience ofdeveloping H ε to order 2 N + 1 because the first appearance of θ occursat 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 dH D dτ = − Θ sinh 2 τ N − (cid:88) j =1 ε j U j j/ ( j + 2)( | Θ | cosh τ ) j +4 − ε N − N Θ N cosh N +1 τ ( N V N − sinh τ + W N − ( N sinh τ cos q ( s , τ ) ∓ ( N −
1) sin q ( s , τ )))+ O ( ε N − ) , with q ( s , τ ) = ( N − (cid:0) s ∓ τ / ± Θ ε (9 sinh τ + sinh 3 τ ) (cid:1) . Now, we observe that by the parity of the derivative with respect to τ , the only term with no zero integral in the derivative is that factorized ESTRICTED N -BODY PROBLEMS 25 by W N − , thus we introduce ε N − M N − = − ε N − N W N − Θ N cosh N +1 τ ( N sinh τ cos q ( s , τ ) ∓ ( N −
1) sin q ( s , τ )) . We perform the change z = sinh τ and define the Melnikov function as M N − ( s ; Θ , ε ) = − N W N − Θ N (cid:90) ∞−∞ z + 1) N +1 ( N z cos ˜ q ( s , z ) ∓ ( N −
1) sin ˜ q ( s , z )) dz, (33)with˜ q ( s , z ) = ( N − (cid:0) s ± Θ ε z ( z + 3) ∓ arcsinh z )) (cid:1) . In order to illustrate how the theory of this paper applies, we par-ticularize the calculations for two specific cases, namely N = 7 , N = 7 we get M ( s ; Θ , ε ) = ± F (Θ , ε ) sin 6 s where F (Θ , ε ) = (cid:90) ∞−∞ z + 1) (cid:16) p ( z ) cos( Θ ε z ( z + 3))+ p ( z ) sin( Θ ε z ( z + 3)) (cid:17) dz, with p ( z ) = 2(45 z − z + 4257 z − z + 2255 z − z + 3) ,p ( z ) = z (7 z − z + 4785 z − z + 8217 z − z + 79) . For N = 8 the corresponding Melnikov function reads as M ( s ; Θ , ε ) = ± F (Θ , ε ) sin 7 s where F (Θ , ε ) = (cid:90) ∞−∞ − z + 1) (cid:16) p ( z ) cos( ε z ( z + 3))+ p ( z ) sin( ε z ( z + 3)) (cid:17) dz, - - Θ ϵ F12 - - Θ ϵ F14
Figure 2.
On the left: graph of F ; on the right, graphof F .with p ( z ) = 119 z − z + 23023 z − z + 37037 z − z +749 z − ,p ( z ) = 2 z (4 z − z + 5278 z − z + 24024 z − z +1638 z − . From the expressions of both Melnikov functions it is clearly deducedthat the equations M = 0 and M = 0 have simple roots provided,respectively, F and F do not vanish. However F , F are of thesame type as the functions F , F , , F , of Section 5 analyzed in theAppendix. Then they have a global maximum and a global minimum,their graphs cut the horizontal axis at a few points and tend asymp-totically to zero as long as Θ tends to ±∞ , see also Figure 2.The asymptotic estimates of the Appendix hold provided ε is smallenough so that F and F do not vanish. Thus, for the polygonalrestricted 7- and 8-body problems, the stable and unstable manifoldsof the parabolic orbits γ intersect transversally.Regarding the case N = 4, corresponding to the equilateral restricted4-body problem of Subsection 6.2 when m = m = m = 1 /
3, theMelnikov function obtained from (33) is a particular case of the function M in (30), thus we can achieve the transversality from the analysisof this subsection. When N = 5, the square configuration studied inSubsection 6.3 is analyzed using the Melnikov function (33) which isof order eight in ε . Proceeding similarly to what we did for N = 7 , M has simple roots provided, thus extending theanalysis performed in Subsection 6.3 when a = b .Examining (33) carefully, it is not difficult to infer that the integralsappearing in the Melnikov function M N − are of the same type as F , ESTRICTED N -BODY PROBLEMS 27 F , and so on. We also take into account that the smallest harmonicappearing in the Melnikov function is sin( N − s , thus we end upwith an expression like M N − ( s ; Θ , ε ) = ± K Θ N F N − (Θ , ε ) sin( N − s , with K a non-null constant. Applying the estimates provided in theAppendix and taking ε small enough, we conclude that ε N − M N − behaves like ε − N − / exp( − ( N − / (3 ε ))(1 + O ( ε )) when Θ > ε − N +1 exp(( N − / (3 ε ))(1 + O ( ε )) for negative Θ . Thus, wecan apply Theorem 5.1, achieving the transversality of the manifoldsof γ in the polygonal restricted N -body problem for all N ≥ Appendix: Qualitative study of functions F (Θ , ε ) , F , (Θ , ε ) and F , (Θ , ε )The function F has been defined in (29) and its graph is givenin Figure 3. Considered as a function in z , the integrand is smoothin R and bounded by above in the intervals in ( −∞ , − ∪ [1 , ∞ ) bythe improperly integrable function 80 / | z | . Calling ˜Θ = Θ /ε , thecomparison test for improper integrals implies that F ( ˜Θ ) is a welldefined smooth function for all ˜Θ ∈ R . In addition it has two zeroes,namely ˜Θ (cid:48) = 0 and ˜Θ ∗ = 0 . ... , and F ( ˜Θ ) < < ˜Θ ∗ (excepting at ˜Θ (cid:48) ), while F ( ˜Θ ) > > ˜Θ ∗ . The function F takes a global maximum and a global minimum values. Furthermorelim ˜Θ →±∞ F ( ˜Θ ) = 0 . - - Θ ϵ F Figure 3.
The graph of the function F (Θ , ε ).The functions F , , F , were introduced in (31) and present an anal-ogous behavior to the function F , as it can be seen in Figure 4. Thecorresponding improper integrals are absolutely convergent. Specifi-cally, F , = 0 has its unique root at ˜Θ = 0 whereas the roots of - - - Θ ϵ - - - - - Θ ϵ - - - - - Figure 4.
The graphs of the functions F , (Θ , ε ) and F , (Θ , ε ). F , = 0 occur at ˜Θ = 0, ˜Θ = 0 . ... , ˜Θ = 0 . ... .Besides, F , > < F , < > F , > < . ... < ˜Θ < . ... , F , < < ˜Θ < . ... and ˜Θ > . ... . As F , the functions F , , F , take a global maximum as well as a global minimum. Finally,lim ˜Θ →±∞ F , ( ˜Θ ) = lim ˜Θ →±∞ F , ( ˜Θ ) = 0 . We remark that similar integrals have been analyzed and can befound in [12, 13]. In fact, due to the highly oscillatory character of theintegrals, an asymptotic analysis of the functions F , F , , F , andother related functions appearing in the Melnikov functions obtainedin Section 5 has to be done.Following [13] we introduce the improper integrals I k ( δ ) = (cid:90) ∞ cos( δ ( z + z / z ) k dz, J k ( δ ) = (cid:90) ∞ z sin( δ ( z + z / z ) k dz. In [12] it is proved that J k can be written in terms of I k through J k +2 ( δ ) = δ k + 1) I k ( δ ) , whereas for δ > I n − ( δ ) = exp( − δ/ (cid:18) π n +1 (2 n − δ n − + O ( δ n − / ) (cid:19) ,I n ( δ ) = exp( − δ/ (cid:18) √ π n +1 (2 n − δ n − / + O ( δ n − ) (cid:19) . (34)The functions F , F , , F , as well as the rest of the functions ap-pearing in M k with k > I k and J k , after ESTRICTED N -BODY PROBLEMS 29 performing a partial fraction decomposition to each of the rationalparts of them.We start with Θ > ε small enough with ε ≥ ε > F , F , , etc. Concerning M we apply the estimates (34) to F , and after arranging the functionconveniently, we conclude that ε M = √ π ε − / Θ / e − ε ( c sin 2 s − c cos 2 s )(1 + O ( ε )) . For M we get ε M = − √ π √ ε − / Θ − / e − Θ303 ε ( d cos s − d sin s )(1 + O ( ε )) − √ π √ ε − / Θ / e − Θ30 ε ( d cos 3 s − d sin 3 s )(1 + O ( ε )) , and similar expressions are obtained for M , M , and so on.From the above calculations it is clear that for ε small enough themost important terms are those related to exp( − Θ / (3 ε )), then thoserelated to exp( − / (3 ε )), next those with exp( − Θ /ε ) and so on.Furthermore, we observe that the terms with asymptotic estimateshaving the factor exp( − Θ / (3 ε )) correspond to the harmonics cos s ,sin s , and in general, the asymptotic expressions with exp( − k Θ / (3 ε ))are related to the harmonics cos ks , sin ks . In addition to this, for k ≥ ks , sin ks are of order ε − k − / exp( − k Θ / (3 ε ))(1 + O ( ε )) while for k = 1 the terms of cos s , sin s have the estimate ε − / exp( − Θ / (3 ε ))(1 + O ( ε )). This in turn implies that the leadingterms in the Melnikov function are the ones depending on the coeffi-cients d , d , the next ones those with estimate ε − / exp( − Θ / (3 ε ))to which follows the rest of terms with harmonics cos s , sin s . Nextwe consider the main terms factorized by cos 2 s , sin 2 s , they are theones depending on c , c . We continue with the higher order terms andso on. The Melnikov function becomes the formal Fourier series M ( s ; Θ , ε ) = ∞ (cid:88) k =1 α k ( ε ) cos ks + β k ( ε ) sin ks with α ( ε ) = ε − / e − Θ303 ε ( A + O ( ε )) , β ( ε ) = ε − / e − Θ303 ε ( B + O ( ε )) ,α k ( ε ) = ε − k − / e − k Θ303 ε ( A k + O ( ε )) , β k ( ε ) = ε − k − / e − k Θ303 ε ( B k + O ( ε )) , for k ≥ A = − √ π √ Θ − / d , B = √ π √ Θ − / d ,A = √ π Θ / c , B = − √ π Θ / c ,A k , B k , k ≥ ε. Regarding the estimate of R we use the ideas of Sanders [21] for thecase of exponentially small estimates. Given a vector field of the form˙ x = f ( x ) + εf ( x, t, ε ) with x ∈ D ⊂ R and ε a small parameter, hedefines the Melnikov integral ∆ ε ( t, x ) = ε − f ( x s,u ( t )) ∧ ( x uε ( t ) − x sε ( t ))where ∧ is the wedge product in R , x s,u refers to the parametrizationof the stable and unstable manifolds of the unperturbed system and x uε , x sε denote solutions on the unstable and stable manifolds, respec-tively. After some assumptions on the smoothness of the vector field,and a lemma regarding the relationships between the unperturbed andperturbed manifolds, Sanders arrives at an expression of the form∆ ε ( t, x ) = ∆ ( x ) + O ( ε (1 + e − µ | t | ) min { , e − µ | t | } ) , where ∆ stands for the usual Melnikov function and µ is the Lipschitzconstant associated to f . Systems of the type ˙ x = εf ( x, t, ε ), afterapplying averaging and rescaling time by τ = εt , are transformed into y (cid:48) = dy/dτ = f ( y )+ εf ( y, τ /ε, ε ), thus admitting the estimate O ( ε (1+exp( − µ | τ | /ε )) min { , exp( − µ | τ | /ε ) } ) = O ( ε exp( − µ | τ | /ε )) for τ (cid:54) = 0.We adopt Sanders’ point of view in our setting as follows. First, wenotice that the hypotheses on the Hamiltonian and the existence of themanifolds are fulfilled. Then we realize that t and τ are related throughthe change of time, assuming that x ( τ ) = √ / Θ sech τ + O ( ε ) we get t = Θ τ / (2 ε ) + O (1). The Lipschitz constant of H D along the homo-clinic ξ ( τ ) is calculated, obtaining µ = √ / Θ . Finally, after adjustingthe factor ε in the whole expression of ∆ ε so that we identify ∆ with M , we get an upper bound on R as O (exp( − Θ | τ | / ( √ ε ))). To con-trol the size of R we compare the estimate with the dominant term of M . When d or d are not zero, due to the presence of the factor ε − / in α , β , it is enough that exp( − Θ / (3 ε )) > exp( − Θ | τ | / ( √ ε ))from where it is deduced that | τ | ≥ √ Θ , which is true in big por-tions of the manifolds as Θ is of moderate size. Next, the transver-sality condition is verified in the part of phase space where this re-striction on τ holds, but it implies that transversality is satisfied forevery τ as this property is preserved through diffeomorphisms. When d = d = 0 one compares exp( − k Θ / (3 ε )) with exp( − Θ | τ | / ( √ ε )),starting with k = 2. ESTRICTED N -BODY PROBLEMS 31 When Θ < I k , J k with δ <
0. Then we notice that I k ( δ ) = I k ( − δ ), J k ( δ ) = − J k ( − δ ) and J k +2 ( δ ) = δ/ (2( k + 1)) I k ( − δ )and the estimates for I k given in (34) apply replacing δ by − δ in theexpressions. Proceeding similarly to the case Θ > ε M = π ε − e ε ( c sin 2 s − c cos 2 s )(1 + O ( ε )) ,ε M = − π Θ − e Θ303 ε ( d cos s − d sin s )(1 + O ( ε ))+ π ε − Θ e Θ30 ε ( d cos 3 s − d sin 3 s )(1 + O ( ε )) . Reasoning as in the positive case we realize that the main term inthe Melnikov function is that of ε M having coefficients d , d , thenthe rest of terms related to cos s , sin s , next the term associated tocos 2 s , sin 2 s beginning with those whose coefficients are c , c , andso on. Moreover, when k ≥ ks , sin ks behave like ε − k exp( k Θ / (3 ε ))(1 + O ( ε )), thus we obtain analogous results to thecase Θ positive although with estimates of different order in ε .The estimate analysis of R is alike the procedure for Θ > Acknowledgements
We appreciate the comments of Prof. M. Guardia, P. Mart´ın andT.M. Seara who pointed out a crucial error in the previous versionof the paper and helped substantially in its improvement. The au-thors have received partial support from Project Grant Red de CuerposAcad´emicos de Ecuaciones Diferenciales, Sistemas Din´amicos y Estabi-lizaci´on. PROMEP 2011-SEP, Mexico and from Projects 2014–59433–C2–1–P of the Ministry of Economy and Competitiveness of Spainand 2017–88137–C2–1–P of the Ministry of Economy, Industry andCompetitiveness of Spain. The facilities provided by Cinvestav – IPN(Mexico) are also acknowledged.
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