Invariant manifolds of L_3 and horseshoe motion in the restricted three-body problem

Invariant manifolds of L3and horseshoe motion in the restricted three-body problem Esther Barrab´es(1) and Merc`e Oll´e(2) 17th May 2006 (1) Dept. Inform`atica i Matem`atica Aplicada, Universitat de Girona, Avd. Llu´ıs Santal´o s/n, 17071 Girona, Spain. E-mail: [email protected] (2) Dept. de Matem`atica Aplicada I, ETSEIB, Universitat Polit`ecnica de Catalunya, Diagonal 647, 08028 Barcelona, Spain. E-mail: [email protected] Abstract In this paper, we consider horseshoe motion in the planar restricted three-body problem. On one hand, we deal with the families of horseshoe periodic orbits (which surround three equilibrium points called L3,L4and L5), when the mass parameter µis positive and small; we describe the structure of such families from the two-body problem (µ= 0). On the other hand, the region of existence of horseshoe periodic orbits for any value of µ∈(0,1/2] implies the understanding of the behaviour of the invariant manifolds of L3. So, a systematic analysis of such manifolds is carried out. As well the implications on the number of homoclinic connections to L3, and on the simple infinite and double infinite period homoclinic phenomena are also analysed. Finally, the relationship between the horseshoe homoclinic orbits and the horseshoe periodic orbits are considered in detail. Keywords: periodic orbits, invariant stable and unstable manifolds, homoclinic orbits, restricted problem. 2On horseshoe motion 1 Introduction Over the past decades the interest in horseshoe periodic orbits (roughly speaking bananashaped orbits) arose when modelling the motion of co-orbital satellites, the most famous being Saturn’s co-orbital satellites Janus (1980S1) and Epimetheus (1980S3), whose existence was confirmed by Voyager flights to Saturn in 1981 (see [7], [8], [16] and references therein). More recently, several near-Earth asteroids have been found to move on horseshoe orbits (see [5] and [3]). The horseshoe motion and the dynamics of co-orbital satellites have been analysed both analytically and numerically using the three-body context. In the framework of singular perturbation theory the motion of co-orbital satellites may be approximated by two independent solutions of a two-body problem when they are far apart. However, this approximation breaks down when the distance of the two satellites becomes small, and a different two-body problem must be used instead. The complete description of the motion requires the matching of both solutions (see [21] and [23]). Cors and Hall [6] studied the same problem by introducing small parameters to the three-body equations, truncating higher order terms and deriving dynamical information from the resulting equations. From a numerical point of view, horseshoe periodic orbits have been explored as invariant objects using Hill’s problem (see [19]), the planar restricted three-body problem – RTBP– (as a simple model to describe Saturn’s coorbitals motion with the mass parameter µ= 3.5×10−9, see [15] and [16]) and the spatial RTBP (where some families of horseshoe periodic orbits were computed, see [1]). In this paper we consider the planar RTBP. It is known that this problem has five equilibrium points (in a rotating system of reference) called Li,i= 1, ..., 5 and a first integral called the Jacobi integral. We will denote by Cthe constant value of the Jacobi integral along a solution. Throughout the paper we study some open questions concerning horseshoe periodic orbits (HPO) not answered in previous papers. We focus our attention on periodic symmetric solutions of the RTBP in a rotating system. In this context, HPO are orbits that surround the equilibrium points L3,L4and L5, and have two orthogonal crossings with the horizontal synodical axis (see Figures 3, 7 and 8). Our contribution in this paper is threefold: (i) for a value of µfixed and small, and for a big interval of values of the Jacobi constant C, the understanding of the diagram of continuous families of HPO follows from the families of periodic orbits obtained from rotating circular and elliptical orbits in the µ= 0 case (in [16] only a mechanism to generate one set of isolated HPO for a fixed value of the Jacobi constant was given). From now on we call such families for µ= 0 generating families of HPO and we give their analytical expression. (ii) Many properties concerning the evolution of the families, for µ > 0 and small, as well as the shape of the HPO within a family, are now easily explained from the generating families of HPO. The diagram of the characteristic curves of the families of HPO is computed for µ= 0.0001 and we remark on the similarities and differences between this diagram and the corresponding one for µ= 0. However, the continuation of this diagram when varying µ > 0 is not straight forward. This E. Barrab´es and M. Oll´e 3 is mainly due to the existence of the stable and unstable invariant manifolds of the collinear equilibrium point L3. In particular the behaviour of such manifolds for C3=C(L3) and the existence of a finite or infinite number of HPO for this value C3are analysed in order to understand how the diagram of families of HPO evolves when increasing µ. (iii) Finally, we study the existence and location of horseshoe periodic orbits and horseshoe homoclinic orbits to L3for any value of µ∈(0,1/2]. Since the invariant manifolds of L3play a key role as a mechanism to generate HPO, we have done a detailed exploration of the behaviour of these manifolds when increasing µfrom 0 onwards. In particular, we study the existence of homoclinic horseshoe orbits to L3 (that is, an orbit that tends to L3in backward and forward time) when varying µ. For each homoclinic orbit there is an infinity of HPO tending to it. Furthermore, we observe that the set of values of µfor which simple homoclinic orbits exist, is a sequence tending to zero. And, tending to each simple isolated homoclinic orbit, a double sequence of double homoclinic orbits can be obtained. (See Section 4 for the explicit definition of simple and double homoclinic orbits.) As a final remark, we must point out that not only L3is important in the horseshoe dynamics, but also the other collinear equilibrium points L1and L2, and the equilateral ones, L4and L5(as well as the periodic orbits around them and their associated invariant manifolds), are also involved. We show some examples. In Section 2 we briefly recall the RTBP. Items (i) and (ii) are developed in Section 3 and item (iii) is done in Section 4. 2 The restricted three-body problem Let us consider a system of three bodies in an inertial (also called sidereal) reference system. Two bodies, called big and small primaries of masses 1 −µand µ,µ∈(0,1/2], (in suitable units), are describing circular orbits about their common centre of mass (the origin of coordinates) in a plane. The third body is a particle of infinitesimal mass which moves under the gravitational effect of the primaries but has negligible effect on their motion. The problem of the description of the motion of the particle is known as the circular restricted three-body problem (RTBP). With suitable units, such as the mean motion of the primaries is the unity, the equations of motion in a rotating (also called synodical) system of coordinates, where the big and small primaries remain fixed at positions (µ, 0) and (µ−1,0) respectively, are (see Szebehely [22]) x′′ −2y′=∂Ω ∂x , y′′ + 2x′=∂Ω ∂y , (1) where Ω(x, y) = 1 2(x2+y2) + 1−µ r1 +µ r2 +1 2µ(1 −µ), 4On horseshoe motion r2 1= (x−µ)2+y2and r2 2= (x−µ+ 1)2+y2are the distances between the particle and the big and small primaries respectively, and ′stands for d/dt. It is well known that these equations have the so called Jacobi first integral x′2+y′2= 2Ω(x, y)−C, (2) and satisfy the symmetry (x, y, x′, y′, t)→(x, −y, −x′, y′,−t).(3) There exist 5 equilibrium points: the collinear points, L1,L2and L3, with positions (xi,0), for i= 1,2,3, and the equilateral ones, L4and L5, located at (µ−1/2,±√3/2). If one computes the value of the Jacobi constant at the equilibrium points Ci=C(Li) for any value of µ∈(0,1/2), one has (see [22]) 3 = C4=C5< C3≤C1< C2, and C3=C1for µ= 1/2. Along the paper, the point L3and its invariant manifolds will play a key role. So we briefly recall that, according to the eigenvalues of the Jacobian matrix at L3, the collinear point L3(also L1and L2), are saddle-centre (that is, the eigenvalues are λ1=a > 0, λ2=−a < 0 and λ3,4=±bı, b∈R). In particular, L3has one-dimensional unstable and stable manifolds associated to the saddle. In particular, an eigenvalue vassociated with the eigenvalue λ1=a > 0 (λ1=a < 0) gives the tangent direction of the unstable (stable) manifold; and we shall distinguish between the two branches of the unstable (or stable) manifold according to vor −v. We will denote by Wu,1 L3,Wu,2 L3the two branches of the unstable manifold of L3and by Ws,1 L3,Ws,2 L3the two branches of the stable one. Actually only Wu,1 L3and Wu,2 L3have to be computed since Ws,1 L3and Ws,2 L3are obtained from the symmetry (3). We also recall that if a branch of the invariant unstable manifold crosses orthogonally the xaxis, that is, in a point (x, 0) in position and (0, y′) in velocity, then due to the symmetry (3), this unstable branch coincides with one of the stable branches, giving rise to a homoclinic orbit that tends asymtotically, in forward and backward time, to L3. Finally, the regions in the (x, y) plane where the motion of the particle is possible are bounded by the zero velocity curves (ZVC) given by the equation 2Ω(x, y)−C= 0. obtained from (2) (see Fig. 1). 3 Families of planar horseshoe periodic orbits According to [16], a planar horseshoe periodic orbit will be a periodic solution of the RTBP in which the particle follows a path which surrounds only the equilibrium points L3,L4and L5and has two orthogonal crossings with the xaxis. It is well known, using symmetry (3), that any solution with two orthogonal crossings with the horizontal axis E. Barrab´es and M. Oll´e 5 becomes symmetric, with respect to the xaxis, and periodic. We can consider that the orthogonal crossings occur at epochs t= 0 and t=T/2, with Tbeing the period. We denote the initial conditions (at t= 0) by (x0,0,0, y′ 0) (or in short (x0, y′ 0)) and the final conditions (at t=T/2) by (xf,0,0, y′ f) ((xf, y′ f)). So, the motion of an HPO will be considered only for t∈[0, T/2] (see Figures 3 right, 7 and 8). For greater clarity, some included plots of the periodic orbits only show a half-period of the motion and the dotted line represents the horizontal axis in the synodical reference system where the primaries are located. An initial question concerns the suitable region in the plane (x, y) to find horseshoe motion. It is clear from the zero velocity curves (see Figure 1) that the equilibrium points Li,i= 1,2,3,4,5, play a role here and that horseshoe motion is only possible for C < C1 (see [22] for the expansions in µof xiand Ci, for i= 1,2,3). Thus, we will begin our numerical explorations by using values of the Jacobi constant between C3and C1. -1.5 -1 -0.5 0 0.5 1 1.5 -1.5 -1 -0.5 0 0.5 1 1.5x y L4 L5 L1L2L3 m2m1 3<C<C3 C3<C<C1 Figure 1: Zero velocity curves for µ > 0. The motion is possible outside the region enclosed by the ZVC. Horseshoe motion takes place for values C < C1. Given a value of µ > 0 and small, a second question will refer to a mechanism that explains the existence of HPO from the µ= 0 case. In this sense, for µ6= 0 and for any HPO when t∈[0, T/2], we will distinguish between the outer solution, the piece of the HPO from t= 0 to the returning point (close to the small primary), and the inner solution from the returning point to t=T/2 (analogously for t∈[T/2, T]). The outer and inner solutions can be approximated by two different solutions of the rotating two-body problem when the infinitesimal mass is far from the small primary; in Figure Figure 2 we show two HPO, for µ= 0.0001, and the corresponding approximating rotating circular orbits (left) and rotating elliptical ones (right), for µ= 0. In Subsection 3.1, we study the families of periodic orbits of the two-body problem that give the outer and inner approximations of the HPO. We call such families generating families of the HPO for µ= 0. We also give analytic expressions that describe the families and the corresponding diagram in suitable variables. Next, we consider the families of HPO for µ > 0 and small (Subsection 3.2). First, 6On horseshoe motion -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 Figure 2: A whole HPO (continuous line) for µ= 10−4and the two-body orbits (discontinuous line) for µ= 0 approximating the outer and inner solutions. a method of numerical computation of such families is given. Secondly, we describe the diagram of characteristic curves (curves such that each point characterises an HPO) for a specific value of µ(0.0001) and we compare it with the diagram of continuous families of rotating ellipses and circular orbits for µ= 0. Finally, it is natural to consider the continuation of this diagram for increasing values of µ. However, this is not a straightforward task. The explanation comes from the behaviour of the invariant manifolds of the equilibrium point L3. In Subsection 3.3, we observe that the loops that appear in the (x, y)-projection of the invariant manifolds and the existence of homoclinic orbits to L3(orbits that tend to this point in backward and forward time) play an important role in order to describe the continuation of these diagrams. 3.1 Generating families of HPO for µ= 0 When µ= 0 we consider the sidereal orbits described by the particle around the big primary located at the origin, as natural candidates to approximate the outer and inner solutions of a horseshoe orbit. We distinguish between a circular sidereal orbit (which becomes also a synodical circular one) or a synodical orbit coming from a rotating sidereal ellipse. In order to characterise the families of sidereal circular and elliptical orbits, we consider the well known relation C 2+h=M, (4) with Cthe Jacobi constant, hthe energy and Mthe angular momentum with sign: positive for direct orbits and negative for retrograde ones. As M=±pa(1 −e2) and h=−1/2a(a,ebeing the semimajor axis and the eccentricity) we obtain C 2−1 2a=±pa(1 −e2),(5) E. Barrab´es and M. Oll´e 7 or equivalently e2+1 4aµC−1 a¶2 = 1.(6) First, we consider the circular orbits. Substituting e= 0 in Equation (5), we obtain C=1 a±2√a(7) with the plus sign (the l−ifamilies in the Str¨omgren’s notation, see for example ) for sidereal direct orbits and the minus sign (the h−mfamilies) for sidereal retrograde ones. Each such orbit will have as initial synodical position the point (x, 0), with x=asatisfying Equation (7). In Figure 3, both curves are represented, as well as the ZVC obtained from Equation (2) considering µ= 0, i.e. x2+ 1/x =C. In order to approximate an HPO, we are interested in the two circular orbits of families l−iclose to the ZVC, for a value of C > 3 fixed (see Figure 3). Two such circular orbits will approximate the outer and inner solutions of an HPO for small µ > 0 (see Figure 2 left). Because of this, this family is called the generating (or also approximating) family of circular orbits. It should be noted that for any sidereal orbit we have, from Kepler’s third law, that n2a3= 1, where nand n−1 are the sidereal and synodical mean motions respectively If n < 0, the orbit is retrograde in both reference systems, while for 0 < n < 1 (a > 1) the orbit is sidereal direct but synodical retrograde and for n > 1 (a < 1) direct in both systems. Therefore, the outer and inner solutions of an HPO are approximated by a retrograde and a direct synodical circular solution respectively. 3 zero velocity curve retrograde orbits direct orbits 0 1 3 a 42 Figure 3: Plane (x, C); the ZVC and the two families of sidereal circular orbits (of radius x=a) of the problem of two bodies given by equation (7). Circular orbits are always symmetric and periodic in both sidereal and synodical reference systems. However, in order for a rotating ellipse to be symmetric and periodic in the synodical system, a rational mean motion n=Q/P is necessary (or equivalently the semimajor axis ahas to verify the condition a= (P/Q)2/3), for P, Q ∈Nrelatively primes), and the two orthogonal crossings in the synodical xaxis must take place at a pericentre or an apocentre of the sidereal orbit. 8On horseshoe motion So, let x > 0 be the value of the synodical xcoordinate of the initial orthogonal crossing. Then x=a(1 + e) if it is at an apocentre or x=a(1 −e) at a pericentre. In both cases, e2=(x−a)2 a2and Equation (6) becomes (x−a)2 a2+1 4aµC−1 a¶2 = 1,(8) which gives a curve the (x, C) plane that, for a fixed value of a, is an ellipse itself with centre (a, 1/a) and semimajor axis aand 2√a. So, for fixed a, the points in the ellipse of Equation (8) give a family of elliptical orbits and each of these points represents an elliptic orbit with semimajor axis aand eccentricity e=|x−a|/a (see Figure 4 left). If x > a (respectively x < a) the orthogonal crossing takes place at the apocentre (resp. pericentre). When x=a, we have a circular orbit (e= 0), and for x= 0 or x= 2a, we get a degenerate ellipse (e= 1) (see Figure 4 right). Also, from Equation (5) and for a fixed value of a, we distinguish between direct and retrograde orbits according to the value of the Jacobi constant: C∈(1 a−2√a, 1 a) sidereal retrograde orbits, C∈(1 a,1 a+ 2√a) sidereal direct ones. Both intervals correspond to the upper and lower semiellipses of the associated ellipse (labelled by a). Again if a < 1 (respectively a > 1) the orbit becomes synodical direct (retrograde). We also note that for any fixed value of a, the corresponding family of elliptical orbits given by Equation (8) intersects the family of circular orbits at two points (see Figure 4 left): (a, 1 a+ 2√a) corresponding to a circular orbit and (x, C) that corresponds both to a circular orbit with radius xand to an ellipse with x=a(1 + e) or x=a(1 −e). A deeper analysis shows that, if a < 1 the initial orthogonal crossing xof the ellipse is at an apocentre while for a > 1 the initial xis at a pericentre. Furthermore, for a < 2−5/3the orbit is sidereal retrograde whereas for 2−5/3< a the orbit is sidereal direct. So we have two different orbits (circular and elliptic) with the same initial condition (x, 0), but with different sign in the initial velocity y′(see also Subsection 3.2 for details). We note that the sign of the velocity does not determine the sense of the synodical orbit: the velocity may be negative for a direct synodical orbit, due to the existence of loops (see for example Figures 7 and 8). As reasoned above, the solutions of Equation (8) will be called the generating families of rotating ellipses. Finally, we remark that Equations (7) and (8) may be regarded as the equations of the characteristic curves of families of symmetrical periodic orbits for µ= 0. For a fixed a, a point of the curve in the plane (x, C) gives rise to the synodical initial condition (x, 0,0, y′), with y′=±v−x(plus or minus sign if the sidereal orbit is direct or retrograde respectively), v=q1±e a(1∓e), and x=a(1 ∓e), where the up (down) sign is taken if the particle is at the pericentre (apocentre). Such characteristic curves will be meaningful when studying the HPO for small µ > 0 in Subsection 3.2. E. Barrab´es and M. Oll´e 9 forbiden region circular orbits 2 4 0 2 4 x 1/a a apocenters pericenters direct orbits retrograde orbits Figure 4: Left. Families of sidereal elliptical orbits in the plane (x, C). Right. One family given by Equation (8) with a fixed a. Points (a, 1/a ±2√a) represent circular orbits. 3.2 Families of HPO for µ > 0and small The aim of this Subsection is to compare the diagram of characteristic curves of families of HPO for µ > 0 and small with the generating families for µ= 0, and to derive some properties of the families for µ > 0. Thus, in this Subsection, we fix a value of µ > 0 and small, for example µ= 10−4. We want to obtain, numerically, families of symmetrical horseshoe periodic orbits. Each periodic orbit is completely determined by its initial condition (x0, y′ 0), and a family can be represented by the set of the initial conditions of its periodic orbits. Thus, each family can be represented by a curve (characteristic curve) for example in the (x0, y′ 0)-plane, or in the (x0, C)-plane, Cobtained from Equation (2). The last, being our choice. In order to compute the families of HPO, we take into account that a family of periodic orbits with initial conditions (x0,0,0, y′ 0) is defined implicitly by the equation x′(T/2, x0, y′ 0) = 0 where T=T(x0, y′ 0) is given by the Poincar´e section y(T/2, x0, y′ 0) = 0. The numerical continuation of the family has been done using the arc step method which we will outline. A family of periodic orbits is regarded as a curve parameterised by the arc parameter s, that is, p(s) = (x0(s), y′ 0(s)) such that x′ f(p(s)) = 0, where x′ fstands for the value of x′when the first (or k-th) crossing with the xaxis takes place. Using the fact that the curve p(s) satisfies a suitable system of differential equations, we predict the successive points on the curve by a (low) order Adams-Bashforth method and we refine them by using a modified Newton’s method (see for example [1], [2] or [20] for details). Of course an initial point (seed) of each family must be taken. To obtain it, we fix a value of C, and for any value x0(greater that x3=x(L3) if C≥C3) we integrate the equations of motion until the first (or k-th) crossing with the horizontal axis is reached. At this point, yf= 0, but in general x′ f6= 0 (is not an orthogonal crossing). Then we 16 On horseshoe motion (1) (2) (3) -3 -2 -1 0 1 2 3 1.025 1.026 1.027 1.028 1.029 1.03 1.031 -0.002 -0.0015 -0.001 -0.0005 0 0.0005 1.026 1.0265 1.027 1.0275 1.028 1.0285 1.029 -0.002 -0.0015 -0.001 -0.0005 0 0.0005 1.026 1.0265 1.027 1.0275 1.028 1.0285 1.029 (4) (5) -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.95 0.96 0.97 0.98 0.99 1 1.01 1.02 1.03 Figure 9: Example of a bifurcation family of double period for µ= 0.0001. (1) Curve (x0, sp) of a family of HPO with a bifurcation orbit with sp=−2; (2) for C= 3.0007512 fixed (corresponding to the bifurcation orbit), the curve (x0, x′ f) at first crossing; (3) for C= 3.0007506 fixed, (x0, x′ f) curve at second crossing; (4) example of a bifurcating HPO of double period in the (x, y) plane for t∈[0, T/2]; (5) a detail of the HPO: not the first but the second crossing is orthogonal (at t=T/2) (this plot is not scaled in order to observe this fact). synodical axis, for x0increasing from x3+δ,δ > 0 very small, on. Let us compare the obtained curves for increasing values of µ. For µ= 0.0001, we see in Figure 10 top left, that there exist many values of x0for which x′ f= 0. Each of these values gives the initial condition (y′ 0obtained from the value C3) of an HPO with two crossings with the horizontal axis in one period. We can also see many intervals for which there are jumps or discontinuities in the curve. In each one there exists a value x0for which x′ f= 0 not at the first cross, but at the second one. This value corresponds to an orbit whose projection in the (x, y) plane has a loop close to the first crossing (see Figure 11), and therefore such x0corresponds to an HPO with 4 crossings with the y= 0 line in one period. However, we might say that the (x0, x′ f) curve for µ= 0.0001 is quite regular (in the sense of being an increasing sinus type curve, say). But when we increase µa little bit, µ= 0.00015, a different behaviour is obtained for the corresponding (x0, x′ f) curve (see Figure 10 top right) although there are still values of x0such that x′ f(x0) = 0, which correspond to HPO with two crossings. For µ= 0.0005 and the corresponding (x0, x′ f) curve we remark that E. Barrab´es and M. Oll´e 17 there are no HPO with only two crossings in one period for any value x0in the interval (x3,1.00024) (see Figure 10 bottom left). Thus, the number of intersections with the horizontal axis can change from one value of µto another one, and this will be a problem when doing the continuation varying the mass parameter. However, for bigger values of µ, we remark that we recover again the regular behaviour of the (x0, x′ f) curve, although the whole curve moves up or down when varying µ. In Figure 12 we plot the curve (x0, x′ f) at first crossing for µ= 0.00697485 (left), µ= 0.007 (middle) and µ= 0.008 (right). In the first case, there seems to be an infinite set of points with x′ f= 0 (this will be confirmed later, in Section 4), whereas for the other two values of µwe have just a finite one. -1e-005 -5e-006 0 5e-006 1e-005 1.00004 1.00005 1.00005 1.00006 1.00006 1.00007 -1e-006 -8e-007 -6e-007 -4e-007 -2e-007 0 2e-007 4e-007 6e-007 8e-007 1e-006 1.00006 1.00006 1.00006 1.00006 1.00006 1.00006 1.00006 -0.0002 -0.00015 -0.0001 -5e-005 0 5e-005 0.0001 0.00015 0.0002 1.00021 1.00022 1.00022 1.00023 1.00023 1.00024 1.00024 1.00025 1.00025 -0.0002 -0.00015 -0.0001 -5e-005 0 5e-005 0.0001 0.00015 0.0002 1.00043 1.00044 1.00045 1.00046 1.00047 1.00048 1.00049 1.0005 Figure 10: Curve (x0, x′ f) for Jacobi constant C=C3and µ= 0.0001 (top left), µ= 0.00015 and x0∈[1.0000627,1.000064] (top right) and µ= 0.0005 (bottom, left and right). -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 -1 -0.5 0 0.5 1 -0.14 -0.12 -0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.84 0.86 0.88 0.9 0.92 0.94 0.96 0.98 1 1.02 Figure 11: Left. Two horseshoe (non periodic) orbits with a loop close to the first crossing with the xaxis (projection in the (x, y) plane). Right. Zoom. 18 On horseshoe motion -0.06 -0.05 -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 1.003 1.004 1.005 1.006 1.007 1.008 1.009 1.01 -0.06 -0.05 -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 1.003 1.004 1.005 1.006 1.007 1.008 1.009 1.01 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 1 1.02 1.04 1.06 1.08 1.1 Figure 12: For C=C3, curve (x0, x′ f) at first crossing for µ= 0.00697485 (left), µ= 0.007 (middle) and µ= 0.008 (right). It is clear now that the obtained diagram of HPO for µ= 0.0001 (Figure 6), will be rather different when the mass parameter increases. In Figure 13 the diagram of characteristic curves in the (x0, C) plane for µ= 0.008 is shown. The families have been computed using the method explained in Subsection 3.2: continuing the families from the HPO obtained fixing a value of C, and looking for orbits with x′ f= 0. When comparing both diagrams (µ= 0.0001 and µ= 0.008) we observe the following differences: first, as explained above and shown in Figure 12 right (curve (x0, x′ f) for C=C3), for the µ= 0.008 case, there are not HPO (with two crossings) near L3. Secondly, the rigid structure of families organised in branches around the skeleton (see Figure 5) disappears completely when µincreases. Actually, the influence of the two body problem disappears as µincreases, but the small primary and the Lypaunov orbits around L1and L2play their role instead. For example to find HPO that enter a neighbourhood of the small primary is easier for µ= 0.008 than for µ= 0.0001, and the HPO become more intricate, having several crossings with the xaxis. In Figure 14 three HPO for µ= 0.008 are shown. Thus, we can conclude that the continuation of the families of HPO is very sensitive with respect to µ. The reasons for the differences in the diagrams of families for different values of µdepend on the behaviour of the invariant manifolds of L3, and the existence of homoclinic orbits to L3, as we will see in Section 4. 4 Horseshoe motion for µ∈(0,1/2]. Our most ambitious goal in this Section would be to analyse the existence of HPO for any value of the mass parameter µ∈(0,1/2] in an interval of values of Cclose to C3. Of course a mechanism suitable to describe the HPO for µ > 0 and small (as explained in Section 3), does not apply for any given value of µ. We are going to show that a natural mechanism that explains the existence of HPO and the differences between the diagrams of the characteristic curves (of the families of HPO) for different values of µ, relies on the behaviour of the one-dimensional manifolds of the collinear point L3. We recall that to compute the two branches of the unstable manifold, Wu,i L3i= 1,2, we have taken as an initial condition PL3+s·vwhere sis a small quantity (usually E. Barrab´es and M. Oll´e 19 3.014 3.016 3.018 3.02 3.022 3.024 1 1.02 1.04 1.06 1.08 1.1 1.12 1.14 Figure 13: Characteristic curves of families of HPO for µ= 0.008. The dotted curve on the bottom corresponds to the Lyapunov family around L3. -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 -1 -0.5 0 0.5 1 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 -1 -0.5 0 0.5 1 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 -1 -0.5 0 0.5 1 Figure 14: Samples of HPO for µ= 0.008 and C= 3.0000029162232198, x0= 0.9462538001607815 (left), C= 3.0128479526207705, x0= 1.162120446968716 (middle) and C= 3.0082900381403035, x0= 1.117289488220401 (right). 10−6) positive or negative according to i= 1,2 respectively and vis the unit eigenvector associated to the eigenvalue λ > 0 of the Jacobian matrix of the vector field at L3. From this initial condition, we follow the invariant manifold numerically (integrating the system of ODE) under the check test that along the integration the Jacobi constant values must be C=C3. Along the Section, two main ideas will play a role: (i) given µ > 0, if the (stable or unstable) invariant manifold of L3has a horseshoe shape, we may expect to have HPO close to it. (ii) If, besides the horseshoe shape, the manifold Wu,i L3,i= 1 or 2, has a first (k-th) orthogonal crossing (close to L3) with the xaxis, i.e y=x′ f= 0, then (by symmetry (3)), the unstable and stable manifolds intersect giving rise to a homoclinic orbit which tends asymptotically, in forward and backward time, to L3; for this µvalue, we expect the existence of an infinite set of HPO tending to it (known as blue sky catastrophe phenomenon after Devaney [9], see also [13] and [14]). So, along this Section, we will concentrate on the C=C3value, where the manifolds of L3exist. First of all, we will apply these two ideas to answer the open question, described 20 On horseshoe motion in Subsection 3.3, concerning the restriction of the diagram (of characteristic curves of families of HPO) for different values of µat C3(Figures 10 and 12). This is done in Subsection 4.1. This exploration, for selected values of µ, leads us to the following natural questions: for which values of µdo the invariant manifolds of L3(i) have a horseshoe shape? (ii) become homoclinic orbits? So a systematic exploration of the shape of the invariant manifolds of L3is carried out for all values of µ∈(0,1/2]. The values of µfor which there is a homoclinic connection are analysed as well. For µ∈(0,01174] we will obtain two different sequences of values of µ,µn,homo,1and µn,homo,2tending both to 0 as n→ ∞. For each value of µin µn,homo,1, the (x, y) projection of the homoclinic orbit has only one crossing (the orthogonal one) with the xaxis, whereas for µin µn,homo,2, the whole homoclinic orbit has a loop (so three crossings with the xaxis, the second one being orthogonal). this is done in Subsection 4.2. For µ > 0.01174, we obtain either horseshoe invariant manifolds which describe a path close to the small primary or even collide with it, or manifolds which have no horseshoe shape. This is explained in Subsection 4.3. Finally, in Subsection ?? we show how the phenomenon of double homoclinic orbits bifurcation around each µn,homo,i, for i= 1,2 takes place. 4.1 Invariant manifolds of L3when varying µ From the invariant manifolds of L3it is now easy to explain the plots in Figure 10. First we plot the (x, y) projection of the unstable manifold Wu,1 L3for µ= 0.0001 (Figure 15 top and bottom left). The simple horseshoe shape of the invariant manifold which is (almost) a homoclinic orbit (the homoclinic connection occurs for µ= 0.00010001) explains again the infinite sequence of values x0(of the initial conditions of HPO) accumulating to x3 (see Figure 10 top left), that is, for µ= 0.00010001 we have a blue sky catastrophe. But as µincreases a little bit from 0.0001, we can see how there appear loops close to the first crossing with the xaxis and the width of the loops grows fast in µ(Figure 15 top middle). It is this width of the loops that has consequences on the dynamics of the orbits with initial position (x0,0) for x0close to x3. More precisely, for µ= 0.0001 the width of the loops is very small and therefore it has no effects in the (x0, x′ f) curve (which had a nice regular behaviour as shown); whereas for µ= 0.0005, the effect of the loops, that appear in both Wu,1 L3and Ws,1 L3(Figure 15 top middle), is twofold: on the one hand, there do not exist HPO with two orthogonal crossing in an interval of values of x0near x3(see curve x′ f(x0) in Figure 10 bottom left). And on the other hand, the loops may also produce the existence of an infinite set of initial conditions of HPO accumulating to a certain x0 far from x3. This is the case for µ= 0.0005: we plot the curve (x0, x′ f) for x0> x3and we concentrate on the interval x0∈[1.00043,1.0005] (Figure 10 bottom right). Near the value x0= 1.00048826 there seems to exist an infinite number of HPO with two orthogonal crossings, but actually only a finite number exists (as a zoom exploration reveals): if we compute the manifold Wu,2 L3, we observe that the first crossing (almost orthogonal) takes place at xf= 1.000489376, x′ f= 8.175 ·10−6. Of course the case x′ f= 0 would imply E. Barrab´es and M. Oll´e 21 a homoclinic orbit to L3and the existence of an infinite set of HPO tending to it. In fact, for µ= 0.0005, we expect a homoclinic orbit to the Lyapunov orbit around L3for a suitable value of C < C3and very close to C3. We finally remark that this loop shape of the manifolds of L3disappears as µincreases, see Figure 15 top and bottom right. So we have the simple plots in Figure 12. The infinity of HPO for µ= 0.00697485 (crossings of the x′ f(x0) curve with x′= 0 in the left plot) with the corresponding initial x0tending to x3, is just a direct consequence of the homoclinic (horseshoe shape) orbit to L3for this value of µ(again a blue sky catastrophe). Instead, for a µa little different, µ= 0.007 (and also µ= 0.008, Figure 12 right), the invariant manifolds describe a simple horseshoe shape but are not homoclinic, and therefore, only a finite number of HPO appears. Therefore, in summary, we have shown that, although the plots of the manifolds are qualitatively similar, the loops have relevant dynamical consequences for the diagram of characteristic curves of HPO at C=C3. (1) (2) (3) -0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.997 0.9975 0.998 0.9985 0.999 0.9995 1 1.0005 1.001 -0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.997 0.9975 0.998 0.9985 0.999 0.9995 1 1.0005 1.001 -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.94 0.95 0.96 0.97 0.98 0.99 1 (4) (5) -1 -0.8 -0.6 -0.4 -0.2 0 -1 -0.5 0 0.5 1 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 -1 -0.5 0 0.5 1 Figure 15: Projection (x, y) of the manifold Wu,1 L3until the first cross with the horizontal axis. Top. Zoom near L3for: (1) µ= 0.0001, (2) µ= 0.0005, (3) µ= 0.001. In order to see the differences between them, these plots are not scaled. Bottom: (4) µ= 0.0001, (5) µ= 0.008. 4.2 Homoclinic orbits to L3when varying µ Motivated by the previous Subsection, we proceed to a systematic exploration, for µ∈ (0,1/2], of the behaviour of the unstable manifolds of L3(a similar analysis follows for the stable ones). Our aim is to explore for which values of µ∈(0,1/2] the manifolds have 22 On horseshoe motion a horseshoe shape and are homoclinic to L3; of course, for each such µ, the dynamics concerning HPO will be similar to the one described in the previous section. Therefore we compute Wu,1 L3for each value of µ > 0 and we keep the xand x′values at the k-th crossing with the xaxis, y= 0, denoted from now on xf,k and x′ f,k. Again we will concentrate on the analysis of the behaviour of the invariant manifolds from L3 to the first crossing, (k= 1), with the xaxis; however some aspects of the dynamics that have to do with other crossings and that will appear in a quite natural way will be also remarked. -2 -1.5 -1 -0.5 0 0.5 1 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 -1 -0.5 0 0.5 1 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 Figure 16: Left: Curve xf(µ) (left) and x′ f(µ) (right) at first crossing. In the figure on the left, the position of L3(x3) and the secondary for each value of µare also plotted (see the text for details). We plot in Figure 16 the (µ, xf,1) curve and the (µ, x′ f,1) one and the positions of L3 and the small primary (µ, µ −1) as well. From these plots we have the following results related to horseshoe motion: When 0 < µ ≤0.01174 the invariant manifold associated with each value of µhas a horseshoe shape (numerically checked through the condition that the manifold crosses the x= 0, y < 0 semi-axis twice –at least– but does not cross the x= 0, y > 0 semiaxis) and the first crossing takes place with xf,1close to L3and on its left (see Figure 16 left). We also remark that in this interval, there are infinitely many values of µsuch that x′ f,1= 0 (see Figure 16 right), each one corresponding to a horseshoe homoclinic orbit to L3. Actually Font ([10]) had computed some values of µtending to 0 for which there is a homoclinic connection to L3. Let us denote this infinity of values µas µn,homo,1 such that when n→ ∞,µn,homo,1→0. In this same interval, each ‘sharp’ point in the (µ, xf,1) curve, associated with a jump or discontinuity in the (µ, x′ f,1) one, corresponds to an invariant manifold whose projection in the (x, y) plane has a loop (see also Figure 11). Therefore the values of xf,2and x′ f,2at the second crossing must be taken into account. As discussed in Section 3, for each discontinuity with the xvalue, there is a nearby horseshoe homoclinic orbit such that the second crossing (the orthogonal one) takes place at the half loop –in the (x, y) plane– (see Figure 19 bottom). Since there is another infinity of such discontinuities in x′ f,1, that means there is another infinity of horseshoe homoclinic, orbits each of them with a loop. Let us denote this sequence as µn,homo,2(also µn,homo,2→0). E. Barrab´es and M. Oll´e 23 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 Figure 17: For µcol,1= 0.02004225 the unstable manifold of L3collides with the small primary (Levi-Civitta position coordinates). 4.3 Collision manifolds and influence of L2 We have just described the evolution of the first crossing curve xf,1(µ) for µ∈(0,0.01174). Let us go on describing the behaviour of such curve for µ > 0.01174 (see Figure 16). Another discontinuity in both curves xf,1(µ) and x′ f,1(µ) which has nothing to do with loops is the jump in xfobserved for µ= 0.01175; it takes place because there is a tangency on the negative xaxis, however for increasing values of µthe invariant manifold keeps its horseshoe shape although invading the x < 0, y > 0 region, therefore not the first but the third crossing with the xaxis will be meaningful for horseshoe motion. However keeping track only of the first crossing xf<0 after the tangency as µincreases, when µhomo,1= 0.0159375, the manifold is a homoclinic orbit to L3, and for µcol,1= 0.02004225 the unstable manifold collides with the small primary at the first crossing (see Figure 16); we have used Levi-Civitta coordinates (see [22]) in order to regularize the binary collision between the particle and the small primary; in Figure 17 we show this manifold in the position Levi-Civitta coordinates (u, v) (the collision corresponds to the origin). After the collision, when µincreases, the xfvalue keeps decreasing on the left of the small primary and for µ≥µhomo,1the unstable manifold Wu,1 L3does not have a horseshoe shape anymore. At this point, let us comment in more detail what happens when the invariant manifold visits the x < 0, y > 0 region when µvaries in the interval (0.01175, µcol,1). Actually, since only the first crossing in the curve (µ, x′ f(µ)) is considered in Figure 16, this plot hides that the shape of the manifold of L3is more intricate. As stated above the invariant manifold for given µ < 0.012 has a horseshoe shape; but as µincreases the Lyapunov orbit around L2and the small primary play a role now: we have followed the manifold up to seven crossings with the xaxis, for varying µ, and our conclusion is that either the homoclinic orbits (to L3) which do not have a horseshoe shape or the collision manifolds (leaving asymptotically from L3and colliding with the small primary m2), are the typical barriers that prevent the manifold Wu,1 L3from having a horseshoe shape. Examples are given in Figure 18: the unstable manifold for µ= 0.0145642 does have a horseshoe shape (we remark the path of the manifold near the shaped Lyapunov orbit of L2), but after the collision with m2, the manifold for µ= 0.01456344 does not. For µ= 0.0145655 the manifold is a simple homoclinic to L3with the second crossing which is orthogonal, so it does not have a horseshoe shape; but after a collision with m2at the six-th crossing, the manifold, for µ= 0.0146968 has a horseshoe shape again. 24 On horseshoe motion -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 -1 -0.5 0 0.5 1 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 -1 -0.5 0 0.5 1 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 -1 -0.5 0 0.5 1 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 -1 -0.5 0 0.5 1 Figure 18: Wu,1 L3in the (x, y) plane for µ= 0.0145642 (top left), µ= 0.01456344 (top right), µ= 0.0145655 (bottom left), µ= 0.0146968 (bottom right). See the text for details. So we can conclude that the dynamics of the unstable manifold Wu,1 L3becomes very rich when m2and the Lyapunov orbit of L2are taken into account. A systematic analysis of such dynamics (considering also the influence of the Lyapunov orbits around L1and L2), for any µ∈(0,1/2] and for a given number of crossings (greater than two), is left for future work. 4.4 Double homoclinic bifurcation Up to now, we have been describing the simple horseshoe manifolds of L3, in the sense that they surround L4and L5just once. However a more detailed numerical exploration reveals that there appear bifurcation of double period homoclinic orbits, although the word ‘period’ does not make sense since a homoclinic orbit has, say, an infinite period, but the nomenclature reminds what happens in the periodic orbits context. From now on we call it double homoclinic orbit. To show this phenomenon, let us now consider one of the values µn,homo,i (described in Subsection 4.2), for a fixed nand i= 1 or 2. In particular, we have taken µn,homo,1= 0.0037258 and µn,homo,2= 0.0041976. We have computed both x′ f,1(µ) and x′ f,2(µ) curves for the interval µ∈[0.0035,0.0044] that contain both values µn,homo,1and µn,homo,2(see Figure 19 top). The interest in the curve x′ f,2(µ) is precisely to show the double homoclinic orbits. We consider a neighbourhood of µn,homo,1and one of µn,homo,2separately. (i) Let us concentrate first on the neighbourhood µ∈I1= [0.0035,0.004] where µn,homo,1= 0.0037258 belongs to. In Figure 19 top, the curves xf,1(µ) and xf,2(µ) (left), as well as x′ f,1(µ) and x′ f,2(µ) (left and right) are represented. We can see E. Barrab´es and M. Oll´e 25 that the first crossing is always close to and less than the xposition of L3, that is xf,1< x3, however xf,2(µ)> x3. We also remark that whereas µ=µn,homo,1is the unique value for which x′ f,1(µ) = 0 in I1, we have infinite values (on the left and right of µn,homo,1) of µsuch that x′ f,2(µ) = 0. And in a similar way as discussed above (see also the infinity of jumps in Figure 19 top), there are also infinite values of µ(on both sides of µn,homo,1) for which x′ f,3(µ) = 0 at a half loop. Therefore we can conclude that there are four sequences of values of µ, say µm,2 n,homo,1< µn,homo,1, µm′,2 n,homo,1> µn,homo,1,µm,3 n,homo,1< µn,homo,1, and µm′,3 n,homo,1> µn,homo,1such that lim m,m′→∞ µm,m′,2 n,homo,1=µn,homo,1,lim m,m′→∞ µm,m′,3 n,homo,1=µn,homo,1, and for each value of each sequence the invariant unstable manifold of L3becomes a homoclinic orbit to L3. The geometric behaviour of such manifolds is the following: the horseshoe homoclinic manifold for µn,homo,1surrounds, in the (x, y) plane, only once the points L4and L5(see Figure 19 middle right). However the horseshoe homoclinic manifolds for µm,m′,2 n,homo,1or µm,m′,3 n,homo,1have 3 or 5 crossings (respectively) with the x axis, the orthogonal one with xf,i > x3once they have surrounded both L4and L5(see Figure 19 middle left), that is the whole homoclinic manifold surrounds twice the points L4 and L5. Actually, when m→ ∞ or m′→ ∞ the shape of the manifold remains almost the same (see Figure 19 middle left), that is, close to y= 0 and once it has surrounded both L4and L5, the (x, y) projection of the manifold has always one loop, so the orthogonal crossing takes place either at the second (outside the loop) or third crossing (at half loop) with y= 0, x′= 0. These two possibilities give rise to the two sequences (at each side of µn,homo,1). Tending to the limit, i.e µn,homo,1, this loop becomes thinner and in the limit there is no loop at all (compare the double period manifold and the limit homoclinic one in Figure 19 middle). (ii) A different behaviour takes place around the horseshoe homoclinic orbit with a loop for µn,homo,2= 0.0041976, nfixed. In this case, we consider the interval I2= [0.004,0.0044] and we also compute the curves xf,1(µ), xf,2(µ), x′ f,1(µ) and x′ f,2(µ) (see Figure 19 top). We can conclude in this case that there are four finite sets of horseshoe homoclinic orbits, with 3 crossings with the x axis, for µclose to µn,homo,2. For values close to µn,homo,2, the corresponding manifolds behave in a similar way as the one described in (a) but the loop (Figure 19 top and bottom left) avoids the infinity of homoclinics. The dynamical consequence of such double homoclinic orbits is that for each value of µfor which there is a double homoclinic orbit, there will exist an infinity of HPO enclosing twice the points L4 and L5, and tending to the homoclinic connection. Finally we remark that the two patterns just described also take place for any µn,homo,i ∈(0,0.01174), i= 1,2, and n∈N; see for instance Figure 19 top right, where only the curves x′ f,1(µ) and x′ f,2(µ) are plotted in the interval µ∈(0,0.05). As we can see, the same patterns of crossings with x′ f= 0 described in detail in Figure