Order-chaos-order and invariant manifolds in the bounded planar Earth-Moon system
Vitor M. de Oliveira, Priscilla A. Sousa-Silva, Iberê L. Caldas
AArXiv manuscript No. (will be inserted by the editor)
Order-chaos-order and invariant manifolds in thebounded planar Earth-Moon system
Vitor M. de Oliveira · Priscilla A.Sousa-Silva · Iberê L. Caldas
Received: date / Accepted: date
Abstract
In this work, we investigate the Earth-Moon system, as modeledby the planar circular restricted three-body problem, and relate its dynamicalproperties to the underlying structure associated to specific invariant mani-folds. We consider a range of Jacobi constant values for which the neck aroundthe Lagrangian point L is always open but the orbits are bounded due to Hillstability. First, we show that the system displays three different dynamicalscenarios in a neighborhood of the Moon: two mixed ones, with regular andchaotic orbits, and an almost entirely chaotic one in between. We then analyzethe transitions between these scenarios using the Monodromy matrix theoryand determine that they are given by two specific types of bifurcations. Afterthat, we illustrate how the phase space configurations, particularly the shapesof stability regions and stickiness, are intrinsically related to the hyperbolicinvariant manifolds of the Lyapunov orbits around L and also to the ones ofsome particular unstable periodic orbits. Lastly, we define transit time in amanner which is useful to depict dynamical trapping and show that the tracedgeometrical structures are also connected to the transport properties of thesystem. Keywords
Restricted three-body problem · Chaos · Invariant manifolds
V. M. de Oliveira · I. L. CaldasUSP - University of São Paulo, Institute of PhysicsRua do Matão, 1371 - Edif. Basílio Jafet - CEP 05508-090Cidade Universitária - São Paulo/SP - BrasilE-mail: [email protected] A. Sousa-SilvaUNESP - São Paulo State University, São João da Boa VistaAvenida Professora Isette Corrêa Fontão, 505 - CEP 13876-750Jardim das Flores - São João da Boa Vista/SP - Brasil a r X i v : . [ n li n . C D ] J un Vitor M. de Oliveira et al.
The dynamics of nonintegrable Hamiltonian systems may be characterizedby the coexistence of both chaos and stable motion (Lichtenberg and Lieber-man, 1992) and a good understanding of this type of systems comes fromanalyzing how said scenario is affected by the constants of motion. A com-plete description, however, also involves the underlying geometrical structures,which are related to hyperbolic invariant manifolds associated with unstableperiodic orbits embedded in the chaotic sea and whose properties can influencethe systems’ dynamics.In Celestial Mechanics, many systems are Hamiltonian or can be well repre-sented by a Hamiltonian function. Thus, chaotic behavior is a common featurein these systems and is related, for example, to the motion of asteroids andof the solar system itself (Poincaré, 1890; Laskar, 1989; Ferraz-Mello, 1999).In this context, invariant manifolds have been investigated and employed ina variety of applications, ranging from natural transport to space mission de-sign (Koon et al, 2008; Gawlik et al, 2009; Perozzi and Ferraz-Mello, 2010).Furthermore, these solutions can also be linked to the occurrence of sticki-ness in the dynamics of spiral galaxies (Contopoulos, 2004; Contopoulos andHarsoula, 2010).In this work, we adopt the planar Circular Restricted Three-body Problem(CRTBP) as a model to investigate the dynamical properties of the Earth-Moon system. We consider a range of values for the constant of motion in whichall orbits analyzed are bounded within the system and we use numerical toolsto obtain particular periodic orbits and their respective invariant manifolds.Our results illustrate how the order-chaos-order scenario relates to geometricalstructures in phase space, thus contributing to understanding the fundamentalconnection between dynamics and geometry in the Earth-Moon system.This paper is organized as follows. In Sec. 2 we present the planar CRTBPand its dynamical features. In Sec. 3 we describe the phase space configurationfor the considered range of Jacobi constant and discuss the bifurcations thatoccur in the stability regions. In Sec. 4 we trace the invariant manifolds asso-ciated with the Lyapunov orbits around L and illustrate their relation to thephase space configuration. Later, we consider the stickiness effect by tracingthe manifolds associated with selected unstable periodic orbits in the mixedscenarios. In Sec. 5 we define transit time in a suitable manner and examinethe transport properties of the system. Finally, we give our conclusions in Sec.6. The framework we use to model the Earth-Moon system is the planarCRTBP, which provides a good approximation to the dynamical behavior ofthis physical system (Murray and Dermott, 1999). It describes the motion of a haos and manifolds in the Earth-Moon system 3 a = 0.188 a = 0.1725
Fig. 1:
Physical model in a vicinity of the Moon. The third body moves in the white area,the Hill region, and Σ represents the Poincaré surface where orbits are analyzed. As theJacobi constant C goes from C to C , the neck around the Lagrangian point L becomeslarger. The Lyapunov orbit is also depicted for both situations. body of negligible mass under the gravitational influence of two massive bodiesmoving in circular orbits around their combined center of mass.We assume that the third body moves in the same plane as the two-bodysystem. This is a useful assumption because both the planar version of theproblem and the geometrical structures that we deal with in this work havethe advantage of being naturally represented in a two-dimensional surface ofsection.In a synodic reference frame, which rotates with the same frequency asthe system formed by the primaries and is centered at its center of mass, thedimensionless equations of motion in terms of the variables ( x, y, ˙ x, ˙ y, t ) are ¨ x − y = ∂Ω∂x , ¨ y + 2 ˙ x = ∂Ω∂y , (1)where the pseudo-potential Ω is given by Ω = 12 ( x + y ) + 1 − µ (cid:112) ( x + µ ) + y + µ (cid:112) ( x − (1 − µ )) + y . (2)The primaries are located at ( − µ, and (1 − µ, , with µ being the massparameter , the ratio between the mass of the less massive primary and thesystem’s total mass. For the Earth-Moon system, we have µ ≈ . × − .An schematic representation of the system around the Moon is presented inFig. 1. There is one unstable Lagrangian equilibrium point on each side of theMoon, namely L (left) and L (right). Fig. 1 also presents some importantconcepts which are addressed later in this paper.The system has one constant of motion, called Jacobi constant C , which isgiven by C = 2 Ω − ˙ x − ˙ y , (3) Vitor M. de Oliveira et al. and therefore the dynamics effectively occurs in a three-dimensional subspace.Additionally, since ˙ x + ˙ y > , eq. (3) defines the area accessible to the thirdbody in the coordinate space x - y for a given C, H = { ( x, y ) ∈ R | Ω − C > } , (4)which is called the Hill region . In Fig. 1, H is represented by the white area.Furthermore, C ≈ . and C ≈ . are the Jacobi constants at theLagrangian points L and L , respectively.It is important to note that, for the range of Jacobi constant chosen in thiswork, C > C > C , H is divided in two disconnected areas. Consequently,the orbits that lie in a vicinity of either primary are bounded and cannot exitthe system ( Hill stability ).Since our analyses involve numerical calculations, it is necessary to dealwith the singularities in eq. (2). This is achieved by using the
Levi-Civitatransformation (Szebehely, 1967). Let ( u, v, u (cid:48) , v (cid:48) , τ ) be the new set of variablesin the system and let us define z = x + iy and ω = u + iv . The transformationsare then given by z = ω − µ + 1 for regularization in a vicinity of the Moonand z = ω − µ for regularization in a vicinity of the Earth. In both cases, therelation between the time variables is given by dt = 4( u + v ) dτ .In the new set of variables, the equations of motion, eq. (1), become u (cid:48)(cid:48) − u + v ) v (cid:48) = ∂V∂u ,v (cid:48)(cid:48) + 8( u + v ) u (cid:48) = ∂V∂v , (5)where the new pseudo-potential V is V M ( u, v ) = 4 µ + 2( u + v ) (cid:40) ( u + v ) + 2(1 − µ )( u − v )+(1 − µ − C ) + 2(1 − µ ) (cid:113) u + v ) + 2( u − v ) (6)for the Moon and, V E ( u, v ) = 4(1 − µ ) + 2( u + v ) (cid:40) ( u + v ) − µ ( u − v )+( µ − C ) + 2 µ (cid:113) u + v ) − u − v ) (7)for the Earth.The regularization procedure is performed locally about the singularities.In practice, we establish two radii with values δ M = 1 . × − around the haos and manifolds in the Earth-Moon system 5 Moon and δ E = 3 . × − around the Earth and we switch between equations(1) and (5) as the orbits enter or exit these regions. In both transformations,the system is re-centered to one of the primaries and we can verify in equations(6) and (7) that V is finite when ( u, v ) → (0 , , thus removing the singularitiesfrom these locations. We now proceed to study the dynamical properties in a vicinity of theMoon. In order to do so we choose a surface of section Σ between the Moonand L defined by Σ = { x = ( x, y, ˙ x, ˙ y ) | − µ < x < x L , y = 0 , ˙ y > } , (8)where x L ≈ . is the position of L in the x -axis, which depends only on µ (Gómez et al, 2001). In Fig. 1, we depict Σ for C (cid:46) C and C (cid:38) C .Fig. 2 shows the system’s phase space x - ˙ x for different values of C . The ini-tial conditions are chosen in a by grid on Σ and the orbits are integratedup to t = 5 × both forward and backward in time. Numerical integrationof the equations of motion are carried out using the explicit embedded Runge-Kutta Prince-Dormand 8(9) (Galassi et al, 2001) and errors associated withthe Jacobi constant along the orbit and with the intersection between orbitand surface of section are kept below − .The first feature we observe is the existence of three different scenarios asthe Jacobi constant is decreased: I. ( C = 3 . , . and . ) the systempresents a mixed phase space as the region of stability decreases in size; II.( C = 3 . and . ) all orbits analyzed are chaotic and hence the formerstability region was destroyed; III. ( C = 3 . , . and . ) the phasespace becomes mixed again with the creation, enlargement and subsequentslight decrease in size of a new stability region.We can use Newton’s Method and the symmetry of the model to calculateboth stable and unstable periodic orbits in the system for adequate initialconditions. We acknowledge here the Celestial Mechanics notes by J. D. MirelesJames . In order to understand then what happens with the stability regionsin both mixed phase space scenarios, we follow the periodic orbits in eachcase and study their stability by computing the eigenvalues of their respectiveMonodromy matrices.The Monodromy matrix has four eigenvalues, two of which are alwaysunitary. The remaining two eigenvalues determine the stability of the periodicorbit as follows: if the orbit is stable, the eigenvalues are complex conjugate toeach other; however, if the orbit is unstable, the eigenvalues are real and oneis the inverse of the other.In scenario I, there is one periodic orbit of period 1 which is located atthe center of the stability region. In Fig. 3a we evaluate the real part of both Available at http://cosweb1.fau.edu/~jmirelesjames/notes.html . Vitor M. de Oliveira et al.
Fig. 2:
Phase space in the surface of section Σ for the selected range of Jacobi constant C .The system goes from and back to a mixed scenario but with different stickiness behavior.haos and manifolds in the Earth-Moon system 7 (a) Scenario I (b)
Scenario III (c)
Scenario III
Fig. 3:
Bifurcation analysis for both mixed phase space scenarios. The real part of theeigenvalues Re ( λ ) as a function of the Jacobi constant C are shown for (a) the center orbitin scenario I and (b) the period-1 stable and unstable orbits in scenario III. In (c), the orbitsin scenario III are shown to collide as their x -axis component tend to the same value. eigenvalues of this orbit which are associated with stability as a function ofthe Jacobi constant. We observe that the orbit is stable for C = 3 . and iteventually becomes unstable as C is lowered. We have, in this case, a director inverse bifurcation (Contopoulos, 2004), which happens at approximately C bif = 3 . .In scenario III, there are two periodic orbits of period 1: the stable oneat the center of the stability region and its unstable counterpart to the leftof it, just outside the stability region and inside the chaotic sea. We performthe same analysis as before for both orbits, but this time we increase theJacobi constant. The the results are shown in Fig. 3b. For C = 3 . , all foureigenvalues are distinct and, as C is increased, they all tend to the same value.We have, in this case, a saddle-node bifurcation (Contopoulos, 2004), whichhappens at approximately C bif = 3 . . After the bifurcation is reached, The direction of the bifurcation determines the stability of a new periodic orbit whichis created outside Σ and hence it is not relevant to our analysis. Vitor M. de Oliveira et al. both periodic orbits are destroyed. In Fig. 3c we present the position in the x -axis of both orbits up until their collision.We note from Fig. 3 that the eigenvalues go through − in scenario I andto in scenario III. Hence, the trace of the Monodromy matrix goes to and , respectively, both of which indicates the occurrence of a bifurcation intwo-degree of freedom Hamiltonian systems (de Aguiar et al, 1987).The second feature which stands out in Fig. 2 is the difference in thestickiness behavior in both mixed phase space scenarios. In scenario III, thereis a higher orbit concentration just about the stability region as is usually thecase. However, for higher values of C in scenario I, the stickiness effect reachesdeep into the chaotic sea and far from the stable portion of phase space, whichsuggests that it is being caused by invariant manifolds associated with unstableperiodic orbits around the stability region (Contopoulos and Harsoula, 2010).We present a summary of the three dynamical scenarios in Tab. 1. The type order indicates the presence of stability regions in the system. As discussedbefore, the Hill region H is composed of two disconnected areas and it isimportant to note here that it remains as such in all scenarios. Table 1:
Overview of the three different scenarios that are present in the system.
Scenario Range Type StickinessI C > C > C bif order non-localizedII C bif > C > C bif chaos absentIII C bif > C > C order localized The Lagrangian point L is the only equilibrium of the system which isinside the Hill region for the range of Jacobi constant that we considered.Furthermore, there exists an uniparametric family of unstable periodic orbitsaround this point, namely the Lyapunov orbits . We are able to calculate aLyapunov orbit for any value of C using a continuation method along thelinear solution around L (Gómez et al, 2001). For illustration, the orbitscorresponding to C = 3 . (cid:46) C and C = 3 . (cid:38) C are shown in Fig. 1.Let p be a point of the unstable periodic orbit α . As described in Sec. 3,the Monodromy matrix calculated at p has a pair of real eigenvalues which de-termine the orbit’s stability. These eigenvalues, with moduli lower and greaterthan one, are related to eigenvectors that define a stable and an unstabledirection, respectively. Therefore, there is a set of orbits originated in a neigh-borhood of p which will tend to it as time goes to ±∞ . If we extend thisset to the whole space, we define the stable manifold W s ( p ) and the unstablemanifold W u ( p ) associated with p . Formally, we write haos and manifolds in the Earth-Moon system 9 W s ( p ) = { x ∈ U ⊂ R | ϕ t ( x ) → p as t → ∞} ,W u ( p ) = { x ∈ U ⊂ R | ϕ t ( x ) → p as t → −∞} , (9)where ϕ t ( x ) is the solution of the system at time t with initial condition x .We can then define the stable manifold W s ( α ) and unstable manifold W u ( α ) associated with the unstable periodic orbit α as W s,u ( α ) = (cid:91) p ∈ α W s,u ( p ) . (10)Due to the fact that the dynamics in our system effectively occurs in athree-dimensional subspace, α is an one-dimensional curve and W ( α ) are two-dimensional surfaces that are locally homeomorphic to cylinders (Ozorio deAlmeida et al, 1990). Furthermore, both W s ( α ) and W u ( α ) have two brancheswhich are associated to an eigenvector and its counterpart of opposite direc-tion.Let us now define Γ as the intersection between the invariant manifoldsand our surface of section, which can be naturally ordered by following thedynamics on W and counting the crossings with Σ . We have Γ s,u ( α ) = W s,u ( α ) ∩ Σ = ∞ (cid:91) i =1 Γ s,ui ( α ) . (11) Γ is a set of one-dimensional curves. If Σ is always transversal to W ,the curves are open, similar to manifolds in two-dimensional maps. Otherwise,some Γ i may have ellipse-like shapes as W crosses Σ in a perpendicular fashion.Hence, the representation of invariant manifolds in phase space depends onhow they intersect the surface of section.To numerically trace W ( α ) , we first calculate one Monodromy matrixeigenvector and then propagate it to the other points of the orbit in a pre-defined discretization using the transition matrix. We then take one initialcondition on each vector with a distance of − from the orbit and integrateit forward or backward in time, depending on the eigenvector stability.With the aforementioned scheme, we can calculate the invariant manifolds W ( L ) associated with a Lyapunov orbit. Fig. 4 shows the first few Γ i ( L ) forthe same Jacobi constant values as in Fig. 2. The first aspect we observe isthat the area enclosed by the manifolds gets bigger as we lower C , thereforeoccupying a larger region in phase space for a similar number of crossings.This is a consequence of the fact that the system is area-preserving and anelement of the family of Lyapunov orbits is larger in length than the otherelements with higher Jacobi constants.The most significant result to be noted here is the fine interplay betweenLyapunov orbit manifolds and phase space configuration. Initially, the mani-folds intersect the surface of section far from the stability region. As we startto lower the Jacobi constant, they begin to travel across a larger area of phasespace and spread towards the stability region, which gets smaller accordingly. Fig. 4:
First few components of Γ s ( L ) (blue) and Γ u ( L ) (red) in phase space. The invariantmanifolds associated with the Lyapunov orbits evolve along the phase space configurationas the Jacobi constant C is lowered.haos and manifolds in the Earth-Moon system 11 Eventually, they cover all the stability region and the stable periodic orbitat its center bifurcates and changes stability. After the global chaos scenario,another region of stability emerges in an area of phase space which is not yetcovered by the invariant manifolds. In the end, they start to ripple around andinvade the new stability region.Another interesting aspect we observe from Fig. 4 is the apparent relationbetween the spatial disposition of the invariant manifolds and the propertiesof the stickiness phenomenon in both mixed phase space scenarios. In scenarioI, the manifolds do not yet occupy a large portion of phase space, which makesit possible for the stickiness to reach far into the chaotic sea. In scenario III,on the other hand, the manifolds are spread around the new stability regionand the stickiness is then confined next to it.As we discussed before, the stickiness effect is likely caused by invariantmanifolds associated with particular unstable periodic orbits in phase space.In order to verify this assertion, we choose suitable values of C for both mixedscenarios and we calculate the main unstable periodic orbit located aroundeach stability region. We then trace the invariant manifolds associated withthese orbits and compare them to the stickiness observed in Fig. 2. For scenarioI, we choose C = 3 . and we calculate an unstable periodic orbit of period7 which we call P I . For scenario III, C = 3 . and the orbit P III has period8. The results are shown in Fig. 5.In Figs. 5a and 5b, we observe that Γ ( P I ) extend deep into the chaoticsea and closely reproduce the structure corresponding to the stickiness effect.Furthermore, Fig. 5c shows that Γ ( P III ) are concentrated around the stabilityregion, as we expected, also repoducing the stickiness behavior. In Fig. 5d, wepresent the manifolds associated with the unstable periodic orbit of period 1 P III that is created in the second bifurcation. The value of the Jacobi constanthere is C = 3 . and we observe that Γ ( P III ) do not have a complex geometryapart from the small oscillation near the saddle. However, it is interesting tonote that a ghost effect is observed before the bifurcation with the same shapeas given by these manifolds, as we can see in Fig. 2 for C = 3 . .Finally, we depict an overview of the system in Fig. 6 for the chosen Jacobiconstant in each mixed phase space scenario. It is clear that each group ofinvariant manifolds contribute differently to the phase space configuration andthat all of them are necessary for a broad description of the system. Another aspect regarding the phase space configuration is the presence ofless dense areas in the chaotic sea. We can observe it more clearly in Fig. 2for C = 3 . . If we compare it to Fig. 4, we note that the less dense areasare the ones enclosed by the traced manifolds. This phenomenon comes fromthe fact that W ( L ) are responsible for transporting orbits between the Moonand Earth realms (Koon et al, 2008). The orbits inside the first few Γ si ( L ) go (a) C = 3 . (b) C = 3 . (c) C = 3 . (d) C = 3 . Fig. 5:
Stable (blue) and unstable (red) manifolds associated with the main unstable peri-odic orbits (black) in the mixed phase space scenarios. In scenario I, we have Γ ( P I ) in (a)full size and (b) zoomed-in. In scenario III, we have (c) Γ ( P III ) and (d) Γ ( P III ) . (a) C = 3 . (b) C = 3 . Fig. 6:
Overview of the system’s geometrical structures in phase space for (a) scenario Iand (b) scenario III. Stable manifolds are depicted in blue and unstable manifolds in red.These structures have a close relation to the phase space configuration.haos and manifolds in the Earth-Moon system 13 through the Lyapunov orbit onto the Earth’s vicinity faster than other areasand hence they are less populated in phase space.In order to dynamically quantify the geometric structures of the system,we first choose orbits that begin in our surface of section and calculate howlong it takes for each of them to transfer to the Earth realm both forward t f and backward t b in time. We then define transit time as the absolute value ofthe product of t f and t b . This is a convenient definition because our transittime highlights orbits that stay at the lunar realm for a very long time andalso for a very short time. Fig. 7 shows the transit time for a grid of × initial conditions in Σ and the same Jacobi constants of Figs. 2 and 4. Thesystem is integrated up to t = ± × and only orbits which do eventuallyexit the lunar realm are considered for analysis.We can readily observe the influence of invariant manifolds in the system’sdynamics. Regions with shorter transit times correspond exactly to the inte-rior of Γ ( L ) , specially inside the intersections between Γ s ( L ) and Γ u ( L ) forthese are the orbits that most rapidly enter and exit the Moon’s realm. In ad-dition, regions with longer transit times correspond to the invariant manifoldsassociated with the main unstable periodic orbits in the mixed phase spacescenarios, namely Γ ( P I ) and Γ ( P III ) .Hence, what we observe is the coexistence of two effects. On the one hand,we have the Lyapunov orbit manifolds which are responsible for the trans-port between the Moon and Earth realms and, on the other hand, we havethe manifolds associated with higher-order unstable periodic orbits which areaccountable for dynamically trapping the orbits.The transit time profiles also give insight into the chaotic properties of thesystem. Even though all orbits in scenario II are chaotic, they have preferredpaths to follow until they reach the Lyapunov orbit and, therefore, the chaosis not uniform. Moreover, these pathways are very different in the two mixedphase space scenarios, see C = 3 . and C = 3 . for example. This is notvisible from the phase space analysis alone.All orbits in the chaotic sea, except for a set of measure zero, move fromone realm to the other for a large enough integration time, which suggeststhat the Lyapunov orbit manifolds are dense in this area. The first few Γ i ( L ) are homeomorphic to circles but they eventually lose this property (Gideaand Masdemont, 2007). This phenomenon is the outcome of the intersectionbetween two-dimensional manifolds of different stabilities. We explore thisfurther in Fig. 8.Fig. 8a shows the Lyapunov orbit manifolds in phase space for C = 3 . .We observe that the first crossing of the unstable manifold Γ u intersects theseventh crossing of the stable manifold Γ s . But, since all orbits inside W s will at some time go through the Lyapunov orbit, the intersection between W s and W u has the following consequence. After the seventh crossing with Σ , the orbits that compose W u are divided in three parts: the ones that areinside W s when the intersection occurs go through the Lyapunov orbit andon to the other realm; the ones that are exactly in the stable manifold are thehomoclinic orbits and go to the Lyapunov orbit; the rest of the orbits cross the Fig. 7:
Profile of the transit time on a logarithmic scale for different Jacobi constants. Theinitial conditions are chosen in the surface of section.haos and manifolds in the Earth-Moon system 15 (a) C = 3 . (b) C = 3 . Fig. 8:
Intersect and break process in the Lyapunov orbit manifolds as seen from the(a) phase space and the (b) coordinate space for different Jacobi constants. The unstablemanifold eventually breaks if it intersects the stable manifold. One part of it moves to theEarth realm while other part crosses Σ again divided in two pieces. defined surface of section again Γ u although this time divided in two piecesthat asymptotically approach Γ u .The described process happens indefinitely for all intersections betweenthe unstable and stable manifolds which, by consequence, fill the chaotic sea.In Fig. 8b we present the same scenario for C = 3 . but now in coordinatespace. In this situation, both manifolds intersect each other at the first crossingand hence the unstable manifold breaks much faster. We can see a part of themanifold crossing the Lyapunov orbit whilst other part revolves around theMoon and crosses Σ again.The structures that emerge from the intersect and break process are visiblein Fig. 7, specially for C = 3 . . Furthermore, it is interesting to note thata somewhat similar situation occurs with the Lyapunov orbit manifolds andthose associated with the higher-order unstable periodic orbits, since thesestructures also intersect each other. For C = 3 . , for example, we can ob-serve the auto-similar structure formed by the intersection between W ( L ) and W ( P I ) .Our final step is to examine what happens when we consider collisionswith the primaries in our model. Since the structures formed by the invariantmanifolds are closely related to the dynamical properties of the system, it isimportant for us to understand their role in this case. In order to mimic theeffects of collision, we define a radius by hand around the Moon and stops theintegration if an orbit reaches this region. In practice, this added feature worksas leaking (de Assis and Terra, 2014) for these orbits have a finite existenceand therefore do not contribute to our analysis.We present the transit time profiles for this situation in Fig. 9. The pa-rameters chosen are the same as before and the radius of collision with theMoon is given by r M = 4 . × − . By comparison to Fig. 7, we can see Fig. 9:
Profile of the transit time on a logarithmic scale for different Jacobi constants, butthis time discarding collisional orbits. The initial conditions are chosen in the surface ofsection and are the same as in Fig. 7.haos and manifolds in the Earth-Moon system 17 that the presence of a collision radius affects the dynamics of the system intwo different ways. First, there is a riddled structure formed by the collisionalorbits which initially covers all the analyzed space and, as we lower the Jacobiconstant, it becomes more localized, mostly around the new stability region.This scheme shows a close relation between the riddled structure and the man-ifolds associated with the main unstable periodic orbits in the mixed phasespace scenarios.The second effect is the appearance of collision areas which grow largeras we lower C , delimiting the space available to the riddled structure. Anal-ogously, this scheme shows a close relation between collision areas and theinvariant manifolds of the Lyapunov orbit. It is worth noting that, had weconsidered collision in Fig. 4 for example, there would be parts of the man-ifolds missing and therefore their relation to the phase space configurationwould be harder to visualize. In this work, we showed that the planar Earth-Moon system, as modeled bythe restricted three-body problem, presents three different scenarios, each onewith its particular dynamical and geometrical properties. Even though theHill region remains topologically unchanged, the system goes from a mixedscenario with far-reaching stickiness, to the absence of stability regions, andback to a mixed scenario but now with localized stickiness, just by varying theJacobi constant. Moreover, the transition between these scenarios are givenby two different type of bifurcations, namely, the direct or inverse and thesaddle-node bifurcation.We also illustrated how some hyperbolic invariant manifolds in the systemevolve along the phase space configuration. On the one hand, we have the man-ifolds associated with the Lyapunov orbits, which determine shape and sizeof stability regions. On the other hand, there are particular unstable periodicorbits whose invariant manifolds determine the behavior of stickiness. Thesegroups of manifolds are all two-dimensional surfaces, although they cross theunidimensional surface of section in different manners, hence defining geomet-rical structures with different properties.Lastly, with a reasonable definition of transit time, we were able to depictthe influence of the invariant manifolds in the system’s transport properties.There is a fine interplay between the Lyapunov orbit manifolds, which areresponsible for the motion between the realms, and the ones associated withthe higher-order unstable periodic orbits, which temporarily trap the orbitsnear the stability regions. In summary, this work provided a broad picture onthe dynamics of the planar Earth-Moon system and reinforced the importanceof better understanding the connection between dynamics and geometry.
Acknowledgements
This study was financed in part by the Coordenação de Aperfeiçoa-mento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001 and the São PauloResearch Foundation (FAPESP, Brazil), under Grant No. 2018/03211-6.8 Vitor M. de Oliveira et al.