Abstract
It has been proved that, in the classical planar circular restricted three-body problem, the degenerate saddle point processes transverse homoclinic orbits. Since the standard Smale-Birkhoff theorem cannot be directly applied to indicate the chaotic dynamics of the Smale horseshoe type, we in this note alternatively apply the Conley-Moser conditions to analytically prove the existence of a Smale horseshoe in this classical restricted three-body problem.
1. Introduction and Preliminaries
Few bodies problems [1–7] have been studied for long time in celestial mechanics, either as simplified models of more complex planetary systems or as benchmark models where new mathematical theories can be tested. The three-body problem has been the source of inspiration and study in celestial mechanics since Newton and Euler [8–14]. Especially, the following classical planar circular restricted three-body model has been extensively studied in the literature. Let two particles and , of mass and , move uniformly in a circular orbit about their common center of mass with angular velocity . The orbit is located in the plane of the inertial frame of reference and the common center of mass is in the origin. The particle of infinitesimal mass moves in the gravitational field generated by and . Note that since the mass of is so small, its effects on other three particles can be ignored. Without loss of generality, assume that, in the plane of the rotating frame of reference, the particles and rest at the points and , respectively. By denoting their polar coordinates by and and using the polar angle as a new independent variable, the equation of motion of the infinitesimal particle can be written as follows: where and are momenta canonically conjugate to the coordinates and , respectively.
The Hamiltonian of the system (1) is
For the above classical model, Xia [4] has showed, by proper coordinate change for transforming the points at infinity to the origin (i.e, the McGehee transformation [2]), that there is a periodic solution at infinity. Moreover, from [2, 4], we know that this periodic solution is a degenerate saddle in the sense [2] that, for the Poincaré map of the periodic orbit introduced at infinity, its derivative (i.e., the Jacobian) at the origin is the identity.
Further, Xia [4] and Zhu and Xiang [12] both proved the existence of transversal homoclinic orbits by the Melnikov method to the periodic solution at infinity, which corresponds to the origin under the coordinate change. However, since the origin is a degenerate fixed point, the standard Smale-Birkhoff theorem [15] cannot be directly applied to indicate the existence of a Smale horseshoe. This problem has also been pointed out by Dankowicz and Holmes [6] and Llibre and Perez-Chavela [8]. Thus, in this present note, we try to alternatively apply the Conley-Moser conditions to analytically prove the existence of a Smale horseshoe in the above classical model. For this, we introduce the Conley-Moser conditions [16] as follows.
Let be an invertible map, where , and is at least . For two given and , let () be an index set, let be the disjoint -horizontal strips, and be the disjoint -vertical strips. For each , denote as and as . Clearly, . Define and . It is also obvious that .
For an arbitrary point , let be a vector emanating from the point in the tangent space of . The stable sector at is then defined as . Similarly, the unstable sector at is defined as . By taking the union of the stable and unstable sectors over all points in and , we can define sector bundles as follows:
Then, the Conley-Moser conditions for the map are described by the following two assumptions.
Assumption 1. and, for each , maps homeomorphically onto ; that is, . Moreover, the horizontal boundaries of are mapped to the horizontal boundaries of and the vertical boundaries of are mapped to the vertical boundaries of .
Assumption 2. and . Moreover, there exists a positive number satisfying such that(1)if and , then ;(2)if and , then .
Based on Assumptions 1 and 2, we directly have the following.
Lemma 3 (see [16]). If the map satisfies Assumptions 1 and 2, then has an invariant Cantor set, on which it is topologically conjugate to a full shift on symbols and has(i)a countable infinity of periodic orbits of arbitrarily high period;(ii)an uncountable infinity of nonperiodic orbits;(iii)a dense orbit.
Remark 4 (see [16–18]). If satisfies Assumption 2, we call that satisfies the -cone condition.
2. Main Result
In this section, we will analytically prove the existence of a Smale horseshoe in the classical planar circular restricted three-body problem introduced in Section 1, arriving at the following theorem.
Theorem 5. For the classical planar circular restricted three-body problem introduced in Section 1, when the mass ratio is sufficiently small, there exists a Smale horseshoe and thus the system (1) processes chaotic dynamics of the Smale horseshoe type.
In order to prove Theorem 5, we will construct an invertible map and then verify that this satisfies the Conley-Moser conditions.
2.1. Construction of an Invertible Map
According to the McGehee transformation , [2], the Hamiltonian of the system (1) can be reformulated as follows:
Thus, the system (1) can be reformulated as
For the energy surface , where is a constant, there exists a -periodic solution with respect to ; that is, . Further, near this periodic solution, by solving the Jacobi integral for , we have , where is second order in and and -periodic with respect to .
Thus, the system (5) can be further reformulated as where and are -periodic with respect to , is the third order in , and is fourth order in .
From [4, 12], the origin can be regarded as a periodic orbit with period with respect to in the system (6). Moreover, the Poincaré map of the periodic orbit has the form where , , and , are real analytic and contain terms of at least second order in .
Using polar coordinates , the Poincaré map can be reformulated as
According to formula (8), by making the following linear transformation: the system (6) can be reformulated as follows: where . Due to the symmetry of the problem, we subsequently restrict our discussion to the positive quadrant.
We neglect the higher order terms of (10) and then obtain that . It is clear that its solution remains on the hyperbolae , where is a constant. We substitute into the first expression of (10) and neglect the higher order terms, arriving at .
Let be a plane transversal to the periodic orbit at the origin and let be a sufficiently small neighborhood of the origin in the plane . For an arbitrary but fixed point , we define with .
Assume that ; then , where is an auxiliary variable. Substituting into , we can obtain where and .
Moreover, we can calculatewhere . Clearly, .
For the approximate system obtained by neglecting the higher order terms in the system (10), we can describe the Poincaré map defined over the plane by using the truncated flow near the degenerate saddle as follows: Since the terms neglected in (10) are both and , we can use this Poincaré map to approximate .
Letting and , then we can obtain
For the system (10), the coordinate axis corresponds to the local stable manifold and corresponds to the unstable manifold , respectively. Moreover, from [4, 12], when the mass ratio is sufficiently small, there exists a transversal homoclinic orbit, denoted as , of the periodic orbit . Thus, there exist two points and such that , , and . For convenience, by introducing a scale transformation, we can further assume that and .
We define and as the corresponding neighborhoods of and , respectively. For sufficiently small positive numbers and , and satisfy , . Let . When is sufficiently large, . Moreover, we also can obtain for . Again let . When is sufficiently large, . The relation between and can be seen from Figure 1.
Since , when and are sufficiently small, every positive half-orbit of the system (10) that starts from intersects a neighborhood of the point at a point, where . This can be depicted by the map . It is clear that is a diffeomorphism. Let . Since the stable manifold and the unstable manifold of the periodic orbit transversally intersect along , we can obtain .
Let , , , and . Then, there exists a sufficiently small such that , , , . Moreover, let and . We can further obtain that there exists a sufficiently small such that , , .
Based on and , we construct a successor map . Further, we define another map over the set such that . Clearly, is also a homeomorphism.
2.2. Proofs of Some Propositions for
In order to prove that satisfies the Conley-Moser conditions, we need to introduce one lemma and then prove four propositions in this subsection.
Lemma 6 (see [17, 18]). Consider two invertible linear operators of into itself: where . Let be a constant such that the following conditions hold: Then, for arbitrary , there exists a positive constant , which is dependent on , and , such that if the following conditions hold: the linear map satisfies the -cone condition.
By Lemma 6, we have the following proposition.
Proposition 7. For two arbitrary constants and with , when is sufficiently large, satisfies the -cone condition.
Proof. Based on the chain rule on the derivative of a composite function, we can obtainwhere . Let . Since and is , we have . Let and . Then .
Let , , , . Similar to the proof of Condition 1 in [19], after some simple calculations, we can obtain that , , , , , and . Further, when , we can obtain that , and .
Thus, there exists a such that, for sufficiently large , inequalities (16) and (17) in Lemma 6 hold. Thus, according to Lemma 6, we obtain that when is large enough, satisfies the -cone condition.
In fact, we can further prove the following.
Proposition 8. When is sufficiently large, satisfies the -cone condition.
Proof. Let be an arbitrary but fixed integer. For sufficiently large , let , , , and , where . Moreover, let and .
For an arbitrary point , let be a vector emanating from the point in the tangent space of . In addition, for given and , let be the unstable sector at and let be the stable sector at . Similar to Section 1, we also have , , , and .
In order to prove that satisfies the -cone condition, by Remark 4, we need to prove that satisfies Assumption 2. That is, we need to prove the following:(1) and ;(2)there exists a constant satisfying such that if and , where is a vector emanating from the point in the tangent space of ; if and , where is a vector emanating from the point in the tangent space of .
First, we want to prove that . For this, it is sufficient to prove that, for an arbitrary with , .
Clearly, and is bounded. According to the definitions of and , , .
Since and for defining and are chosen to be sufficiently small, and . Letting and , then . According to in Section 2.1, when is sufficiently large, we have
Clearly, . Moreover, when , . So, for sufficiently large and sufficiently small and , . Thus, . This directly implies that .
Second, following the proof of , we want to prove that there exists a constant satisfying such that .
Similarly, for the above and , when is sufficiently large,
For given and , . So, for any constant satisfying , when is sufficiently large, . Thus, we have , implying that .
Third, we want to prove and . For this, let be the inverse map of . Then,
For any with , , . Similarly, we can prove that and . Thus, and .
Based on all above analysis and Remark 4, we can then obtain that when is sufficiently large, satisfies the -cone condition.
Based on Proposition 8, we can prove that satisfies the boundary condition.
Proposition 9. When and are sufficiently large, and , where is the horizontal boundary of and is the vertical boundary of .
Proof. Due to Proposition 8, when is sufficiently large, satisfies the -cone condition. This implies that contracts in the horizontal direction and expands in the vertical direction. Moreover, -vertical curves are mapped to -vertical curves under the map and -horizontal curves are mapped to -horizontal curves under the map . Thus, for sufficiently large and , , , and . In addition, . Thus, according to the expression in Section 2.1, . Similarly, since , we can obtain that .
Finally, we can prove that satisfies the intersection condition as follows.
Proposition 10. When and are sufficiently large, .
Proof. Let be the family of vertical curves in , where and . From Proposition 8, with sufficiently large satisfies the -cone condition. Thus, infinitely approaches when . Similarly, letting be the family of horizontal curves in , where , we can obtain that infinitely approaches when .
Since is , infinitely approaches when . By the expression in Section 2.1, . Thus, .
Remark 11. In fact, we can prove that when , and intersect at a unique point near .
2.3. Proof of Our Theorem 5
In order to prove our Theorem 5, similar to [19], we try to use Propositions 7, 9, and 10 to verify that satisfies Assumptions 1 and 2. Then, from Lemma 3, we can obtain that is a horseshoe map as follows.
Proof. From Proposition 7, when is sufficiently large, satisfies the -cone condition. Thus, the map contracts in the horizontal direction and expands in the vertical direction. Moreover, -vertical curves are mapped to -vertical curves under the map and -horizontal curves are mapped to -horizontal curves under the map . Therefore, from Propositions 9 and 10, for sufficiently large and , is a -curved rectangle and satisfies and . Similarly, for sufficiently large and , is also a -curved rectangle and satisfies and .
Let be an arbitrary but fixed positive integer. For sufficiently large , , and , by letting , , , and , where , we can obtain that satisfies Assumption 1. In addition, due to Remark 4, obviously satisfies Assumption 2.
Thus, for an arbitrary but fixed , when is sufficiently large, the map over the set satisfies Assumptions 1 and 2; that is, satisfies the Conley-Moser conditions.
By Lemma 3, when is sufficiently large, has an invariant Cantor set, on which it is topologically conjugate to a full shift on symbols. This directly implies that is a horseshoe map.
3. Conclusions
In this present note, we studied the existence of a Smale horseshoe in a planar circular restricted three-body problem by first defining an invertible map and then proving that this satisfies the Conley-Moser conditions. This implies that the planar circular restricted three-body problem processes chaotic dynamics of the Smale horseshoe type.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work was partly supported by NSFC-11422111, NSFC-11290141, NSFC-11371047, and SKLSDE-2013ZX-10.