Abstract and Applied Analysis

Volume 2015, Article ID 965829, 8 pages

http://dx.doi.org/10.1155/2015/965829

## A Note on the Existence of a Smale Horseshoe in the Planar Circular Restricted Three-Body Problem

SKLSDE, LMIB and School of Mathematics and Systems Science, Beihang University, Beijing 100191, China

Received 26 June 2014; Accepted 24 September 2014

Academic Editor: Tonghua Zhang

Copyright © 2015 Xuhua Cheng and Zhikun She. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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.*