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

Authors: Xuhua Cheng; Zhikun She
Journal: Abstract and Applied Analysis
 (2015)

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## 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.


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## Body

## 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 P1 and P2, of mass 1-μ and μ, move uniformly in a circular orbit about their common center of mass with angular velocity ω. The orbit is located in the Oxy plane of the inertial frame of reference and the common center of mass is in the origin. The particle P3 of infinitesimal mass m3 moves in the gravitational field generated by P1 and P2. Note that since the mass of P3 is so small, its effects on other three particles can be ignored. Without loss of generality, assume that, in the Ox¯y¯ plane of the rotating frame of reference, the particles P1 and P2 rest at the points (μ,0) and (μ-1,0), respectively. By denoting their polar coordinates by ρ and φ and using the polar angle τ=ωt as a new independent variable, the equation of motion of the infinitesimal particle P3 can be written as follows:(1)dρdτ=pρ,dpρdτ=pφ2ρ3-(1-μ)(ρ-μcos⁡φ)(ρ2+μ2-2ρμcos⁡φ)3/2-μ[ρ+(1-μ)cos⁡φ](1-μ)[ρ2+(1-μ)2+2ρ(1-μ)cos⁡φ]3/2,dφdτ=pφρ2-1,dpφdτ=-μ(1-μ)×ρsinφ[1[ρ2+(1-μ)2+2ρ(1-μ)cos⁡φ]3/21[ρ2+μ2-2ρμcos⁡φ]3/2kkkkkkkkk-1[ρ2+(1-μ)2+2ρ(1-μ)cos⁡φ]3/2],
where pρ and pφ are momenta canonically conjugate to the coordinates ρ and φ, respectively.The Hamiltonian of the system (1) is
(2)H=12(pρ2+pφ2ρ2-2pφ)-1-μ(ρ2+μ2-2ρμcos⁡φ)1/2-μ(ρ2+(1-μ)2+2ρ(1-μ)cos⁡φ)1/2.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.Letf:D↦R2 be an invertible map, where D={(x,y)∈R2∣0≤x≤1,0≤y≤1}, and f is at least C1. For two given μv>0 and μh>0, let K={1,2,…,N} (N≥2) be an index set, let H1,…,HN be the N disjoint μh-horizontal strips, and V1,…,VN be the N disjoint μv-vertical strips. For each i,j∈K, denote f(Hi)⋂‍Hj as Vji and Hi⋂‍f-1(Hj) as Hij. Clearly, Hij=f-1(Vji). Define H=⋃i,j∈K‍Hij and V=⋃i,j∈K‍Vji. It is also obvious that f(H)=V.For an arbitrary pointz0=(x0,y0)∈H⋃‍V, let (ξz0,ηz0) be a vector emanating from the point z0 in the tangent space of z0. The stable sector at z0 is then defined as Sz0s={(ξz0,ηz0)∈R2∣|ηz0|≤μh|ξz0|}. Similarly, the unstable sector at z0 is defined as Sz0u={(ξz0,ηz0)∈R2∣|ξz0|≤μv|ηz0|}. By taking the union of the stable and unstable sectors over all points in H and V, we can define sector bundles as follows:
(3)SHs=⋃z0∈H‍Sz0s,SVs=⋃z0∈V‍Sz0s;SHu=⋃z0∈H‍Sz0u,SVu=⋃z0∈V‍Sz0u.Then, the Conley-Moser conditions for the mapf are described by the following two assumptions.Assumption 1.
0
≤
μ
v
μ
h
<
1 and, for each i∈{1,2,…,N}, f maps Hi homeomorphically onto Vi; that is, f(Hi)=Vi. Moreover, the horizontal boundaries of Hi are mapped to the horizontal boundaries of Vi and the vertical boundaries of Hi are mapped to the vertical boundaries of Vi.Assumption 2.
D
f
(
S
H
u
)
⊂
S
V
u and Df-1(SVs)⊂SHs. Moreover, there exists a positive number λ satisfying 0<λ<1-μvμh such that(1)
if(ξz0,ηz0)∈Sz0u and (ξf(z0),ηf(z0))≐Df(z0)(ξz0,ηz0)∈Sf(z0)u, then |ηf(z0)|≥(1/λ)|ηz0|;(2)
if(ξz0,ηz0)∈Sz0s and (ξf-1(z0),ηf-1(z0))≐Df-1(z0)(ξz0,ηz0)∈Sf-1(z0)s, then |ξf-1(z0)|≥(1/λ)|ξz0|.Based on Assumptions1 and 2, we directly have the following.Lemma 3 (see [16]).
If the mapf satisfies Assumptions 1 and 2, then f has an invariant Cantor set, on which it is topologically conjugate to a full shift on N 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]).
Iff satisfies Assumption 2, we call that f satisfies the (μh,μv)-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 Section1, arriving at the following theorem.Theorem 5.
For the classical planar circular restricted three-body problem introduced in Section1, when the mass ratio μ 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 Theorem5, we will construct an invertible map f and then verify that this f satisfies the Conley-Moser conditions.


### 2.1. Construction of an Invertible Mapf

According to the McGehee transformationρ=1/x2, pρ=y [2], the Hamiltonian of the system (1) can be reformulated as follows:
(4)H=12(y2+x4pφ2-2pφ)-(1-μ)x2(1+x4μ2-2x2μcos⁡φ)1/2-μx2[1+x4(1-μ)2+2x2(1-μ)cos⁡φ]1/2.Thus, the system (1) can be reformulated as
(5)dxdτ=-12x3y,dydτ=pφ2x6-(1-μ)(1-μx2cos⁡φ)x4(1+μ2x4-2μx2cos⁡φ)3/2-μ[1+(1-μ)x2cos⁡φ]x4[1+(1-μ)2x4+2(1-μ)x2cos⁡φ]3/2,dφdτ=pφx4-1,dpφdτ=μ(1-μ)x4×sinφ[1[1+(1-u)2x4+2(1-μ)x2cos⁡φ]3/2kkkkkkkkk-1(1+μ2x4-2μx2cos⁡φ)3/21[1+(1-u)2x4+2(1-μ)x2cos⁡φ]3/2].For the energy surfaceH=h, where h is a constant, there exists a 2π-periodic solution with respect to φ; that is, (x,y,pφ)=(0,0,-h). Further, near this periodic solution, by solving the Jacobi integral for pφ, we have pφ=-h+ν1(x,y,φ), where ν1(x,y,φ) is second order in x and y and 2π-periodic with respect to φ.Thus, the system (5) can be further reformulated as
(6)dxdτ=-12x3y,dydτ=-(1-2μ)(x4+g1(x,y,φ,μ)),dφdτ=-1+g2(x,y,φ,μ),
where g1 and g2 are 2π-periodic with respect to φ, g1 is the third order in (x,y), and g2 is fourth order in (x,y).From [4, 12], the origin (0,0) can be regarded as a periodic orbit γμ with period 2π with respect to φ in the system (6). Moreover, the Poincaré map of the periodic orbit (x,y)=(0,0) has the form
(7)P0:x⟶x+k1x3(y+r1(x,y))y⟶y+k2x3(x+r2(x,y)),
where k1=π, k2=2π(1-2μ), and r1, r2 are real analytic and contain terms of at least second order in (x,y).Using polar coordinates(ρ,θ), the Poincaré map P0 can be reformulated as
(8)P0:r⟶r-k1r4cos⁡4θ((4μ-3)sinθ+o(r))θ⟶θ-k2r3cos⁡3θkkkkkk×(12(1-2μ)sin2θ-cos⁡2θ+o(r)).According to formula (8), by making the following linear transformation:
(9)x=u+v,y=-2(1-2μ)(u-v),
the system (6) can be reformulated as follows:
(10)dudτ=(u+v)3k3u,dvdτ=-(u+v)3(k3v+h1(u,v,φ,μ)),dφdτ=-1+h2(u,v,φ,μ),
where k3=2(1-2μ)/2. 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 du/dv=-u/v. It is clear that its solution remains on the hyperbolae uv=c0>0, where c0 is a constant. We substitute v=c0/u into the first expression of (10) and neglect the higher order terms, arriving at du/dφ=-k3((u2+c0)3/u2).LetΣ be a plane transversal to the periodic orbit γμ at the origin (0,0) and let U0 be a sufficiently small neighborhood of the origin (0,0) in the plane Σ. For an arbitrary but fixed point (u0,v0)∈U0∖{(0,0)}, we define Tφ(u0,v0)=(uφ,vφ) with T0(u0,v0)=(u0,v0).Assume thatuφ=c0tan(ϕφ/4); then vφ=c0cot(ϕφ/4), where ϕφ is an auxiliary variable. Substituting uφ=c0tan(ϕφ/4) into du/dφ=-k3((u2+c0)3/u2), we can obtain
(11)ϕφ-sinϕφ=k0-32kc03/2φ,k0=ϕ0-sinϕ0,
where c0=u0v0 and ϕ0=4arctanu0/v0.Moreover, we can calculate(12)DTφ=[∂uφ∂u0∂uφ∂v0∂vφ∂u0∂vφ∂v0]=[uφ2u0(1+Δ-3k3(uφ+vφ)3φ)u0uφ2c(1-Δ-3k3(uφ+vφ)3φ)c2u0uφ(1-Δ+3k3(uφ+vφ)3φ)u02uφ(1+Δ+3k3(uφ+vφ)3φ)],where Δ=((uφ+vφ)/(u0+v0))3. Clearly, det⁡DTφ=Δ≠0.For the approximate system obtained by neglecting the higher order terms in the system (10), we can describe the Poincaré map P defined over the plane Σ by using the truncated flow near the degenerate saddle as follows:
(13)P:(u0,v0)⟼(u2π,v2π),where(u0,v0)∈U0.
Since the terms neglected in (10) are both o(u4,v4) and O(μ), we can use this Poincaré map P to approximate P0.Lettingu0=c0tan(ϕ0/4) and v0=c0cot(ϕ0/4), then we can obtain
(14)Pk(u0,v0)≗(uk,vk)=(c0tanϕ2kπ4,c0cotϕ2kπ4).For the system (10), the coordinate axis v=0 corresponds to the local stable manifold Wloc⁡s(γμ) and u=0 corresponds to the unstable manifold Wloc⁡u(γμ), respectively. Moreover, from [4, 12], when the mass ratio μ is sufficiently small, there exists a transversal homoclinic orbit, denoted as γ, of the periodic orbit γμ. Thus, there exist two points p and q such that p∈Wloc⁡s(γμ), q∈Wloc⁡u(γμ), and p,q∈Σ⋂‍γ. For convenience, by introducing a scale transformation, we can further assume that p=(1,0) and q=(0,1).We defineB={(u,v)∣|u-1|≤δ1,|v|≤δ2} and B~={(u,v)∣|u|≤δ2,|v-1|≤δ1} as the corresponding neighborhoods of p and q, respectively. For sufficiently small positive numbers δ1 and δ2, B and B~ satisfy PB⋂‍B=∅, P-1B~⋂‍B~=∅. Let Dk=P-kB~⋂‍B. When k is sufficiently large, Dk≠∅. Moreover, we also can obtain Dk⋂‍Dm=∅ for k≠m. Again let D~k=PkDk. When k is sufficiently large, D~k≠∅. The relation between Dk and D~k can be seen from Figure 1.Figure 1
The relation betweenDk and D~k.Sincep,q∈Σ⋂‍γ, when δ1 and δ2 are sufficiently small, every positive half-orbit of the system (10) that starts from B~ intersects a neighborhood Up of the point p at a point, where Up⊂Σ. This can be depicted by the map F:B~→Up. It is clear that F is a C1 diffeomorphism. Let F(u,v)=(FU,FV). Since the stable manifold and the unstable manifold of the periodic orbit γμ transversally intersect along γ, we can obtain (∂FV/∂v)∣q≠0.LetBh=B⋂‍{v=0}, B~v=B~⋂‍{u=0}, ∂Bh={u=1±δ1,v=0}, and ∂B~v={u=0,v=1±δ1}. Then, there exists a sufficiently small δ1 such that FB~v⋂‍Bh={p}, (∂FV/∂v)∣B~v≠0, FB~v⋂‍∂Bh=∅, F∂B~v⋂‍Bh=∅. Moreover, let ∂vB={(u,v)∈B∣u=1±δ1} and ∂hB~={(u,v)∈B~∣v=1±δ1}. We can further obtain that there exists a sufficiently small δ2 such that (∂FV/∂v)∣B~≠0, FB~⋂‍∂vB=∅, F(∂hB~)⋂‍B=∅.Based onP and F, we construct a successor map Δk=F∘Pk:Dk→Up. Further, we define another map f over the set ⋃k‍Dk such that f∣Dk=Δk. Clearly, f is also a homeomorphism.


### 2.2. Proofs of Some Propositions forf

In order to prove thatf 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 ofR1×R1 into itself:
(15)I=[abcd],J=[ΛEGM],
where dM≠0. Let L>0 be a constant such that the following conditions hold:
(16)‖I‖<L,‖I-1‖<L,|d-1|<L,|EM-1|<L.
Then, for arbitrary 0<μh<μv-1≪1, there exists a positive constant δ0, which is dependent on L, μh and μv, such that if the following conditions hold:
(17)|M-1|<δ0,|Λ-EM-1G|<δ0,|ΛM-1|<δ0,|GM-1|<δ0,|cEM-1|<δ0,
the linear map A=IJ satisfies the (μh,μv)-cone condition.By Lemma6, we have the following proposition.Proposition 7.
For two arbitrary constantsμh and μv with 0<μh<μv-1≪1, when k is sufficiently large, f|Dk satisfies the (μh,μv)-cone condition.Proof.
Based on the chain rule on the derivative of a composite function,  we can obtain(18)Df∣Dk=DF·DPk=DF·DTφ=[∂FU∂u∂FU∂v∂FV∂u∂FV∂v].[∂uφ∂u0∂uφ∂v0∂vφ∂u0∂vφ∂v0],
whereφ=2kπ. Let L=sup⁡β∈B~{‖DF(β)‖,‖DF(β)-1‖,|∂FV/∂v|-1}. Since ∂FV/∂v|B~≠0 and F is C1, we have L<+∞. Let u0=δ≈1 and v0=c0/δ≪1. Then (uk,vk)≈(c0/δ,δ).
LetE=∂uφ/∂v0, Λ=∂uφ/∂u0, G=∂vφ/∂u0, M=∂vφ/∂v0. Similar to the proof of Condition 1 in [19], after some simple calculations, we can obtain that lim⁡k→+∞|EM-1|≈c0/δ2, lim⁡k→+∞|M-1|=0, lim⁡k→+∞|Λ-EM-1G|=0, lim⁡k→+∞|ΛM-1|≈c0/δ4, lim⁡k→+∞|GM-1|≈c0/δ2, and lim⁡k→+∞|EM-1|≈c0/δ2. Further, when c0→0, we can obtain that c0/δ2→0, c0/δ4→0 and (∂FV/∂u)(c0/δ2)→0.
Thus, there exists aδ>0 such that, for sufficiently large k, inequalities (16) and (17) in Lemma 6 hold. Thus, according to Lemma 6, we obtain that when k is large enough, f∣Dk satisfies the (μh,μv)-cone condition.In fact, we can further prove the following.Proposition 8.
Whenk is sufficiently large, Pk satisfies the (μh,μv)-cone condition.Proof.
LetN≥2 be an arbitrary but fixed integer. For sufficiently large k, let Hl=Dl+k-1, Vl=f(Dl+k-1), Vji=PkHi⋂‍Hj, and Hij=Hi⋂‍P-kHj, where 1≤l,i,j≤N. Moreover, let H=⋃i,j‍Hij and V=⋃i,j‍Vij.
For an arbitrary pointz0∈H⋃‍V, let (ξz0,ηz0) be a vector emanating from the point z0 in the tangent space of z0. In addition, for given μh and μv, let Sz0u={(ξz0,ηz0)∣|ξz0|<μv|ηz0|} be the unstable sector at z0 and let Sz0s={(ξz0,ηz0)∣|ηz0|<μh|ξz0|} be the stable sector at z0. Similar to Section 1, we also have SHu, SVu, SHs, and SVs.
In order to prove thatPk satisfies the (μh,μv)-cone condition, by Remark 4, we need to prove that Pk satisfies Assumption 2. That is, we need to prove the following:(1)
DPk(SHu)⊂SVu and DP-k(SVs)⊂SHs;(2)
there exists a constantλ satisfying 0<λ<1-μhμv such that |ηPk(z0)|≥λ-1|ηz0| if (ξz0,ηz0)∈Sz0u and (ξPk(z0),ηPk(z0))≐DPk(z0)(ξz0,ηz0)∈SPk(z0)u, where (ξPk(z0),ηPk(z0)) is a vector emanating from the point Pk(z0) in the tangent space of Pk(z0); |ξP-k(z0)|≥λ-1|ξz0| if (ξz0,ηz0)∈Sz0s and (ξP-k(z0),ηP-k(z0))≐DP-k(z0)(ξz0,ηz0)∈SP-k(z0)s, where (ξP-k(z0),ηP-k(z0)) is a vector emanating from the point P-k(z0) in the tangent space of P-k(z0).
First, we want to prove thatDPk(SHu)⊂SVu. For this, it is sufficient to prove that, for an arbitrary z0=(u0,v0)∈H with (ξz0,ηz0)=(1,ϑ)∈Sz0u, (ξPk(z0),ηPk(z0))=DPk(z0)(ξz0,ηz0)∈SVu.
Clearly,Pk(z0)∈V and ϑ is bounded. According to the definitions of Tφ and P, DPk(z0)(ξz0,ηz0)= DT2kπ(z0)(ξz0,ηz0)= ((∂u2kπ/∂u0)+(∂u2kπ/∂v0)ϑ, (∂v2kπ/∂u0)+(∂v2kπ/∂v0)ϑ).
Sinceδ1 and δ2 for defining B and B~ are chosen to be sufficiently small, u0≈1 and v0≤δ2≪1. Letting u0=δ and c=u0v0, then (u2kπ,v2kπ)≈(c/δ,δ). According to DTφ in Section 2.1, when k is sufficiently large, we have
(19)|(∂v2kπ/∂u0)+(∂v2kπ/∂v0)ϑ(∂u2kπ/∂u0)+(∂u2kπ/∂v0)ϑ|=cu2kπ2×|kkkkkkkkk×(v0+u0ϑ))-1kkkkkkkkk×(v0+u0ϑ)(1-6k3(u2kπ+v2kπ)3kπ))-1(Δ(u0ϑ-v0)+(1+6k3(u2kπ+v2kπ)3kπ)kkkkkkk×(v0+u0ϑ)ef22eΔ(u0ϑ-v0)+(1+6k3(u2kπ+v2kπ)3kπ))kkk×((v0+u0ϑ)Δ(u0ϑ-v0)-(1-6k3(u2kπ+v2kπ)3kπ)kkkkkkkkk×(v0+u0ϑ)(1-6k3(u2kπ+v2kπ)3kπ))-1|≈δ2c|×(δϑ+cδ(1-6k3((δ2+c)/δ)3kπ))(1-6k3(δ2+cδ)3kπ))-1(Δ(δϑ-cδ)+(1+6k3(δ2+cδ)3kπ)kkkkkkkk×(δϑ+cδΔ(δϑ-(c/δ))+(1+6k3((δ2+c)/δ)3kπ))(1+6k3(δ2+cδ)3kπ))kkkkkkk×(Δ(δϑ-cδ)-(1-6k3(δ2+cδ)3kπ)kkkkkkkkk×(δϑ+cδ(1-6k3((δ2+c)/δ)3kπ))(1-6k3(δ2+cδ)3kπ))-1(Δ(δϑ-(cδ))+(1+6k3((δ2+c)δ)3kπ)|≈δ2c|1+13k3δ3kπ|.
Clearly,lim⁡k→+∞(δ2/c)|1+1/3k3δ3kπ|=δ2/c. Moreover, when c→0, δ2/c→+∞. So, for sufficiently large k and sufficiently small δ1 and δ2, |((∂v2kπ/∂u0)+(∂v2kπ/∂v0)ϑ)/((∂u2kπ/∂u0)+(∂u2kπ/∂v0)ϑ)|>1/μv. Thus, DPk(z0)(ξz0,ηz0)∈SVu. This directly implies that DPk(SHu)⊂SVu.
Second, following the proof ofDPk(SHu)⊂SVu, we want to prove that there exists a constant λ satisfying 0<λ<1-μhμv such that |ηPk(z0)|≥λ-1|ηz0|.
Similarly, for the above(u0,v0) and (u2kπ,v2kπ), when k is sufficiently large,
(20)|∂v2kπ∂u0+∂v2kπ∂v0ϑ|=12u2kπ|Δ(u0ϑ-v0)+(1+6k3(u2kπ+v2kπ)3kπ)kkkkkk×(v0+u0ϑ)|≈12c|(1+6k3(δ2+cδ)3kπ)Δ(δ2ϑ-c)kkkkkkkk+(1+6k3(δ2+cδ)3kπ)kkkkkkkk×(c+δ2)(1+6k3(δ2+cδ)3kπ)|≈δ2c|(1+3k3δ3kπ)ϑ|.
For givenB and B~, lim⁡k→+∞(δ2/c)|1+3k3δ3kπ|=∞. So, for any constant λ satisfying 0<λ<1-μhμv, when k is sufficiently large, (δ2/c)|1+3k3δ3kπ|>λ-1. Thus, we have |(∂v2kπ/∂u0)+(∂v2kπ/∂v0)ϑ|>λ-1|ϑ|, implying that |ηPk(z0)|≥λ-1|ηz0|.
Third, we want to proveDP-k(SVs)⊂SHs and |ξP-k(z0)|≥λ-1|ξz0|. For this, let Tφ-1(uφ,vφ)=(u0,v0) be the inverse map of Tφ. Then,
(21)DTφ-1=(DTφ)-1=[∂uφ∂u0∂uφ∂v0∂vφ∂u0∂vφ∂v0]-1=1△[∂vφ∂v0-∂uφ∂v0-∂vφ∂u0∂uφ∂u0].
For any z0=(u0,v0)∈V with (ξz0,ηz0)=(ϖ,1)∈Sz0s, (ξP-k(z0),ηP-k(z0))= DP-k(ξz0,ηz0)= DT2kπ-1(ξz0,ηz0)= (1/△)((∂v2kπ/∂v0)ϖ-(∂u2kπ/∂v0), (-∂v2kπ/∂u0)ϖ+(∂u2kπ/∂u0)). Similarly, we can prove that |(∂v2kπ/∂v0)ϖ-(∂u2kπ/∂v0)/(-∂v2kπ/∂u0)ϖ+(∂u2kπ/∂u0)|>μh and |(∂v2kπ/∂v0)ϖ-(∂u2kπ/∂v0)|>λ-1|ϖ|. Thus, DP-k(SVs)⊂SHs and |ξP-k(z0)|≥λ-1|ξz0|.
Based on all above analysis and Remark4, we can then obtain that when k is sufficiently large, Pk satisfies the (μh,μv)-cone condition.Based on Proposition8, we can prove that f satisfies the boundary condition.Proposition 9.
Wheni and j are sufficiently large, f(∂hDi)⋂‍Dj=∅ and fDi⋂‍∂vDj=∅, where ∂hDi is the horizontal boundary of Di and ∂vDj is the vertical boundary of Dj.Proof.
Due to Proposition8, when k is sufficiently large, Pk satisfies the (μh,μv)-cone condition. This implies that Pk contracts in the horizontal direction and expands in the vertical direction. Moreover, μv-vertical curves are mapped to μv-vertical curves under the map Pk and μh-horizontal curves are mapped to μh-horizontal curves under the map P-k. Thus, for sufficiently large i and j, Dj⊂B, PiDi⊂B~, ∂vDj⊂∂vB and Pi(∂hDi)⊂∂hB~. In addition, f(∂hDi)⋂‍Dj= (F∘Pi(∂hDi))⋂‍Dj=(F∘(Pi(∂hDi)))⋂‍Dj. Thus, according to the expression F(∂hB~)⋂‍B=∅ in Section 2.1, f(∂hDi)⋂‍Dj=∅. Similarly, since FB~⋂‍∂vB=∅, we can obtain that fDi⋂‍∂vDj=∅.Finally, we can prove thatf satisfies the intersection condition as follows.Proposition 10.
Wheni and j are sufficiently large, fDi⋂‍Dj≠∅.Proof.
LetCvi(u)={u}×(Di)v be the family of vertical curves in Di, where u∈Bh and (Di)v={v∣(u,v)∈B,Pi(u,v)∈B~}. From Proposition 8, Pi with sufficiently large i satisfies the (μh,μv)-cone condition. Thus, PiCvi(u) infinitely approaches B~v when i→+∞. Similarly, letting Chi(v)=Bh×{v} be the family of horizontal curves in Di, where v∈(Di)v, we can obtain that Chj(v) infinitely approaches Bh when j→+∞.
SinceF is C1, F(PiCvi(u)) infinitely approaches F(B~v) when i→+∞. By the expression FB~v⋂‍Bh={p} in Section 2.1, F(PiCvi(u))⋂‍Chj(v)≠∅. Thus, fDi⋂‍Dj≠∅.Remark 11.
In fact, we can prove that wheni,j→+∞, fCvi(u) and Chj(v) intersect at a unique point near p.


### 2.3. Proof of Our Theorem5

In order to prove our Theorem5, similar to [19], we try to use Propositions 7, 9, and 10 to verify that f satisfies Assumptions 1 and 2. Then, from Lemma 3, we can obtain that f is a horseshoe map as follows.Proof.
From Proposition7, when k is sufficiently large, f∣Dk satisfies the (μh,μv)-cone condition. Thus, the map f contracts in the horizontal direction and expands in the vertical direction. Moreover, μv-vertical curves are mapped to μv-vertical curves under the map f and μh-horizontal curves are mapped to μh-horizontal curves under the map f-1. Therefore, from Propositions 9 and 10, for sufficiently large i and j, fDi⋂‍Dj≠∅ is a (μh,μv)-curved rectangle and satisfies ∂h(fDi⋂‍Dj)⊂∂hDj and ∂v(fDi⋂‍Dj)⊂∂v(fDi). Similarly, for sufficiently large i and j, f-1(fDi⋂‍Dj)≠∅ is also a (μh,μv)-curved rectangle and satisfies ∂vf-1(fDi⋂‍Dj)⊂∂vDi and ∂hf-1(fDi⋂‍Dj)⊂∂h(f-1Dj).
LetN≥2 be an arbitrary but fixed positive integer. For sufficiently large k, i, and j, by letting Hl=Dl+k-1, Vl=f(Dl+k-1), fDi⋂‍Dj=V(j-k+1)(i-k+1), and Di⋂‍f-1Dj=H(i-k+1)(j-k+1), where 1≤l≤N, we can obtain that f satisfies Assumption 1. In addition, due to Remark 4, f obviously satisfies Assumption 2.
Thus, for an arbitrary but fixedN≥2, when k is sufficiently large, the map f over the set ⋃i=0N-1‍Dk+i satisfies Assumptions 1 and 2; that is, f satisfies the Conley-Moser conditions.
By Lemma3, when k is sufficiently large, f has an invariant Cantor set, on which it is topologically conjugate to a full shift on N symbols. This directly implies that f is a horseshoe map.


## 2.1. Construction of an Invertible Mapf

According to the McGehee transformationρ=1/x2, pρ=y [2], the Hamiltonian of the system (1) can be reformulated as follows:
(4)H=12(y2+x4pφ2-2pφ)-(1-μ)x2(1+x4μ2-2x2μcos⁡φ)1/2-μx2[1+x4(1-μ)2+2x2(1-μ)cos⁡φ]1/2.Thus, the system (1) can be reformulated as
(5)dxdτ=-12x3y,dydτ=pφ2x6-(1-μ)(1-μx2cos⁡φ)x4(1+μ2x4-2μx2cos⁡φ)3/2-μ[1+(1-μ)x2cos⁡φ]x4[1+(1-μ)2x4+2(1-μ)x2cos⁡φ]3/2,dφdτ=pφx4-1,dpφdτ=μ(1-μ)x4×sinφ[1[1+(1-u)2x4+2(1-μ)x2cos⁡φ]3/2kkkkkkkkk-1(1+μ2x4-2μx2cos⁡φ)3/21[1+(1-u)2x4+2(1-μ)x2cos⁡φ]3/2].For the energy surfaceH=h, where h is a constant, there exists a 2π-periodic solution with respect to φ; that is, (x,y,pφ)=(0,0,-h). Further, near this periodic solution, by solving the Jacobi integral for pφ, we have pφ=-h+ν1(x,y,φ), where ν1(x,y,φ) is second order in x and y and 2π-periodic with respect to φ.Thus, the system (5) can be further reformulated as
(6)dxdτ=-12x3y,dydτ=-(1-2μ)(x4+g1(x,y,φ,μ)),dφdτ=-1+g2(x,y,φ,μ),
where g1 and g2 are 2π-periodic with respect to φ, g1 is the third order in (x,y), and g2 is fourth order in (x,y).From [4, 12], the origin (0,0) can be regarded as a periodic orbit γμ with period 2π with respect to φ in the system (6). Moreover, the Poincaré map of the periodic orbit (x,y)=(0,0) has the form
(7)P0:x⟶x+k1x3(y+r1(x,y))y⟶y+k2x3(x+r2(x,y)),
where k1=π, k2=2π(1-2μ), and r1, r2 are real analytic and contain terms of at least second order in (x,y).Using polar coordinates(ρ,θ), the Poincaré map P0 can be reformulated as
(8)P0:r⟶r-k1r4cos⁡4θ((4μ-3)sinθ+o(r))θ⟶θ-k2r3cos⁡3θkkkkkk×(12(1-2μ)sin2θ-cos⁡2θ+o(r)).According to formula (8), by making the following linear transformation:
(9)x=u+v,y=-2(1-2μ)(u-v),
the system (6) can be reformulated as follows:
(10)dudτ=(u+v)3k3u,dvdτ=-(u+v)3(k3v+h1(u,v,φ,μ)),dφdτ=-1+h2(u,v,φ,μ),
where k3=2(1-2μ)/2. 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 du/dv=-u/v. It is clear that its solution remains on the hyperbolae uv=c0>0, where c0 is a constant. We substitute v=c0/u into the first expression of (10) and neglect the higher order terms, arriving at du/dφ=-k3((u2+c0)3/u2).LetΣ be a plane transversal to the periodic orbit γμ at the origin (0,0) and let U0 be a sufficiently small neighborhood of the origin (0,0) in the plane Σ. For an arbitrary but fixed point (u0,v0)∈U0∖{(0,0)}, we define Tφ(u0,v0)=(uφ,vφ) with T0(u0,v0)=(u0,v0).Assume thatuφ=c0tan(ϕφ/4); then vφ=c0cot(ϕφ/4), where ϕφ is an auxiliary variable. Substituting uφ=c0tan(ϕφ/4) into du/dφ=-k3((u2+c0)3/u2), we can obtain
(11)ϕφ-sinϕφ=k0-32kc03/2φ,k0=ϕ0-sinϕ0,
where c0=u0v0 and ϕ0=4arctanu0/v0.Moreover, we can calculate(12)DTφ=[∂uφ∂u0∂uφ∂v0∂vφ∂u0∂vφ∂v0]=[uφ2u0(1+Δ-3k3(uφ+vφ)3φ)u0uφ2c(1-Δ-3k3(uφ+vφ)3φ)c2u0uφ(1-Δ+3k3(uφ+vφ)3φ)u02uφ(1+Δ+3k3(uφ+vφ)3φ)],where Δ=((uφ+vφ)/(u0+v0))3. Clearly, det⁡DTφ=Δ≠0.For the approximate system obtained by neglecting the higher order terms in the system (10), we can describe the Poincaré map P defined over the plane Σ by using the truncated flow near the degenerate saddle as follows:
(13)P:(u0,v0)⟼(u2π,v2π),where(u0,v0)∈U0.
Since the terms neglected in (10) are both o(u4,v4) and O(μ), we can use this Poincaré map P to approximate P0.Lettingu0=c0tan(ϕ0/4) and v0=c0cot(ϕ0/4), then we can obtain
(14)Pk(u0,v0)≗(uk,vk)=(c0tanϕ2kπ4,c0cotϕ2kπ4).For the system (10), the coordinate axis v=0 corresponds to the local stable manifold Wloc⁡s(γμ) and u=0 corresponds to the unstable manifold Wloc⁡u(γμ), respectively. Moreover, from [4, 12], when the mass ratio μ is sufficiently small, there exists a transversal homoclinic orbit, denoted as γ, of the periodic orbit γμ. Thus, there exist two points p and q such that p∈Wloc⁡s(γμ), q∈Wloc⁡u(γμ), and p,q∈Σ⋂‍γ. For convenience, by introducing a scale transformation, we can further assume that p=(1,0) and q=(0,1).We defineB={(u,v)∣|u-1|≤δ1,|v|≤δ2} and B~={(u,v)∣|u|≤δ2,|v-1|≤δ1} as the corresponding neighborhoods of p and q, respectively. For sufficiently small positive numbers δ1 and δ2, B and B~ satisfy PB⋂‍B=∅, P-1B~⋂‍B~=∅. Let Dk=P-kB~⋂‍B. When k is sufficiently large, Dk≠∅. Moreover, we also can obtain Dk⋂‍Dm=∅ for k≠m. Again let D~k=PkDk. When k is sufficiently large, D~k≠∅. The relation between Dk and D~k can be seen from Figure 1.Figure 1
The relation betweenDk and D~k.Sincep,q∈Σ⋂‍γ, when δ1 and δ2 are sufficiently small, every positive half-orbit of the system (10) that starts from B~ intersects a neighborhood Up of the point p at a point, where Up⊂Σ. This can be depicted by the map F:B~→Up. It is clear that F is a C1 diffeomorphism. Let F(u,v)=(FU,FV). Since the stable manifold and the unstable manifold of the periodic orbit γμ transversally intersect along γ, we can obtain (∂FV/∂v)∣q≠0.LetBh=B⋂‍{v=0}, B~v=B~⋂‍{u=0}, ∂Bh={u=1±δ1,v=0}, and ∂B~v={u=0,v=1±δ1}. Then, there exists a sufficiently small δ1 such that FB~v⋂‍Bh={p}, (∂FV/∂v)∣B~v≠0, FB~v⋂‍∂Bh=∅, F∂B~v⋂‍Bh=∅. Moreover, let ∂vB={(u,v)∈B∣u=1±δ1} and ∂hB~={(u,v)∈B~∣v=1±δ1}. We can further obtain that there exists a sufficiently small δ2 such that (∂FV/∂v)∣B~≠0, FB~⋂‍∂vB=∅, F(∂hB~)⋂‍B=∅.Based onP and F, we construct a successor map Δk=F∘Pk:Dk→Up. Further, we define another map f over the set ⋃k‍Dk such that f∣Dk=Δk. Clearly, f is also a homeomorphism.


## 2.2. Proofs of Some Propositions forf

In order to prove thatf 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 ofR1×R1 into itself:
(15)I=[abcd],J=[ΛEGM],
where dM≠0. Let L>0 be a constant such that the following conditions hold:
(16)‖I‖<L,‖I-1‖<L,|d-1|<L,|EM-1|<L.
Then, for arbitrary 0<μh<μv-1≪1, there exists a positive constant δ0, which is dependent on L, μh and μv, such that if the following conditions hold:
(17)|M-1|<δ0,|Λ-EM-1G|<δ0,|ΛM-1|<δ0,|GM-1|<δ0,|cEM-1|<δ0,
the linear map A=IJ satisfies the (μh,μv)-cone condition.By Lemma6, we have the following proposition.Proposition 7.
For two arbitrary constantsμh and μv with 0<μh<μv-1≪1, when k is sufficiently large, f|Dk satisfies the (μh,μv)-cone condition.Proof.
Based on the chain rule on the derivative of a composite function,  we can obtain(18)Df∣Dk=DF·DPk=DF·DTφ=[∂FU∂u∂FU∂v∂FV∂u∂FV∂v].[∂uφ∂u0∂uφ∂v0∂vφ∂u0∂vφ∂v0],
whereφ=2kπ. Let L=sup⁡β∈B~{‖DF(β)‖,‖DF(β)-1‖,|∂FV/∂v|-1}. Since ∂FV/∂v|B~≠0 and F is C1, we have L<+∞. Let u0=δ≈1 and v0=c0/δ≪1. Then (uk,vk)≈(c0/δ,δ).
LetE=∂uφ/∂v0, Λ=∂uφ/∂u0, G=∂vφ/∂u0, M=∂vφ/∂v0. Similar to the proof of Condition 1 in [19], after some simple calculations, we can obtain that lim⁡k→+∞|EM-1|≈c0/δ2, lim⁡k→+∞|M-1|=0, lim⁡k→+∞|Λ-EM-1G|=0, lim⁡k→+∞|ΛM-1|≈c0/δ4, lim⁡k→+∞|GM-1|≈c0/δ2, and lim⁡k→+∞|EM-1|≈c0/δ2. Further, when c0→0, we can obtain that c0/δ2→0, c0/δ4→0 and (∂FV/∂u)(c0/δ2)→0.
Thus, there exists aδ>0 such that, for sufficiently large k, inequalities (16) and (17) in Lemma 6 hold. Thus, according to Lemma 6, we obtain that when k is large enough, f∣Dk satisfies the (μh,μv)-cone condition.In fact, we can further prove the following.Proposition 8.
Whenk is sufficiently large, Pk satisfies the (μh,μv)-cone condition.Proof.
LetN≥2 be an arbitrary but fixed integer. For sufficiently large k, let Hl=Dl+k-1, Vl=f(Dl+k-1), Vji=PkHi⋂‍Hj, and Hij=Hi⋂‍P-kHj, where 1≤l,i,j≤N. Moreover, let H=⋃i,j‍Hij and V=⋃i,j‍Vij.
For an arbitrary pointz0∈H⋃‍V, let (ξz0,ηz0) be a vector emanating from the point z0 in the tangent space of z0. In addition, for given μh and μv, let Sz0u={(ξz0,ηz0)∣|ξz0|<μv|ηz0|} be the unstable sector at z0 and let Sz0s={(ξz0,ηz0)∣|ηz0|<μh|ξz0|} be the stable sector at z0. Similar to Section 1, we also have SHu, SVu, SHs, and SVs.
In order to prove thatPk satisfies the (μh,μv)-cone condition, by Remark 4, we need to prove that Pk satisfies Assumption 2. That is, we need to prove the following:(1)
DPk(SHu)⊂SVu and DP-k(SVs)⊂SHs;(2)
there exists a constantλ satisfying 0<λ<1-μhμv such that |ηPk(z0)|≥λ-1|ηz0| if (ξz0,ηz0)∈Sz0u and (ξPk(z0),ηPk(z0))≐DPk(z0)(ξz0,ηz0)∈SPk(z0)u, where (ξPk(z0),ηPk(z0)) is a vector emanating from the point Pk(z0) in the tangent space of Pk(z0); |ξP-k(z0)|≥λ-1|ξz0| if (ξz0,ηz0)∈Sz0s and (ξP-k(z0),ηP-k(z0))≐DP-k(z0)(ξz0,ηz0)∈SP-k(z0)s, where (ξP-k(z0),ηP-k(z0)) is a vector emanating from the point P-k(z0) in the tangent space of P-k(z0).
First, we want to prove thatDPk(SHu)⊂SVu. For this, it is sufficient to prove that, for an arbitrary z0=(u0,v0)∈H with (ξz0,ηz0)=(1,ϑ)∈Sz0u, (ξPk(z0),ηPk(z0))=DPk(z0)(ξz0,ηz0)∈SVu.
Clearly,Pk(z0)∈V and ϑ is bounded. According to the definitions of Tφ and P, DPk(z0)(ξz0,ηz0)= DT2kπ(z0)(ξz0,ηz0)= ((∂u2kπ/∂u0)+(∂u2kπ/∂v0)ϑ, (∂v2kπ/∂u0)+(∂v2kπ/∂v0)ϑ).
Sinceδ1 and δ2 for defining B and B~ are chosen to be sufficiently small, u0≈1 and v0≤δ2≪1. Letting u0=δ and c=u0v0, then (u2kπ,v2kπ)≈(c/δ,δ). According to DTφ in Section 2.1, when k is sufficiently large, we have
(19)|(∂v2kπ/∂u0)+(∂v2kπ/∂v0)ϑ(∂u2kπ/∂u0)+(∂u2kπ/∂v0)ϑ|=cu2kπ2×|kkkkkkkkk×(v0+u0ϑ))-1kkkkkkkkk×(v0+u0ϑ)(1-6k3(u2kπ+v2kπ)3kπ))-1(Δ(u0ϑ-v0)+(1+6k3(u2kπ+v2kπ)3kπ)kkkkkkk×(v0+u0ϑ)ef22eΔ(u0ϑ-v0)+(1+6k3(u2kπ+v2kπ)3kπ))kkk×((v0+u0ϑ)Δ(u0ϑ-v0)-(1-6k3(u2kπ+v2kπ)3kπ)kkkkkkkkk×(v0+u0ϑ)(1-6k3(u2kπ+v2kπ)3kπ))-1|≈δ2c|×(δϑ+cδ(1-6k3((δ2+c)/δ)3kπ))(1-6k3(δ2+cδ)3kπ))-1(Δ(δϑ-cδ)+(1+6k3(δ2+cδ)3kπ)kkkkkkkk×(δϑ+cδΔ(δϑ-(c/δ))+(1+6k3((δ2+c)/δ)3kπ))(1+6k3(δ2+cδ)3kπ))kkkkkkk×(Δ(δϑ-cδ)-(1-6k3(δ2+cδ)3kπ)kkkkkkkkk×(δϑ+cδ(1-6k3((δ2+c)/δ)3kπ))(1-6k3(δ2+cδ)3kπ))-1(Δ(δϑ-(cδ))+(1+6k3((δ2+c)δ)3kπ)|≈δ2c|1+13k3δ3kπ|.
Clearly,lim⁡k→+∞(δ2/c)|1+1/3k3δ3kπ|=δ2/c. Moreover, when c→0, δ2/c→+∞. So, for sufficiently large k and sufficiently small δ1 and δ2, |((∂v2kπ/∂u0)+(∂v2kπ/∂v0)ϑ)/((∂u2kπ/∂u0)+(∂u2kπ/∂v0)ϑ)|>1/μv. Thus, DPk(z0)(ξz0,ηz0)∈SVu. This directly implies that DPk(SHu)⊂SVu.
Second, following the proof ofDPk(SHu)⊂SVu, we want to prove that there exists a constant λ satisfying 0<λ<1-μhμv such that |ηPk(z0)|≥λ-1|ηz0|.
Similarly, for the above(u0,v0) and (u2kπ,v2kπ), when k is sufficiently large,
(20)|∂v2kπ∂u0+∂v2kπ∂v0ϑ|=12u2kπ|Δ(u0ϑ-v0)+(1+6k3(u2kπ+v2kπ)3kπ)kkkkkk×(v0+u0ϑ)|≈12c|(1+6k3(δ2+cδ)3kπ)Δ(δ2ϑ-c)kkkkkkkk+(1+6k3(δ2+cδ)3kπ)kkkkkkkk×(c+δ2)(1+6k3(δ2+cδ)3kπ)|≈δ2c|(1+3k3δ3kπ)ϑ|.
For givenB and B~, lim⁡k→+∞(δ2/c)|1+3k3δ3kπ|=∞. So, for any constant λ satisfying 0<λ<1-μhμv, when k is sufficiently large, (δ2/c)|1+3k3δ3kπ|>λ-1. Thus, we have |(∂v2kπ/∂u0)+(∂v2kπ/∂v0)ϑ|>λ-1|ϑ|, implying that |ηPk(z0)|≥λ-1|ηz0|.
Third, we want to proveDP-k(SVs)⊂SHs and |ξP-k(z0)|≥λ-1|ξz0|. For this, let Tφ-1(uφ,vφ)=(u0,v0) be the inverse map of Tφ. Then,
(21)DTφ-1=(DTφ)-1=[∂uφ∂u0∂uφ∂v0∂vφ∂u0∂vφ∂v0]-1=1△[∂vφ∂v0-∂uφ∂v0-∂vφ∂u0∂uφ∂u0].
For any z0=(u0,v0)∈V with (ξz0,ηz0)=(ϖ,1)∈Sz0s, (ξP-k(z0),ηP-k(z0))= DP-k(ξz0,ηz0)= DT2kπ-1(ξz0,ηz0)= (1/△)((∂v2kπ/∂v0)ϖ-(∂u2kπ/∂v0), (-∂v2kπ/∂u0)ϖ+(∂u2kπ/∂u0)). Similarly, we can prove that |(∂v2kπ/∂v0)ϖ-(∂u2kπ/∂v0)/(-∂v2kπ/∂u0)ϖ+(∂u2kπ/∂u0)|>μh and |(∂v2kπ/∂v0)ϖ-(∂u2kπ/∂v0)|>λ-1|ϖ|. Thus, DP-k(SVs)⊂SHs and |ξP-k(z0)|≥λ-1|ξz0|.
Based on all above analysis and Remark4, we can then obtain that when k is sufficiently large, Pk satisfies the (μh,μv)-cone condition.Based on Proposition8, we can prove that f satisfies the boundary condition.Proposition 9.
Wheni and j are sufficiently large, f(∂hDi)⋂‍Dj=∅ and fDi⋂‍∂vDj=∅, where ∂hDi is the horizontal boundary of Di and ∂vDj is the vertical boundary of Dj.Proof.
Due to Proposition8, when k is sufficiently large, Pk satisfies the (μh,μv)-cone condition. This implies that Pk contracts in the horizontal direction and expands in the vertical direction. Moreover, μv-vertical curves are mapped to μv-vertical curves under the map Pk and μh-horizontal curves are mapped to μh-horizontal curves under the map P-k. Thus, for sufficiently large i and j, Dj⊂B, PiDi⊂B~, ∂vDj⊂∂vB and Pi(∂hDi)⊂∂hB~. In addition, f(∂hDi)⋂‍Dj= (F∘Pi(∂hDi))⋂‍Dj=(F∘(Pi(∂hDi)))⋂‍Dj. Thus, according to the expression F(∂hB~)⋂‍B=∅ in Section 2.1, f(∂hDi)⋂‍Dj=∅. Similarly, since FB~⋂‍∂vB=∅, we can obtain that fDi⋂‍∂vDj=∅.Finally, we can prove thatf satisfies the intersection condition as follows.Proposition 10.
Wheni and j are sufficiently large, fDi⋂‍Dj≠∅.Proof.
LetCvi(u)={u}×(Di)v be the family of vertical curves in Di, where u∈Bh and (Di)v={v∣(u,v)∈B,Pi(u,v)∈B~}. From Proposition 8, Pi with sufficiently large i satisfies the (μh,μv)-cone condition. Thus, PiCvi(u) infinitely approaches B~v when i→+∞. Similarly, letting Chi(v)=Bh×{v} be the family of horizontal curves in Di, where v∈(Di)v, we can obtain that Chj(v) infinitely approaches Bh when j→+∞.
SinceF is C1, F(PiCvi(u)) infinitely approaches F(B~v) when i→+∞. By the expression FB~v⋂‍Bh={p} in Section 2.1, F(PiCvi(u))⋂‍Chj(v)≠∅. Thus, fDi⋂‍Dj≠∅.Remark 11.
In fact, we can prove that wheni,j→+∞, fCvi(u) and Chj(v) intersect at a unique point near p.


## 2.3. Proof of Our Theorem5

In order to prove our Theorem5, similar to [19], we try to use Propositions 7, 9, and 10 to verify that f satisfies Assumptions 1 and 2. Then, from Lemma 3, we can obtain that f is a horseshoe map as follows.Proof.
From Proposition7, when k is sufficiently large, f∣Dk satisfies the (μh,μv)-cone condition. Thus, the map f contracts in the horizontal direction and expands in the vertical direction. Moreover, μv-vertical curves are mapped to μv-vertical curves under the map f and μh-horizontal curves are mapped to μh-horizontal curves under the map f-1. Therefore, from Propositions 9 and 10, for sufficiently large i and j, fDi⋂‍Dj≠∅ is a (μh,μv)-curved rectangle and satisfies ∂h(fDi⋂‍Dj)⊂∂hDj and ∂v(fDi⋂‍Dj)⊂∂v(fDi). Similarly, for sufficiently large i and j, f-1(fDi⋂‍Dj)≠∅ is also a (μh,μv)-curved rectangle and satisfies ∂vf-1(fDi⋂‍Dj)⊂∂vDi and ∂hf-1(fDi⋂‍Dj)⊂∂h(f-1Dj).
LetN≥2 be an arbitrary but fixed positive integer. For sufficiently large k, i, and j, by letting Hl=Dl+k-1, Vl=f(Dl+k-1), fDi⋂‍Dj=V(j-k+1)(i-k+1), and Di⋂‍f-1Dj=H(i-k+1)(j-k+1), where 1≤l≤N, we can obtain that f satisfies Assumption 1. In addition, due to Remark 4, f obviously satisfies Assumption 2.
Thus, for an arbitrary but fixedN≥2, when k is sufficiently large, the map f over the set ⋃i=0N-1‍Dk+i satisfies Assumptions 1 and 2; that is, f satisfies the Conley-Moser conditions.
By Lemma3, when k is sufficiently large, f has an invariant Cantor set, on which it is topologically conjugate to a full shift on N symbols. This directly implies that f 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 mapf and then proving that this f satisfies the Conley-Moser conditions. This implies that the planar circular restricted three-body problem processes chaotic dynamics of the Smale horseshoe type.



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*Source: 965829-2015-03-19.xml*