Periodic solution of the nonlinear Sitnikov restricted three-body problem

Elbaz. I. Abouelmagd(a, b), Juan Luis Garcıa Guiraoc and Ashok Kumar Pald

ABSTRACT

The objective of this paper is to find a semi–analytical periodic solution of the circular

Sitnikov problem. In this context the multiple scales method is used to remove the secular

terms and find the periodic solutions in closed forms. A comparisons among a numerical

solution (NS), the first approximated solution (FA) and the second approximated solution

(SA) via multiple scales method are investigated graphically under different initial conditions.

Finally we demonstrate that the motion is periodic and the changes may be in the difference

of its amplitude. The behaviour of the chaotic motion may appear due to the provided

solutions via multiple scales method when the infinitesimal body starts its motion from a

point is far from the barycenter of the primaries. Moreover the numerical solution may is not

convergence, thereby the solutions of multiple scales techniques are more realistic than the

numerical solution.

Subject headings: Sitnikov restricted three–body problem; Multiple scales method; Periodic

solution.

Contents

1 Introduction 2

2 Model description 3

3 Properties of Sitnikov motion 5

4 Multiple scale solution 7

aCelestial Mechanics and Space Dynamics Research Group (CMSDRG), Astronomy Department, National Research

Institute of Astronomy and Geophysics (NRIAG), Helwan 11421 - Cairo – Egypt.

Email(Abouelmagd): eabouelmagd@gmail.com or elbaz.abouelmagd@nriag.sci.eg.

b Nonlinear Analysis and Applied Mathematics Research Group (NAAM),

Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia.

c Departamento de Matematica Aplicada y Estadıstica. Universidad Politecnica de Cartagena, Hospital de Marina,

30203-Cartagena, Region de Murcia, Spain.

Email(Guirao): juan.garcia@upct.es

d Department of Applied Mathematics, Indian Institute of Technology (Indian School of Mines), Dhanbad–826004, Jhark-

hand, India.

Email(Ashok): ashokpalism@gmail.com.

– 2 –

5 First order solution 8

5.1 Periodicity condition of first approximation . . . . . . . . . . . . . . . . . . . . . . . . . . 9

6 Second order solution 10

6.1 Periodicity condition of second approximation . . . . . . . . . . . . . . . . . . . . . . . . . 10

7 Numerical results 12

8 Conclusion 16

1. Introduction

One of the most important dynamical systems in celestial mechanics and space mechanics is the

three-body problem. This is according to vary its versions and applications in either stellar dynamics or

astrodynamics. The general model of this problem is the three-body problem in which the three bodies

move in three dimensions under their mutual gravitational interactions, within frame Newton’s laws of

motion and of universal gravitation, without any mandatory constraints on the initial positions and ve-

locities vectors of these bodies. An extended and a comprehensive review on the three–body problem is

stated by Musielak and Quarles (2014).

Within framework of imposed restrictions on the third body, where it has negligible mass with respect

to the other two bodies and moves under the gravitational effects of these two bodies, the three–body

problem will be reduced to the restricted problem, in this case the third body is called the infinitesimal

body and the others bodies called the primaries. With more restrictions, if the primaries motion is in

the same plane with circular or elliptical orbits, we will get the spatial circular or elliptical restricted

three–body problem in which the third body moves in three dimensional space. But in the case of the

infinitesimal body and the primaries move in the same plane, we will obtain the most familiar dynamical

system in celestial mechanics. An important works in this context are addressed by Abouelmagd and

Sharaf (2013); Abouelmagd et al. (2014a,b); Abouelmagd and Mostafa (2015); Abouelmagd et al. (2015);

Elshaboury et al. (2016); Meyer and Schmidt (2000); Luo and Xu (2018).

One of the reduced models of the three–body problem is the Sitnikov problem, which is considered a

sub–case of the spatial elliptic restricted three–body problem, where the infinitesimal body (third body)

has oscillation motion along Z−axis, which is perpendicular to the orbital plane of the primaries. In this

model the primaries whose equal masses and rotate in elliptical orbits around their common center of

mass. This model is called elliptical Sitnikov problem, that is constructed by Pavanini (1907). But a case

of the primaries move in circular orbits, which is called the circular Sitnikov problem was explained by

MacMillan (1911).

The Sitnikov models are considered the simplest sub–cases of general N−body problem which can be

used as a first approximation in many case of real situation in the astronomical problems. The existence

– 3 –

of oscillation motion for these problem is proved at the first time by Sitnikov (1961), thereby it is called

Sitnikov problem. It has also been studied by several researcher such as Liu and Sun (1990); Dvorak

(1993); Faruque (2003); Suraj and Hassan (2014); Ortega (2016); Beltritti et al. (2018).

In the elliptical Sitnikov problem, the equation of motion for the massless body turns out to be

dimensional and depends on time t and the eccentricity parameter e of the primaries. The problem is

a direct generalization of the so-called MacMillan problem which is the integrable approximation of the

Sitnikov problem for e = 0. For e 6= 0 the dynamical system is chaotic, despite its simple form, and

allows all different kinds of motions that are found in chaotic dynamical systems. The simplicity of its

mathematical formulation together with the complexity of its possible kinds of motions make the Sitnikov

problem a unique dynamical problem in the context of celestial mechanics.

Bountis and Papadakis (2009) studied about the stability and bifurcation of vertical motion into

families of three-dimensional (3D) periodic orbits in the Sitnikov restricted N-body problem. Sidorenko

(2011) studied the circular Sitnikov problem by alternation of stability and instability within the fam-

ily of periodic vertical motions with consideration of varying continuous monotonic amplitude. Faruque

(2003) computed the solution by the Lindstedt–Poincare method and compared with existing solutions.

Llibre and Ortega (2008) have computed analytically the families of symmetric periodic orbits in the

elliptic Sitnikov problem. They also provide a qualitative information on the bifurcation diagram of such

families of periodic orbits. Kalantonis et al. (2008) studied the photogravitational Sitnikov problem in

the restricted three–body problem with oblateness. Ortega (2016) provided the existence criteria of odd

periodic solutions with a prescribed number of zeros.

This paper is organized in the following way: A brief overview on literatures review of Sitnikov

problem in Section 1. While in Section 2 the model description of the three-body problem to Sitnikov

model is addressed. In Section 3, the dynamics characteristics of motion are investigated. In Section 4,

the general structure of multiple scales method is stated. The first and second approximated solutions

via multiple scales method and the periodicity conditions are constructed and explained in Sections (5,

6). In addition the numerical results and a comparisons among the obtained solutions are investigated in

Section 7. Finally the conclusion is drew in Section 8.

2. Model description

Let m1, m2 and m are three masses of the bodies b1, b2 and b3 where b1 and b2 are the primaries,

while b is the third body. One of the specialization of the three–body problem is the spatial circular

restricted three-body problem. In which the primaries rotate in the same plane in circular orbits around

their barycenter with the normalized angular velocity to one. The motion of third body, which called

the infinitesimal is free to move in the three-dimensional space (3D) and its motion dose not affect the

primaries motion. We also assume that the total mass of the primaries is one, in which m1 = 1− µ and

m2 = µ. By choosing a synodic coordinates system, then the origin is at the center of mass, as well

as the primaries b1 and b2 are fixed on the x− axis at (−µ, 0, 0) and (1 − µ, 0, 0) respectively. Now we

assume that r, r1 and r2 are the positions vectors of the infinitesimal body with respect to the origin

of the synodic coordinates and the primaries. Furthermore the scale of the time is chosen so that the

– 4 –

gravitational constant is unity. Then the equations of motion of the infinitesimal body will be controlled

by

x− 2y = Wx,

y + 2x = Wy,

z = Wz,

(1)

where

W (x, y, z) =1

2(x2 + y2) +

µ

r1+

1− µ

r2,

and

r =√

x2 + y2 + z2,

r1 =√

(x+ µ)2 + y2 + z2,

r2 =√

(x+ µ− 1)2 + y2 + z2,

(2)

here r = |r|, r1 = |r1|, and r2 = |r2|.

The spatial restricted three–body problem is not only the special model from three–body problem,

but also the Sitnikov problem, which can be considered it as a special model from the spatial restricted

problem, when the primaries are equal in mass and the infinitesimal body moves only on z−axis. Hence

x = y = 0 , µ = 1/2 and r1 = r2 =√

1/4 + z2, see Fig. 1. In this context, Eqs. (1–2) can be reduced to

Fig. 1.— The configuration of Sitnikov problem.

the circular Sitnikov problem or the motion of the test particle along the vertical z−axis

z =d

dzV (z), (3)

where

V (z) =1

√

1/4 + z2, (4)

– 5 –

3. Properties of Sitnikov motion

The motion of a test particle in Sitnikov model is governed by a nonlinear deferential equation as in

Eq. (3) where this equation has no exact solution. The only possible solution with mathematical form for

this equation is an approximate solution by using the perturbation techniques. Although the perturbation

techniques are considered effective tools in obtaining a solution of nonlinear differential equations, but

these techniques have their own limitations. Because they depend on a very small physical parameter,

which does not appear in Eq. (3). In fact the most applications of perturbation methods are not applied

without the existence of this parameter. It is clear that from the Fig. 2, the potential is symmetric around

z = 0, which means that the potential may be represented the motion of a harmonic oscillator. Let E is

-4 -2 0 2 4

-2.0

-1.5

-1.0

-0.5

z

V(z)

Fig. 2.— The curve of the potential V (z).

the total mechanical energy, then1

2z2 −

1√

1/4 + z2= E. (5)

The region of motion can be investigated with a simple analysis of Eq. (5). Since the potential value

V (z) ∈ [−2, 0) for all values of z. Hence the motion will undefined in the rang E < −2 and bounded

for −2 < E < 0, while it will be unbounded in the rang E > 0. The region of motion in 3–dimensional

and its phase portrait are drawn in in Fig. 3 and Fig. 4. In phase portrait figure, the regions of periodic

motion are determined by closed contours when −2 < E < 0, and the region of unbounded motion will

be determined when E ≥ 0.

Furthermore the third body will has zero velocity z = 0 in three cases. The first two cases when it

tends to infinity (z → ±∞) with zero energy (E = 0), then there are two stationary points at positive

(negative) infinity. In these two case the motion is considered unbounded and the third body will be

enforced to remain stationary at infinity. While the third case when total energy E = −2 and z = 0,

then the motion is bounded and the third body will also enforced to remain at the center of masses, this

is clear from the phase portrait in Fig. 4 when the value of total energy E is tends to −2.

Within frame the total energy is negative and it is close to −2, then the third body will be near and

close to the center of masses, thereby |z| ≪ 1. Then with a help of Eqs. (3 , 4) the equation of motion up

O(z3) can be written as

z + ω2z −2z3 = 0, (6)

– 6 –

where ω2 = 8 and 2 = 48.

To find the periodic solution of Eq. (6), an approximation may still be found by introducing a very

small parameter into the problem, but in the end its value will replace by one. This method is acceptable

when the terms recalling the differential equation are quasi–linear or small in of themselves. Parameters

introduced in this way are always called “place keeping parameters”. Let z ≡ εz where ε ≪ 1, then

-5

0

5

z

-5

0

5

dzêdt

-2.0

-1.5

-1.0

-0.5

0.0

E

Fig. 3.— The region of possible motion in the 3–dimensional when E < 0.

-1.5

-1

-0.5

0

0

0

0.5

0.5

0.5

0.5

1

1

1

1

-2 b E < 0

E r 0

-2 -1 0 1 2

-2

-1

0

1

2

z

dz�d

t

Fig. 4.— The phase portrait of motion for different values of total energy E.

– 7 –

Eq.(6) will take the following form

z + ω2z −εz3 = 0. (7)

Now equation of motion is written in standard form of applying one of the perturbation methods, like

multiple scale method or Lindstedt–Poincare

4. Multiple scale solution

Let the initial conditions of motion are

z(0) = Z,

z(0) = 0.(8)

To continue in our procedures, We will write the solution z(t) in the form of a perturbation series:

z(t) =∞∑

n=0

εnzn(t, τ, σ), (9)

where τ = εt and σ = ε2t.

In fact, the provided solution by multiple scales method includes the fast and slow variables. Although

these variables are dependent, the processing strategy of multiple scales method for removing the secular

is handling the two variables as independent variables. Hence the ordinary derivative with respect to the

fast variable t will be expanded to the differential operator Dt. Which includes the partial derivatives

with respect to the fast variable t and the slow variables τ and σ. With the help of the chain rule, the

differential operator Dt is defined as

d

dt= Dt := (

∂

∂t+ ε

∂

∂τ+ ε2

∂

∂σ), (10)

henced

dtz(t) = Dt

∞∑

n=0

εnzn(t, τ, σ). (11)

With the conditions that zn is continuous and differentiable with respect to t, τ and σ, and substituting

Eq.(10) into Eq.(11) with some simple calculations, one obtains

dz

dt=

∂z0∂t

+ ε

(

∂z1∂t

+∂z0∂τ

)

+ε2(

∂z2∂t

+∂z1∂τ

+∂z0∂σ

)

+O(ε3),

d2z

dt2=

∂2z0∂t2

+ ε

(

∂2z1∂t2

+ 2∂2z0∂τ∂t

)

+ε2(

∂2z2∂t2

+ 2∂2z1∂τ∂t

+ 2∂2z0∂σ∂t

+∂2z0∂τ2

)

+O(ε3).

(12)

Substituting Eqs.(9, 12), into Eq.(7) with a help of the condition in Eq.(8) and equating the coef-

ficients which has the same order in ε, we will get a series of linear partial differential equations which

– 8 –

govern the functions of zi, for i = 0, 1, 2, we get

∂2z0∂t2

+ ω2z0 = 0,

z0(0, 0, 0) = Z,∂

∂tz0(0, 0, 0) = 0.

(13)

∂2z1∂t2

+ ω2z1 = 48z30 − 2∂2z0∂τ∂t

,

z1(0, 0, 0) = 0 ,∂

∂tz1(0, 0, 0) = −

∂

∂τz0(0, 0, 0).

(14)

∂2z2∂t2

+ ω2z2 = 144z20z1 − 2∂2z1∂τ∂t

− 2∂2z0∂σ∂t

−∂2z0∂τ2

,

z2(0, 0, 0) = 0 ,∂

∂tz2(0, 0, 0) = −

∂

∂τz1(0, 0, 0) −

∂

∂σz0(0, 0, 0).

(15)

To accomplish our steps, we have to find the solutions of Eqs.(13–15) without including secular terms,

specifically, the solutions are periodic functions in the fast variable t. This means that we have to

determine the formulas of the functions zi , (i = 0, 1, 2) where zi(t + 2π, τ, σ) = zi(t, τ, σ). Now we will

assume that the solution of Eq.(13) in the independently handled variables t, τ and σ is

z0(t, τ, σ) = Z0(τ, σ)eiωt + Z0(τ, σ)e

−iωt (16)

where Z0(τ, σ) is an arbitrary complex function in the slow variable τ , σ and Z0(τ, σ) is its complex

conjugate.

5. First order solution

Insert Eq.(16) into Eq.(14), we get

∂2z1∂t2

+ w2z1 =48[

Z30 (τ, σ)e

i3ωt + Z30 (τ, σ)e

−i3ωt]

+ s1(τ, σ)eiωt + s1(τ, σ)e

−iωt,

z1(0, 0, 0) = 0,∂

∂tz1(0, 0, 0) = −

∂

∂τz0(0, 0, 0).

(17)

where

s1(τ, σ) = 2

(

72Z20 Z0 − iω

∂Z0

∂τ

)

, (18)

s1(τ, σ) = 2

(

72Z20Z0 + iω

∂Z0

∂τ

)

. (19)

It is clear that the solution of Eq.(17) will contain a secular terms, if the coefficients of the functions

Exp (iωt) and Exp (−iωt) have nonzero values. Because these functions are the solutions of the homoge-

neous equation associated to Eq.(17). In order to avoid the secular solutions for Eq.(17) with respect to

the fast variable t, the coefficients s1(τ, σ) and s1(τ, σ) have to equal zero.

– 9 –

5.1. Periodicity condition of first approximation

The equating of coefficients s1(τ, σ) and s1(τ, σ) by zero will help us to determine the function Z0(τ, σ)

and Z0(τ, σ) with an elegant way, but s1(τ, σ) and s1(τ, σ) are just two conjugate quantities. Thereby it

is enough to solve Eq.(18) with s1(τ, σ) = 0 because the solution of Eq.(19) for s2(τ, σ) = 0 will give the

same results for the solution of Eq.(18). This processing will give the warranty that the solution of z1 will

not involve secular terms and at least there are no secularities appear up to the first order approximation

in the perturbation series.

To carry out our purpose, we will solve Eq.(18) with a help of the polar coordinates (R, θ). Hence

we suppose that

Z0(τ, σ) = R(τ, σ)eiθ(τ,σ),

R, θ : R → R,(20)

insert Eq.(20) into Eq.(18), and use the condition of s1(τ, σ) = 0, one obtains

Z0(τ, σ) = α(σ)ei

f(σ)−72α2(σ)

ωτ

,

Z0(τ, σ) = α(σ)e−i

f(σ)−72α2(σ)

ωτ

,

(21)

where α(σ) and f(σ) are arbitrary functions in the slow variable σ, with

R(τ, σ) = α(σ),

θ(τ, σ) = f(σ)−72α2(σ)

ωτ.

(22)

Substituting Eqs.(21, 22 ) into Eq.(16) with help of the initial conditions in Eq.(13), we get the zero

order periodic solution in form

z0(t, τ, σ) = α(σ)[

eiγ(t,τ,σ) + e−iγ(t,τ,σ)]

, (23)

where α(0) = Z/2, f(0) = 0 and γ(t, τ, σ) = ωt− (72α2(σ)/ω)τ + f(σ).

Now Eq.(17) can be written as

∂2z1∂t2

+ w2z1 = Z30 (τ, σ)e

i3ωt + Z30 (τ, σ)e

−i3ωt,

z1(0, 0, 0) = 0 ,∂

∂tz1(0, 0, 0) = −

∂

∂τz0(0, 0, 0),

and its general solution is

z1(t, τ, σ) =Z1(τ, σ)eiωt + Z1(τ, σ)e

−iωt,

−6α3(σ)

ω2

[

ei3γ(t,τ,σ) + e−i3γ(t,τ,σ)]

,

z1(0, 0, 0) =0 ,∂

∂tz1(0, 0, 0) = −

∂

∂τz0(0, 0, 0).

(24)

– 10 –

6. Second order solution

Substituting Eq.(23) and Eq.(24) into Eq.(15), then the second approximated solution z2 is controlled

by

∂2z2∂t2

+ w2z2 =864α5(σ)

ω2

(

ei5γ(t,τ,σ) + e−i5γ(t,τ,σ))

−6048α5(σ)

ω2

(

ei3γ(t,τ,σ) + e−i3γ(t,τ,σ))

+ 144α2(σ)[

Z1(τ, σ)ei(ωt+2θ(τ,σ)) + Z1(τ, σ)e

−i(ωt+2θ(τ,σ))]

+ s2(τ, σ)eiωt + s2(τ, σ)e

−iωt,

z2(0, 0, 0) = 0,∂

∂tz2(0, 0, 0) = −

∂

∂τz1(0, 0, 0) −

∂

∂σz0(0, 0, 0).

(25)

where

s2(τ, σ) = 288α2(σ)Z1(τ, σ) − 2iω∂Z1

∂τ(τ, σ)

+144α2(σ)Z1(τ, σ)e2iθ(τ,σ) (26)

−2

ω2eiθ(τ,σ)

iω2dα(σ)

dσ

(

ω − 144iτα2(σ))

−3024α5(σ)− ω3α(σ)df(σ)

dσ

,

s2(τ, σ) = 288α2(σ)Z1(τ, σ) + 2iω∂Z1

∂τ(τ, σ)

+144α2(σ)Z1(τ, σ)e−2iθ(τ,σ) (27)

−2

ω2e−iθ(τ,σ)

−iω2dα(σ)

dσ

(

ω + 144iτα2(σ))

−3024α5(σ)− ω3α(σ)df(σ)

dσ

.

As a similar way in the first approximated solution, the solution of Eq. (25) will contain a secular

terms, if the coefficients s2(τ, σ) and s2(τ, σ) of the functions Exp (iωt) and Exp (−iωt) are not vanish.

Because these functions are also the solutions of the homogeneous equation associated to Eq. (25). Again

to avoid the secular solutions for Eq. (25) with respect to the fast variable t, the coefficients s2(τ, σ) and

s2(τ, σ) have to take a zero values.

6.1. Periodicity condition of second approximation

The periodicity condition of obtaining periodic solution for the second approximation z2 is available

through the solution of Eq. (26) with s2(τ, σ) = 0 and Eq. (27) with s2(τ, σ) = 0. But it is also enough

to solve Eq. (26) with s2(τ, σ) = 0. Because the output solutions of s2(τ, σ) = 0 and s2(τ, σ) = 0 will be

the same. This processing will also give the warranty that the solution of z2 will not include secular or

unbounded terms and at least there are no secularities appear up to the second order approximation in

the perturbation series.

– 11 –

Furthermore we can construct the arbitrary functions Z1(τ, σ) and Z1(τ, σ). One has to note that

we arrange the equation of z1, within frame its solution has no secular terms or is not resonance with the

solution of homogenous part to find the arbitrary functions Z0(τ, σ) and Z0(τ, σ). The same processing

will be done to find the arbitrary functions Z1(τ, σ) and Z1(τ, σ), within frame the solution of z2 has no

also secular terms. That will be repeated (n+ 1) times to get the periodic function, which represent the

solution zn.

Let us go back to periodicity condition s2(τ, σ) = 0, which leads to a system of partial differential

equations, and there is extra difficulty to find its analytical solution in most cases. This is considered the

major defect of multiple scales method. In order to avoid this difficulty, we have to note that we search

for an approximated analytical solution of the first approximation z1, we mean that the solution dose

not general or exact solution. Hence the particular solution of proper selection of the arbitrary functions

which make a convenient with the initial conditions may be acceptable, if its validation is check.

The general solutions of the zero z0 and the first approximation z1 can be written in the following

form

z0(t, τ, σ) =1

2α(σ) cos

(

ωt−72α2(σ)

ωτ + f(σ)

)

, (28)

where α(0) = Z/2 and f(0) = 0,

z1(t, τ, σ) =Z1(τ, σ)eiωt + Z1(τ, σ)e

−iωt

−12α3(σ)

ω2cos 3

(

ωt−72α2(σ)

ωτ + f(σ)

)

,(29)

with initial conditions

z1(0, 0, 0) = 0 ,∂

∂tz1(0, 0, 0) = −

∂

∂τz0(0, 0, 0), (30)

in slow variables, the solution in Eq. (28) and Eq. (29) include the three arbitrary functions Z1(τ, σ), α(σ)

and f(σ) in slow variables τ and σ. But these two solutions must be convenient with the initial conditions

in Eq. (30). It is clear that the only restriction on the function f(σ) is f(0) = 0 and there is no restriction

on α(σ), but for simplicity we will chose these two functions as constants, where α(σ) = Z/2 = α0 and

f(σ) = 0. To make a complete convenient or a harmony among the solutions and the initial conditions

we have to chose Z1(τ, σ) = Z1(τ, σ) = (12α3(σ)/ω2) = (12α30/ω

2). Thereby the first two solutions in the

perturbation series are

z0(t, τ) =1

2α0 cos ω

(

t−72α2

0

ω2τ

)

, (31)

z1(t, τ) =12α3

0

ω2

[

cosωt− cos 3ω

(

t−72α2

0

ω2τ

)]

. (32)

Since the slow variable τ = εt, but with the method of “place keeping parameters”, the parameter

ε will be replace by one. Using Eqs. (31 , 32) then the approximated solution of multiple scales method

can be written as

– 12 –

z(t) =1

2α0 cos ω

(

1−72α2

0

ω2

)

t+12α3

0

ω2

[

cosωt− cos 3ω

(

1−72α2

0

ω2

)

t

]

. (33)

7. Numerical results

In this section a comparison will be investigated among the numerical solution, the first and second

approximated solutions of the Sitnikov problem. The investigations include the numerical solutions of

Eq. (3), the first and second approximated solutions of Eq. (7) via multiple scales method, which are given

in Eq. (31) and Eq. (33) respectively.

We construct the comparison within frame a three different initial conditions. In general the in-

finitesimal body will start its motion with zero velocity (z(0) = 0), for a three different positions

(z(0) = 0.4, 0.8, 1.2), therefor (α0 = 0.2, 0.4, 0.6). The investigation for the motion of the infinitesi-

mal body will be constructed in two main groups. In the first main groups a three versions for the same

solution will be showed, for a three different initial conditions. Then we mean that the changes with

respect to the initial conditions, not the kind of obtained solution. This group will include Figs. (5, 6, 7),

in this figures the red, green and blue curves refer to α0 = 0.2, α0 = 0.4 and α0 = 0.6 respectively.

While in the Second main groups a three different solutions will be showed, for the same initial condition.

Therefor we mean that the changes with respect to the kind of obtained solution. This group will include

Figs. (8, 9, 10), but in this figures the red, green and blue curves indicate to the numerical solution (NS),

first approximated solution (FA) and second approximated solution (SA) of multiple scales method re-

spectively.

In Fig. 5, a comparison among a three version for the numerical solutions are showed when the in-

finitesimal starts its motion from a three different positions with respect to the barycenter of the primaries.

It is clear that the motion is periodic and the number of periods of motion will be increased when the

infinitesimal body starts its motion from a nearer position due to the barycenter of the primaries, and

vice versa. More reading for the numerical solution the initial conditions play a role in the behavior of

motion, and may be a solution is not convergence when the infinitesimal body starts its motion from a

point is very far from the barycenter of the primaries. But the solution may be periodic and the motion

has a regular in its periodicity for an appropriate initial conditions.

In Fig. 6, a comparison among a three versions for the first approximated solutions (FA) when the

infinitesimal starts its motion under the same conditions of the numerical solution. The behaviour of mo-

tion within frame of the first approximated solution is the opposite of motion within frame of numerical

solution. Such that the number of periods of motion will be decreased when the infinitesimal body starts

its motion from a nearer position due to the barycenter, vice versa. In general the motion is periodic and

regular, but the changes may be in the amplitude of motion (maximum displacement from the equilibrium

position).

However the behaviour of motion within frame of the second Approximated solution (SA) seems to the

behaviour of motion within frame of the first approximated solution (FA), in particulary when the initial

– 13 –

0 5 10 15 20

-1.0

-0.5

0.0

0.5

1.0

t

z(t)

α0=0.2 α0=0.4 α0=0.6

Fig. 5.— Numerical solution for three different values of initial conditions.

0 5 10 15 20

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

t

z(t)

α0=0.2 α0=0.4 α0=0.6

Fig. 6.— First approximated solution for three different values of initial conditions.

conditions are small, the motion may be regular and periodic. While the motion may be a chaotic when

the infinitesimal body starts its motion from a point is far from the barycenter of the primaries, see Fig. 7.

In Figs. (8, 9, 9) comparisons among the numerical solution (NS), the first approximated solution

(FA) and the second approximated solution (SA) are showed within frame the same initial conditions. It

is clear that the motion are periodic due to the numerical solutions but the number of motion periods

is increased when the infinitesimal body starts its motion from a nearer point due to the barycenter of

the primaries. While the behavior of motion is the same and may be coincident within frame the first

and second approximated solutions, in particular when the infinitesimal body starts its motion from a

– 14 –

0 5 10 15 20

-0.5

0.0

0.5

t

z(t)

α0=0.2 α0=0.4 α0=0.6

Fig. 7.— Second approximated solution for three different values of initial conditions.

nearer point due to the barycenter of the primaries. In general the motion is periodic and the changes

may be in the difference of its amplitude, and the behaviour of the chaotic motion is appears due to the

provided solutions via multiple scales method when the infinitesimal body starts its motion from a point

is far from the barycenter of the primaries.

0 5 10 15 20

-0.4

-0.2

0.0

0.2

0.4

t

z(t)

NS FA SA

Fig. 8.— Comparison among numerical, first and second approximated solutions at α0 = 0.2.

– 15 –

0 5 10 15 20

-0.5

0.0

0.5

t

z(t)

NS FA SA

Fig. 9.— Comparison among numerical, first and second approximated solutions at α0 = 0.4.

0 5 10 15 20

-1.0

-0.5

0.0

0.5

1.0

t

z(t)

NS FA SA

Fig. 10.— Comparison among numerical, first and second approximated solutions at α0 = 0.6.

– 16 –

8. Conclusion

In this paper the semi–analytical periodic solutions of Sitnikov problem are constructed . The mul-

tiple scales method is used to remove the secular terms and find periodic solutions in closed forms. In

the same time of removing the secular terms the periodicity conditions for the approximated solutions

are constructed.

A comparisons among a numerical solution (NS), the first approximated solution (FA) and the second

approximated solution (SA) are investigated graphically in two main groups. In the first group a com-

parison among three issues for the same solution is constructed by depending on a three different initial

conditions, in small word the infinitesimal body starts its motion from three different positions from the

barycenter of the two primaries. While in the second main group the comparisons are constructed among

three different solution at the same initial conditions.

Finally we demonstrate that the motion is periodic, the changes may be in the difference of its

amplitude and the behaviour of the chaotic motion is appears due to the provided solutions via multiple

scales method when the infinitesimal body starts its motion from a point is far from the barycenter of

the primaries. Furthermore the obtained solutions by multiple scales method is more realistic than the

numerical solution, because the numerical solution may is divergence.

Acknowledgements

The first and second authors are partially supported by Fundacion Seneca (Spain), grant 20783/PI/18.,

and Ministry of Science, Innovation and Universities (Spain), grant PGC2018-097198-B-100

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This preprint was prepared with the AAS LATEX macros v5.2.