# The Orbital Dynamics of Synchronous Satellites: Irregular Motions in the 2 : 1 Resonance

Authors: Jarbas Cordeiro Sampaio; Rodolpho Vilhena de Moraes; Sandro da Silva Fernandes
Journal: Mathematical Problems in Engineering
 (2012)

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## Abstract

The orbital dynamics of synchronous satellites is studied. The 2 : 1 resonance is considered; in other words, the satellite completes two revolutions while the Earth completes one. In the development of the geopotential, the zonal harmonicsJ20 and J40 and the tesseral harmonics J22 and J42 are considered. The order of the dynamical system is reduced through successive Mathieu transformations, and the final system is solved by numerical integration. The Lyapunov exponents are used as tool to analyze the chaotic orbits.


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## Body

## 1. Introduction

Synchronous satellites in circular or elliptical orbits have been extensively used for navigation, communication, and military missions. This fact justifies the great attention that has been given in literature to the study of resonant orbits characterizing the dynamics of these satellites since the 60s [1–14]. For example, Molniya series satellites used by the old Soviet Union for communication form a constellation of satellites, launched since 1965, which have highly eccentric orbits with periods of 12 hours. Another example of missions that use eccentric, inclined, and synchronous orbits includes satellites to investigate the solar magnetosphere, launched in the 90s [15].The dynamics of synchronous satellites are very complex. The tesseral harmonics of the geopotential produce multiple resonances which interact resulting significantly in nonlinear motions, when compared to nonresonant orbits. It has been found that the orbital elements show relatively large oscillation amplitudes differing from neighboring trajectories [11].Due to the perturbations of Earth gravitational potential, the frequencies of the longitude of ascending nodeΩ and of the argument of pericentre ω can make the presence of small divisors, arising in the integration of equation of motion, more pronounced. This phenomenon depends also on the eccentricity and inclination of the orbit plane. The importance of the node and the pericentre frequencies is smaller when compared to the mean anomaly and Greenwich sidereal time. However, they also have their contribution in the resonance effect. The coefficients l, m, p which define the argument ϕlmpq in the development of the geopotential can vary, producing different frequencies within the resonant cosines for the same resonance. These frequencies are slightly different, with small variations around the considered commensurability.In this paper, the 2 : 1 resonance is considered; in other words, the satellite completes two revolutions while the Earth carries one. In the development of the geopotential, the zonal harmonicsJ20 and J40 and the tesseral harmonics J22 and J42 are considered. The order of the dynamical system is reduced through successive Mathieu transformations, and the final system is solved by numerical integration. In the reduced dynamical model, three critical angles, associated to the tesseral harmonics J22 and J42, are studied together. Numerical results show the time behavior of the semimajor axis, argument of pericentre and of the eccentricity. The Lyapunov exponents are used as tool to analyze the chaotic orbits.


## 2. Resonant Hamiltonian and Equations of Motion

In this section, a Hamiltonian describing the resonant problem is derived through successive Mathieu transformations.Consider (2.1) to the Earth gravitational potential written in classical orbital elements [16, 17](2.1)V=μ2a+∑l=2∞∑m=0l∑p=0l∑q=+∞-∞μa(aea)lJlmFlm(I)Glpq(e)cos⁡(ϕlmpq(M,ω,Ω,θ)),
where μ is the Earth gravitational parameter, μ=3.986009×1014 m3/s2, a, e, I, Ω, ω, M are the classical keplerian elements: a is the semimajor axis, e is the eccentricity, I is the inclination of the orbit plane with the equator, Ω is the longitude of the ascending node, ω is the argument of pericentre, and M is the mean anomaly, respectively; ae is the Earth mean equatorial radius, ae=6378.140 km, Jlm is the spherical harmonic coefficient of degree l and order m, Flmp(I) and Glpq(e) are Kaula’s inclination and eccentricity functions, respectively. The argument ϕlmpq(M,ω,Ω,θ) is defined by(2.2)ϕlmpq(M,ω,Ω,θ)=qM+(l-2p)ω+m(Ω-θ-λlm)+(l-m)π2,
where θ is the Greenwich sidereal time, θ=ωet (ωe is the Earth’s angular velocity, and t is the time), and λlm is the corresponding reference longitude along the equator.In order to describe the problem in Hamiltonian form, Delaunay canonical variables are introduced,(2.3)L=μa,G=μa(1-e2),H=μa(1-e2)cos⁡(I),l=M,g=ω,h=Ω.L, G, and H represent the generalized coordinates, and ℓ, g, and h represent the conjugate momenta.Using the canonical variables, one gets the HamiltonianF̂,(2.4)F̂=μ22L2+∑l=2∞∑m=0lRlm,with the disturbing potentialRlm given by(2.5)Rlm=∑p=0l∑q=-∞+∞Blmpq(L,G,H)cos⁡⁡(ϕlmpq(l,g,h,θ)).The argumentϕlmpq is defined by(2.6)ϕlmpq(l,g,h,θ)=ql+(l-2p)g+m(h-θ-λlm)+(l-m)π2,
and the coefficient Blmpq(L,G,H) is defined by(2.7)Blmpq=∑l=2∞∑m=0l∑p=0l∑q=+∞-∞μ2L2(μaeL2)lJlmFlmp(L,G,H)Glpq(L,G).The HamiltonianF̂ depends explicitly on the time through the Greenwich sidereal time θ. A new term ωeΘ is introduced in order to extend the phase space. In the extended phase space, the extended Hamiltonian Ĥ is given by(2.8)Ĥ=F̂-ωeΘ.For resonant orbits, it is convenient to use a new set of canonical variables. Consider the canonical transformation of variables defined by the following relations:(2.9)X=L,Y=G-L,Z=H-G,Θ=Θ,x=l+g+h,y=g+h,z=h,θ=θ,whereX,Y,Z,Θ,x,y,z,θ are the modified Delaunay variables.The new HamiltonianH′̂, resulting from the canonical transformation defined by (2.9), is given by(2.10)H′̂=μ22X2-ωeΘ+∑l=2∞∑m=0lRlm′,where the disturbing potentialRlm′ is given by(2.11)Rlm′=∑p=0l∑q=-∞+∞Blmpq′(X,Y,Z)cos⁡(ϕlmpq(x,y,z,θ)).Now, consider the commensurability between the Earth rotation angular velocityωe and the mean motion n=μ2/X3. This commensurability can be expressed as(2.12)qn-mωe≅0,
considering q and m as integers. The ratio q/m defining the commensurability will be denoted by α. When the commensurability occurs, small divisors arise in the integration of the equations of motion [9]. These periodic terms in the Hamiltonian H′̂ with frequencies qn-mωe are called resonant terms. The other periodic terms are called short- and long-period terms.The short- and long-period terms can be eliminated from the HamiltonianH′̂ by applying an averaging procedure [12, 18]:(2.13)〈H′̂〉=14π2∫02π∫02πH′̂dξspdξlp.
The variables ξsp and ξlp represent the short- and long-period terms, respectively, to be eliminated of the Hamiltonian H′̂.The long-period terms have a combination in the argumentϕlmpq which involves only the argument of the pericentre ω and the longitude of the ascending node Ω. From (2.10) and (2.11), these terms are represented by the new variables in the following equation:(2.14)Hlp′̂=∑l=2∞∑m=0l∑p=0l∑q=-∞+∞Blmpq′(X,Y,Z)cos⁡((l-2p)(y-z)+mz).The short-period terms are identified by the presence of the sidereal timeθ and mean anomaly M in the argument ϕlmpq; in this way, from (2.10) and (2.11), the term Hsp′̂ in the new variables is given by the following equations:(2.15)Hsp′̂=∑l=2∞∑m=0l∑p=0l∑q=-∞+∞Blmpq′(X,Y,Z)cos⁡(q(x-y)-mθ+ζp).The termζp represents the other variables in the argument ϕlmpq, including the argument of the pericentre ω and the longitude of the ascending node Ω, or, in terms of the new variables, y-z and z, respectively.A reduced HamiltonianĤr is obtained from the Hamiltonian H′̂ when only secular and resonant terms are considered. The reduced Hamiltonian Ĥr is given by(2.16)Ĥr=μ22X2-ωeΘ+∑j=1∞B2j,0,j,0′(X,Y,Z)+∑l=2∞∑m=2l∑p=0lBlmp(αm)′(X,Y,Z)cos⁡(ϕlmp(αm)(x,y,z,θ)).Several authors, [11, 15, 19–22], also use this simplified Hamiltonian to study the resonance.The dynamical system generated from the reduced Hamiltonian, (2.16), is given by(2.17)d(X,Y,Z,Θ)dt=∂Ĥr∂(x,y,z,θ),d(x,y,z,θ)dt=-∂Ĥr∂(X,Y,Z,Θ).The equations of motiondX/dt, dY/dt, and dZ/dt defined by (2.17) are(2.18)dXdt=-α∑l=2∞∑m=2l∑p=0lmBlmp(αm)′(X,Y,Z)sin⁡(ϕlmp(αm)(x,y,z,θ)),(2.19)dYdt=-∑l=2∞∑m=2l∑p=0l(l-2p-mα)Blmp(αm)′(X,Y,Z)sin⁡(ϕlmp(αm)(x,y,z,θ)),(2.20)dZdt=∑l=2∞∑m=2l∑p=0l(l-2p-m)Blmp(αm)′(X,Y,Z)sin⁡(ϕlmp(αm)(x,y,z,θ)).From (2.18) to (2.20), one can determine the first integral of the system determined by the Hamiltonian Ĥr.Equation (2.18) can be rewritten as(2.21)1αdXdt=-∑l=2∞∑m=2l∑p=0lmBlmp(αm)′(X,Y,Z)sin⁡(ϕlmp(αm)(x,y,z,θ)).Adding (2.19) and (2.20),(2.22)dYdt+dZdt=(α-1)∑l=2∞∑m=2l∑p=0lmBlmp(αm)′(X,Y,Z)sin⁡(ϕlmp(αm)(x,y,z,θ)),and substituting (2.21) and (2.22), one obtains(2.23)dYdt+dZdt=-(α-1)1αdXdt.Now, (2.23) is rewritten as(2.24)(1-1α)dXdt+dYdt+dZdt=0.In this way, the canonical system of differential equations governed byĤr has the first integral generated from (2.24):(2.25)(1-1α)X+Y+Z=C1,whereC1 is an integration constant.Using this first integral, a Mathieu transformation(2.26)(X,Y,Z,Θ,x,y,z,θ)⟶(X1,Y1,Z1,Θ1,x1,y1,z1,θ1)can be defined.This transformation is given by the following equations:(2.27)X1=X,Y1=Y,Z1=(1-1α)X+Y+Z,Θ1=Θ,x1=x-(1-1α)z,y1=y-z,z1=z,θ1=θ.
The subscript 1 denotes the new set of canonical variables. Note that Z1=C1, and the z1 is an ignorable variable. So the order of the dynamical system is reduced in one degree of freedom.Substituting the new set of canonical variables,X1, Y1, Z1, Θ1, x1, y1, z1, θ1, in the reduced Hamiltonian given by (2.16), one gets the resonant Hamiltonian. The word “resonant” is used to denote the Hamiltonian Hrs which is valid for any resonance. The periodic terms in this Hamiltonian are resonant terms. The Hamiltonian Hrs is given by(2.28)Hrs=μ22X12-ωeΘ1+∑j=1∞B2j,0,j,0(X1,Y1,C1)+∑l=2∞∑m=2l∑p=0lBlmp,(αm)(X1,Y1,C1)cos⁡(ϕlmp(αm)(x1,y1,θ1)).The HamiltonianHrs has all resonant frequencies, relative to the commensurability α, where the ϕlmp(αm) argument is given by(2.29)ϕlmp(αm)=m(αx1-θ1)+(l-2p-αm)y1-ϕlmp(αm)0,with(2.30)ϕlmp(αm)0=mλlm-(l-m)π2.The secular and resonant terms are given, respectively, byB2j,0,j,0(X1,Y1,C1) and Blmp(αm)(X1,Y1,C1).Each one of the frequencies contained indx1/dt, dy1/dt, dθ1/dt is related, through the coefficients l, m, to a tesseral harmonic Jlm. By varying the coefficients l, m, p and keeping q/m fixed, one finds all frequencies dϕ1,lmp(αm)/dt concerning a specific resonance.FromHrs, taking, j=1,2, l=2,4, m=2, α=1/2, and p=0,1,2,3, one gets(2.31)Ĥ1=μ22X12-ωeΘ1+B1,2010(X1,Y1,C1)+B1,4020(X1,Y1,C1)+B1,2201(X1,Y1,C1)cos⁡(x1-2θ1+y1-2λ22)+B1,2211(X1,Y1,C1)cos⁡(x1-2θ1-y1-2λ22)+B1,2221(X1,Y1,C1)cos⁡(x1-2θ1-3y1-2λ22)+B1,4211(X1,Y1,C1)cos⁡(x1-2θ1+y1-2λ42+π)+B1,4221(X1,Y1,C1)cos⁡(x1-2θ1-y1-2λ42+π)+B1,4231(X1,Y1,C1)cos⁡(x1-2θ1-3y1-2λ42+π).The HamiltonianĤ1 is defined considering a fixed resonance and three different critical angles associated to the tesseral harmonic J22; the critical angles associated to the tesseral harmonic J42 have the same frequency of the critical angles associated to the J22 with a difference in the phase. The other terms in Hrs are considered as short-period terms.Table1 shows the resonant coefficients used in the Hamiltonian Ĥ1.Table 1
Resonant coefficients.
Degree (l)Order (m)pq220122112221421142214231Finally, a last transformation of variables is done, with the purpose of writing the resonant angle explicitly. This transformation is defined by(2.32)X4=X1,Y4=Y1,Θ4=Θ1+2X1,x4=x1-2θ1,y4=y1,θ4=θ1.So, considering (2.31) and (2.32), the Hamiltonian H4 is found to be(2.33)H4=μ22X42-ωe(Θ4-2X4)+B4,2010(X4,Y4,C1)+B4,4020(X4,Y4,C1)+B4,2201(X4,Y4,C1)cos⁡(x4+y4-2λ22)+B4,2211(X4,Y4,C1)cos⁡(x4-y4-2λ22)+B4,2221(X4,Y4,C1)cos⁡(x4-3y4-2λ22)+B4,4211(X4,Y4,C1)cos⁡(x4+y4-2λ42+π)+B4,4221(X4,Y4,C1)cos⁡(x4-y4-2λ42+π)+B4,4231(X4,Y4,C1)cos⁡(x4-3y4-2λ42+π),
with ωeΘ4 constant and
(2.34)B4,2010=μ4X46ae2J20(-34(C1+2X4)2(X4+Y4)2+14)(1+32-Y42-2X4Y4X42),(2.35)B4,4020=μ6X410ae4J40(10564(1-(C1+2X4)2(X4+Y4)2)2-32+158(C1+2X4)2(X4+Y4)2)×(1+5-Y42-2X4Y4X42),(2.36)B4,2201=218X47μ4ae2J22(1+C1+2X4X4+Y4)2-Y42-2X4Y4,(2.37)B4,2211=32X47μ4ae2J22(32-32(C1+2X4)2(X4+Y4)2)-Y42-2X4Y4,(2.38)B4,2221=-38X47μ4ae2J22(1-C1+2X4X4+Y4)2-Y42-2X4Y4,(2.39)B4,4211=92X411μ6ae4J42(3527(1-(C1+2X4)2(X4+Y4)2)(C1+2X4)×(1+C1+2X4X4+Y4)(X4+Y4)-1-158(1+C1+2X4X4+Y4)2)-Y42-2X4Y4(2.40)B4,4221=52X411μ6ae4J42(10516(1-(C1+2X4)2(X4+Y4)2)(1-3(C1+2X4)2(X4+Y4)2)+154-154(C1+2X4)2(X4+Y4)2)-Y42-2X4Y4,(2.41)B4,4231=μ6X410ae4J42(-3527(1-(C1+2X4)2(X4+Y4)2)(C1+2X4)×(1-C1+2X4X4+Y4)(X4+Y4)-1-158(1-C1+2X4X4+Y4)2)×(12-Y42-2X4Y4X4+3316-Y42-2X4Y4X42).Since the termωeΘ4 is constant, it plays no role in the equations of motion, and a new Hamiltonian can be introduced,(2.42)Ĥ4=H4+ωeΘ4.The dynamical system described byĤ4 is given by(2.43)d(X4,Y4)dt=∂Ĥ4∂(x4,y4),d(x4,y4)dt=-∂Ĥ4∂(X4,Y4).The zonal harmonics used in (2.34) and (2.35) and the tesseral harmonics used in (2.36) to (2.41) are shown in Table 2.Table 2
The zonal and tesseral harmonics.
Zonal harmonicsTesseral harmonicsJ20=1.0826×10-3J22=1.8154×10-6J40=-1.6204×10-6J42=1.6765×10-7The constant of integrationC1 in (2.34) to (2.41) is given, in terms of the initial values of the orbital elements, ao, eo, and Io, by(2.44)C1=μao(1-eo2cos⁡(Io)-2),
or, in terms of the variables X4 and Y4,(2.45)C1=X4(cos⁡(Io)-2)+Y4cos⁡(Io).In Section4, some results of the numerical integration of (2.43) are shown.


## 3. Lyapunov Exponents

The estimation of the chaoticity of orbits is very important in the studies of dynamical systems, and possible irregular motions can be analyzed by Lyapunov exponents [23].In this work, “Gram-Schmidt’s method,” described in [23–26], will be applied to compute the Lyapunov exponents. A brief description of this method is presented in what follows.The dynamical system described by (2.43) can be rewritten as
(3.1)dX4dt=P1(X4,Y4,x4,y4;C1),dY4dt=P2(X4,Y4,x4,y4;C1),dx4dt=P3(X4,Y4,x4,y4;C1),dy4dt=P4(X4,Y4,x4,y4;C1).Introducing(3.2)z=(X4Y4x4y4),Z=(P1P2P3P4).Equations (3.2) can be put in the form(3.3)dzdt=Z(z).
The variational equations, associated to the system of differential equations (3.3), are given by(3.4)dζdt=Jζ,
where J=(∂Z/∂z) is the Jacobian.The total number of differential equations used in this method isn(n+1), n represents the number of the motion equations describing the problem, in this case four. In this way, there are twenty differential equations, four are motion equations of the problem and sixteen are variational equations described by  (3.4)The dynamical system represented by (3.3) and (3.4) is numerically integrated and the neighboring trajectories are studied using the Gram-Schmidt orthonormalization to calculate the Lyapunov exponents.The method of the Gram-Schmidt orthonormalization can be seen in [25, 26] with more details. A simplified denomination of the method is described as follows.Considering the solutions to (3.4) as uκ(t), the integration in the time τ begins from initial conditions uκ(t0)=eκ(t0), an orthonormal basis.At the end of the time interval, the volumes of theκ-dimensional (κ=1,2,…,N) produced by the vectors uκ are calculated by(3.5)Vκ=‖⋀j=1κuj(t)‖,
where ⋀ is the outer product and ∥·∥ is a norm.In this way, the vectorsuκ are orthonormalized by Gram-Schmidt method. In other words, new orthonormal vectors eκ(t0+τ) are calculated, in general, according to(3.6)eκ=uκ-∑j=1κ-1(uκ⋅ej)ej‖uκ-∑j=1κ-1(uκ⋅ej)ej‖.The Gram-Schmidt method makes invariant theκ-dimensional subspace produced by the vectors u1,u2,u3,…,uκ in constructing the new κ-dimensional subspace spanned by the vectors e1,e2,e3,…,eκ.With new vectoruκ(t0+τ)=eκ(t0+τ), the integration is reinitialized and carried forward to t=t0+2τ. The whole cycle is repeated over a long-time interval. The theorems guarantee that the κ-dimensional Lyapunov exponents are calculated by [25, 26]:(3.7)λ(κ)=lim⁡n→∞1nτ∑j=1nln⁡(Vκ(t0+jτ))ln⁡(Vκ(t0+(j-1)τ)).The theory states that if the Lyapunov exponent tends to a positive value, the orbit is chaotic.In the next section are shown some results about the Lyapunov exponents.


## 4. Results

Figures1, 2, 3, and 4 show the time behavior of the semimajor axis, x4 angle, argument of perigee and of the eccentricity, according to the numerical integration of the motion equations, (2.43), considering three different resonant angles together: ϕ2201, ϕ2211, and ϕ2221 associated to J22, and three angles, ϕ4211, ϕ4221, and ϕ4231 associated to J42, with the same frequency of the resonant angles related to the J22, but with different phase. The initial conditions corresponding to variables X4 and Y4 are defined for eo=0.001, Io=55°, and ao given in Table 3. The initial conditions of the variables x4 and y4 are 0° and 0°, respectively. Table 3 shows the values of C1 corresponding to the given initial conditions.Table 3
Values of the constant of integrationC1 for e=0.001, I=55° and different values for semimajor axis.
a(0)×103(m)C1×1011(m2/s)26555.000−1.46754315826561.700−1.46772828226562.400−1.46774762326563.500−1.46777801326565.000−1.467819454Figure 1
Time behavior of the semimajor axis for different values ofC1 given in Table 3.Figure 2
Time behavior ofx4 angle for different values of C1 given in Table 3.Figure 3
Time behavior of the argument of pericentre for different values ofC1 given in Table 3.Figure 4
Time behavior of the eccentricity for different values ofC1 given in Table 3.Figures5, 6, 7, and 8 show the time behavior of the semimajor axis, x4 angle, argument of perigee and of the eccentricity for two different cases. The first case considers the critical angles ϕ2201, ϕ2211, and ϕ2221, associated to the tesseral harmonic J22, and the second case considers the critical angles associated to the tesseral harmonics J22 and J42. The angles associated to the J42, ϕ4211, ϕ4221, and ϕ4231, have the same frequency of the critical angles associated to the J22 with a different phase. The initial conditions corresponding to variables X4 and Y4 are defined for eo=0.05, Io=10°, and ao given in Table 4. The initial conditions of the variables x4 and y4 are 0° and 60°, respectively. Table 4 shows the values of C1 corresponding to the given initial conditions.Table 4
Values of the constant of integrationC1 for e=0.05, I=10°, and different values for semimajor axis.
a(0)×103(m)C1×1011(m2/s)26555.000−1.04572433126565.000−1.04592121026568.000−1.04598026726574.000−1.046098370Figure 5
Time behavior of the semimajor axis for different values ofC1 given in Table 4.Figure 6
Time behavior ofx4 angle for different values of C1 given in Table 4.Figure 7
Time behavior of the argument of pericentre for different values ofC1 given in Table 4.Figure 8
Time behavior of the eccentricity for different values ofC1 given in Table 4.Analyzing Figures5–8, one can observe a correction in the orbits when the terms related to the tesseral harmonic J42 are added to the model. Observing, by the percentage, the contribution of the amplitudes of the terms B4,4211, B4,4221, and B4,4231, in each critical angle studied, is about 1,66% up to 4,94%. In fact, in the studies of the perturbations in the artificial satellites motion, the accuracy is important, since adding different tesseral and zonal harmonics to the model, one can have a better description about the orbital motion.Figures9, 10, 11, and 12 show the time behavior of the semimajor axis, x4 angle, argument of perigee and of the eccentricity, according to the numerical integration of the motion equations, (2.43), considering three different resonant angles together; ϕ2201, ϕ2211, and ϕ2221 associated to J22 and three angles ϕ4211, ϕ4221, and ϕ4231 associated to J42. The initial conditions corresponding to variables X4 and Y4 are defined for eo=0.01, Io=55°, and ao given in Table 5. The initial conditions of the variables x4 and y4 are 0° and 60°, respectively. Table 5 shows the values of C1 corresponding to the given initial conditions.Table 5
Values of the constant of integrationC1 for e=0.01, I=55°, and different values for semimajor axis.
a(0)×103(m)C1×1011(m2/s)26555.000−1.46757237026558.000−1.46765526526562.000−1.46776578626564.000−1.46782104326568.000−1.467931552Figure 9
Time behavior of the semimajor axis for different values ofC1 given in Table 5.Figure 10
Time behavior ofx4 angle for different values of C1 given in Table 5.Figure 11
Time behavior of the argument of pericentre for different values ofC1 given in Table 5.Figure 12
Time behavior of the eccentricity for different values ofC1 given in Table 5.Analyzing Figures1–12, one can observe possible irregular motions in Figures 1–4, specifically considering values for C1=-1.467778013×1011 m2/s and C1=-1.467819454×1011 m2/s, and, in Figures 9–12, for C1=-1.467765786×1011 m2/s and C1=-1.467821043×1011 m2/s. These curves will be analyzed by the Lyapunov exponents in a specified time verifying the possible regular or chaotic motions.Figures13 and 14 show the time behavior of the Lyapunov exponents for two different cases, according to the initial values of Figures 1–4 and 9–12. The dynamical system involves the zonal harmonics J20 and J40 and the tesseral harmonics J22 and J42. The method used in this work for the study of the Lyapunov exponents is described in Section 3. In Figure 13, the initial values for C1, x4, and y4 are C1=-1.467778013×1011 m2/s and C1=-1.467819454×1011 m2/s, x4=0° and y4=0°, respectively. In Figure 14, the initial values for C1, x4, and y4 are C1=-1.467765786×1011 m2/s and C1=-1.467821043×1011 m2/s, x4=0° and y4=60°, respectively. In each case are used two different values for semimajor axis corresponding to neighboring orbits shown previously in Figures 1–4 and 9–12.Figure 13
Lyapunov exponentsλ(1) and λ(2), corresponding to the variables X4 and Y4, respectively, for C1=-1.467778013×1011 m2/s and C1=-1.467819454×1011 m2/s, x4=0° and y4=0°.Figure 14
Lyapunov exponentsλ(1) and λ(2), corresponding to the variables X4 and Y4, respectively, for C1=-1.467765786×1011 m2/s and C1=-1.467821043×1011 m2/s, x4=0° and y4=60°.Figures13 and 14 show Lyapunov exponents for neighboring orbits. The time used in the calculations of the Lyapunov exponents is about 150.000 days. For this time, it can be observed in Figure 13 that λ(1), corresponding to the initial value a(0)=26565.0 km, tends to a positive value, evidencing a chaotic region. On the other hand, analyzing the same Figure 13, λ(1), corresponding to the initial value a(0)=26563.5 km, does not show a stabilization around the some positive value, in this specified time. Probably, the time is not sufficient for a stabilization in some positive value, or λ(1), initial value a(0)=26563.5 km, tends to a negative value, evidencing a regular orbit. Analyzing now Figure 14, it can be verified that λ(1), corresponding to the initial value a(0)=26564.0 km, tends to a positive value, it contrasts with λ(1), initial value a(0)=26562.0 km. Comparing Figure 13 with Figure 14, it is observed that the Lyapunov exponents in Figure 14 has an amplitude of oscillation greater than the Lyapunov exponents in Figure 13. Analyzing this fact, it is probable that the necessary time for the Lyapunov exponent λ(2), in Figure 14, to stabilize in some positive value is greater than the necessary time for the λ(2) in Figure 13.Rescheduling the axes of Figures13 and 14, as described in Figures 15 and 16, respectively, the Lyapunov exponents tending to a positive value can be better visualized.Figure 15
Lyapunov exponentsλ(1) and λ(2), corresponding to the variables X4 and Y4, respectively, for C1=-1.467778013×1011 m2/s and C1=-1.467819454×1011 m2/s, x4=0° and y4=0°.Figure 16
Lyapunov exponentsλ(1) and λ(2), corresponding to the variables X4 and Y4, respectively, for C1=-1.467765786×1011 m2/s and C1=-1.467821043×1011 m2/s, x4=0° and y4=60°.


## 5. Conclusions

In this work, the dynamical behavior of three critical angles associated to the 2 : 1 resonance problem in the artificial satellites motion has been investigated.The results show the time behavior of the semimajor axis, argument of perigee and e-ccentricity. In the numerical integration, different cases are studied, using three critical angles together:ϕ2201, ϕ2211, and ϕ2221 associated to J22 and ϕ4211, ϕ4221, and ϕ4231 associated to the J42.In the simulations considered in the work, four cases show possible irregular motions forC1=-1.467778013×1011 m2/s, C1=-1.467819454×1011 m2/s, C1=-1.467765786×1011 m2/s, and C1=-1.467821043×1011 m2/s. Studying the Lyapunov exponents, two cases show chaotic motions for C1=-1.467819454×1011 m2/s and C1=-1.467821043×1011 m2/s.Analyzing the contribution of the terms related to theJ42, it is observed that, for the value of C1=-1.045724331×1011 m2/s, the amplitudes of the terms B4,4211, B4,4221, and B4,4231 are greater than the other values of C1. In other words, for bigger values of semimajor axis, it is observed a smaller contribution of the terms related to the tesseral harmonic J42.The theory used in this paper for the 2 : 1 resonance can be applied for any resonance involving some artificial Earth satellite.



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*Source: 405870-2012-01-17.xml*