Research Article Fourier Series Approximations to -Bounded ...downloads.hindawi.com/journals/mpe/2014/568318.pdf · Such pseudo-elliptic bounded orbit displays a rosette shape in

Research ArticleFourier Series Approximations to 119869

2-Bounded Equatorial Orbits

Wei Wang1 Jianping Yuan23 Yanbin Zhao1 Zheng Chen2 and Changchun Chen1

1 Research and Development Center Shanghai Institute of Satellite Engineering Shanghai 200240 China2 College of Astronautics Northwestern Polytechnical University Xirsquoan 710072 China3 Science and Technology on Aerospace Flight Dynamics Laboratory Xirsquoan 710072 China

Correspondence should be addressed to Wei Wang 418362467qqcom

Received 17 August 2013 Accepted 2 October 2013 Published 23 February 2014

Academic Editor Piermarco Cannarsa

Copyright copy 2014 Wei Wang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The current paper offers a comprehensive dynamical analysis and Fourier series approximations of 1198692-bounded equatorial orbits

The initial conditions of heterogeneous families of 1198692-perturbed equatorial orbits are determined first Then the characteristics

of two types of 1198692-bounded orbits namely pseudo-elliptic orbit and critical circular orbit are studied Due to the ambiguity of

the closed-form solutions which comprise the elliptic integrals and Jacobian elliptic functions showing little physical insight intothe problem a new scheme termed Fourier series expansion is adopted for approximation herein Based on least-squares fittingto the coefficients the solutions are expressed with arbitrary high-order Fourier series since the radius and the flight time varyperiodically as a function of the polar angle As a consequence the solutions can be written in terms of elementary functions such ascosines rather than complexmathematical functions Simulations enhance the proposed approximationmethod showing boundedand negligible deviations The approximation results show a promising prospect in preliminary orbits design determination andtransfers for low-altitude spacecrafts

1 Introduction

Themotion of a particle is known to be integrable by quadra-tures in a central force field wherein the potential displays aspherical symmetry in the mathematical expression writtenin the following form via a finite or infinite expansion

119881 (119903) =

+infin

sum

119899=minusinfin

119886119899119903119899 119899 isin Z 0 (1)

thus the motion of the particle depends on its distance119903 from the center of the planet only Such potentials aredetermined by the density of the planets usually taken toestablish the dynamic models in stellar and galactic systems[1] Several typical ones are the Hernquist-Newton potentialthe Plummer potential the spherical harmonic potentialthe power-law potential the logarithmic potential and theKepler potential [2] Interestingly whatever formats thepotentials are and despite the differences in mathematicalexpressions analogous rosette-shaped bounded orbits alwaysrise under the specific conditions Moreover potentials in

some special cases are also equivalent to be sphericallysymmetric amongwhich are the radial acceleration problemsfor orbital transfers in spacecrafts mission design [3ndash14] andthe problems of a particle in equatorial orbits considering the1198692term studied by some researchers recently [15ndash18]In general the total gravitational potential including

1198692perturbative potential and Kepler potential is axially

symmetric and it works significantly on a low-orbitingparticle that moves around an ellipsoidal planet While theparticlersquos orbit is confined to the equatorial plane the 119869

2

potential reduces to be spherically symmetric and the prob-lem becomes integrable Due to its integrability in the senseof Liouville [17] the closed-form solutions of 119869

2-bounded

equatorial orbits were given in terms of elliptic integrals[15 18] based on which the periodic and pseudo-ellipticbounded relative orbits were obtained Lately the solutionsof 1198692-unbounded equatorial orbits were also studied with the

help of elliptic functions and two types of new unboundedorbits in equatorial plane namely pseudo-parabolic orbitsand pseudo-hyperbolic orbits were unveiled [16]

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 568318 8 pageshttpdxdoiorg1011552014568318

2 Mathematical Problems in Engineering

The dynamical analysis of the 1198692-bounded equatorial

orbits which may be of deep insight to the problem isessential but scarce in the literature Besides although theelliptic integrals and elliptic functions may present compactclosed-form solutions according to the previous studiesthesemathematical functions complicate the practical use forpreliminarymission design due to the lack of physical insight

In view of this the present paper first offers a compre-hensive study on dynamical characteristics and initial condi-tions of 119869

2-perturbed equatorial orbits Then we establish a

methodology to obtain arbitrary high-order approximationsof the 119869

2-bounded equatorial orbits by means of Fourier

series expansions such as trigonometric functions since thedistance varies periodically as a function of time or polarangle In this paper we utilize the Fourier series of order 1 andorder 2 via least-squares fitting to the coefficients The devi-ations are quantifiable and negligible and are guaranteed tobe bounded by imposing a constraint that the approximationsolutions coincide with the exact solutions at both endpointsof the half period

As a result the solutions are expressed in terms ofelementary functions such as cosines rather than complexmathematical functions and the proposed methods aredemonstrated to be effective in simulation

2 Problem Formulationand Mathematic Model

The equatorial geopotential comprising 1198692

perturbativepotential and Kepler potential can be expressed as [19]

119881 (119903) = minus120583

119903[1 +

1198692

2(119877119890

119903)

2

] (2)

where 119877119890represents the mean equatorial radius of the

planet 1198692is the second zonal harmonics coefficient 120583 is

the gravitational parameter and 119903 denotes the modulus ofposition vector r

Since we are dealing only with gravitational forces inthis paper the orbit of a particle in a given field does notdepend on its mass Hence we examine the dynamics of aparticle of unit mass so the quantities such as momentumangular momentum and energy and functions such asthe Lagrangian and Hamiltonian are normally written perunit mass The Lagrangian 119871 of the system is formed bysubtracting the potential energy119881(119903) from the kinetic energy119879 where polar coordinates (119903 120579) are mostly convenientlyused as follows

119871 = 119879 minus 119881 (119903) =1

2[ 1199032+ (119903 120579)

2

] +120583

119903[1 +

1198692

2(119877119890

119903)

2

] (3)

Obviously 120579 is an ignorable coordinate from (3) thecanonical momentum also referred to as the first integralconstant yields from

120597119871

120597 120579

= 1199032 120579 = ℎ = const (4)

TheEuler-Lagrange equation for 119903 therefore becomes theradial force equation

119903 minus 119903 1205792= 119865 (119903) (5)

where119865(119903) = minus120583(1+311986921198772

11989021199032)1199032 denoting the acceleration

per mass that resulted from the center force or imposed byKepler and 119869

2perturbative forces along r Also the second

integral constant is easily obtained from the Hamiltonian

120576 = 119879 + 119881 =1

21199032+1

2(119903 120579)2

minus120583

119903minus11986921205831198772

119890

21199033

= const (6)

as the energy is conservedDenote the effective potential energy of the radial motion

as

119881eff =1

2(119903 120579)2

+ 119881 (119903) =ℎ2

21199032minus120583

119903minus11986921205831198772

119890

21199033 (7)

Substituting (6) into (7) yields

119903 = plusmnradic2 (120576 minus 119881eff) (8)

3 Initial Conditions of 1198692-Perturbed

Equatorial Orbits

The particle motion in equatorial plane subject to 1198692pertur-

bation is therefore established in Section 2 Itmay be checkedthat if we set 119903 = 0 a unified equation is obtained accordingto (8)

1205761199033+ 1205831199032minusℎ2

2119903 +

12058311986921198772

119890

2= 0 (9)

comprising three main cases that need to be discussed Notethat (9) is a cubic equation if 120576 = 0 To have three real rootsthe discriminant of (9) satisfies

Δ = (minus120583ℎ2

121205762minus

1205833

271205763minus11986921205831198772

119890

4120576)

2

+ (minusℎ2

6120576minus1205832

91205762)

3

le 0

(10)

where the equality sign indicates the existence of a couple ofidentical roots

Mathematical Problems in Engineering 3

The three roots are given by

1199091= minus

120583

3120576+3radicminus

120583ℎ2

121205762minus

1205833

271205763minus11986921205831198772

119890

4120576+ radicΔ

+3radicminus

120583ℎ2

121205762minus

1205833

271205763minus11986921205831198772

119890

4120576minus radicΔ

1199092= minus

120583

3120576+minus1 + radic3119894

2

3radicminus

120583ℎ2

121205762minus

1205833

271205763minus11986921205831198772

119890

4120576+ radicΔ

+minus1 minus radic3119894

2

3radicminus

120583ℎ2

121205762minus

1205833

271205763minus11986921205831198772

119890


1199093= minus

120583

3120576+minus1 minus radic3119894

2

3radicminus

120583ℎ2

121205762minus

1205833

271205763minus11986921205831198772

119890

4120576+ radicΔ

+minus1 + radic3119894

2

3radicminus

120583ℎ2

121205762minus

1205833

271205763minus11986921205831198772

119890


(11)

The other special case is 120576 = 0 and (9) reduces to be aquadratic equation

1199032minusℎ2

2120583119903 +

11986921198772

119890

2= 0 (12)

Accordingly the discriminant of (9) satisfies

Δ = (ℎ2

2120583)

2

minus 211986921198772

119890gt 0 (13)

The two roots are

11990912=

ℎ22120583 plusmn radicℎ

441205832minus 211986921198772

119890

2

(14)

Assume that the 1198692zonal harmonics is taken as 119869

2= 108263times

10minus3 herein 119903

0denotes the initial radius and (sdot) represents the

normalized variables 119903 = 119903119877119890 = 119905radic119877

3

119890120583 ℎ = ℎradic119877e120583

120576 = 120576radic119877119890120583 As the total energy 120576 and angular momentum

ℎ are the two integral constant values (4) and (6) can betranslated in a graphical form in (119903

0sim 120576) plane for different

values of the fixed ℎ as shown in Figure 1Observe that all the contour lines illustrated in Figure 1

pass through the horizontal line 120576 = 0 In other wordsit implies the existence of heterogeneous families of orbitsbounded orbits for 120576 lt 0 and unbounded obits for 120576 = 0 and120576 gt 0 the orbital shape is totally determined by the initialconditionsThis is confirmed by Figure 2 in which the phasespace portrait is depicted and the corresponding potentialwell is outlined

Generally the curve of the potential energy intersects theconstant energy line twice for 120576 ge 0 and three times for120576 lt 0 However the points associated with the minimumroots of (8) are found to be located below the surface ofthe planet and are of less physical meaning so they are

05 06 07 08 09 1 11 12 13 14 15

0

05

1

15

2

25

3

09222

1120513189

15172

minus1

minus05

r0

h

Figure 1 Representation of motion in the plane (1199030sim 120576) with

different values of the fixed ℎ

120576 lt 0

r

r

120576 = 0

120576 = 0

120576 gt 0

120576 gt 0

120576 lt 0

Veff

r

Figure 2 Orbital motions in the phase space (119903 119903) and potentialenergy wellignored in Figure 2 If 120576 lt 0 and 120597119881eff120597119903 = 0 the particlewill be trapped in the well oscillating radially and the radiusrange is 119903min le 119903 le 119903max Here 119903min and 119903max correspond to119903 = 0 in the phase space (119903 119903) referred to as the lower limitand upper limit As the energy decreases until the effectiveenergy arrives at its local minimum namely 120597119881eff120597119903 = 0the orbital shape degenerates from a pseudo-ellipse to a circlewith a radius 119903

0 corresponding to the fixed point in Figure 2

Otherwise if the energy increases till 120576 ge 0 pseudo-parabolicor pseudo-hyperbolic unbounded orbits will occur whichlead to bifurcations in phase space In this paper we onlyfocus on the behaviors of the bounded orbits whereas theunbounded orbits are left for the further research

4 Fundamental Characteristics of 1198692-Bounded

Equatorial Orbits

A comprehensive analysis on initial conditions of heteroge-neous families of 119869

2-perturbed equatorial orbits is discussed

In this section the focus is 1198692-bounded equatorial orbitsrsquo


fundamental characteristics To begin radial and azimuthalkinetic energy are defined for convenience

120576119903=1

21199032 120576

120579=1

2(119903 120579)2

(15)

41 Pseudo-Elliptic Orbit For pseudo-elliptic bounded orbit120576119903

= 0 is satisfied for a certain 119903 and the equation119903 = 0 normally has two roots 119903min and 119903max provided that120597119881eff120597119903 = 0 known as pericenter and apocenter betweenwhich the particle oscillates radially as it revolves The radialperiod and the azimuthal angle increases by an amounttogether with the azimuthal period and the distance r can becomputed as [2]

119879119903= radic

2

minus120576[1205821119865(

120587

2 119896) + 120582

2119864(

120587

2 119896) + 120582

3Π(

120587

2 1205722 119896)]

Δ120579 =2ℎ

1205822

radic2

minus120576119865(

120587

2 119896)

119879120579=2120587

Δ120579119879119903= (120587120582

2[1205821119865(

120587

2 119896) + 120582

2119864(

120587

2 119896)

+1205823Π(

120587

2 1205722 119896)])

times (ℎ119865(120587

2 119896))

minus1

119903 (120579) = (119903min119903max)

times (119903min + (119903max minus 119903min) 1199041198992

times [119865(120587

2 119896) minus

1205822radicminus2120576

2ℎ120579 119896])

minus1

(16)

where 119865(1205872 119896) 119864(1205872 119896) Π(1205872 1205722 119896) are the completeelliptic integrals of the first second and third kind respec-tively The coefficients are as follows

1205821= minus119903maxradic

119903min119903max minus 119903lowast

1205822= radic119903min (119903max minus 119903lowast)

1205823=119903max (119903max + 119903min + 119903lowast)

radic119903min (119903max minus 119903lowast)

(17)

Here 119903lowastis another root of (9) and

119896 = radic(119903max minus 119903min) 119903lowast

(119903max minus 119903lowast) 119903min 120572 = radic1 minus

119903max119903min

(18)

where sn(sdot) is the Jacobian elliptic functionSuch pseudo-elliptic bounded orbit displays a rosette

shape in polar frame that bifurcates from the circular orbitand wraps around it enclosed in between and tangential totwo concentric circles of radii 119903min and 119903max

The flight time less than one-half radial period as afunction of the polar angle is given by

119905 = int

119903

119903min

119889119905

119889119903119889119903 = int

119903

119903min

1

radic2 (120576 minus 119881eff)119889119903 =

1

radicminus2120576

times [radic120576 minus 119881eff119903

+ 1205821119865 (120579 119896)

+1205822119864 (120579 119896) + 120582

3Π(120579 120572

2 119896) ]

(19)

One should take modulo if 119905 gt 1198792 namely

119905 =

119899119879119903+ int

119903

119903min

119889119905

119889119903119889119903 119905 isin [119899119879

119903 (119899 +

1

2)119879]

(119899 + 1) 119879119903minus int

119903max

119903

119889119905

119889119903119889119903 119905 isin [(119899 +

1

2)119879119903 (119899 + 1) 119879]

120579 =

119899Δ120579 + int

119903

119903min

119889120579

119889119903119889119903 119905isin[119899119879

119903 (119899 +

1

2)119879]

(119899 + 1) Δ120579 minus int

119903max

119903

119889120579

119889119903119889119903 119905 isin [(119899 +

1

2)119879119903(119899 + 1) 119879]

(20)

where 119899 isin N

42 Critical Circular Orbit Consider a special case in whichthe curve of the potential energy is tangent to the constantenergy line indicating two equivalent roots that appear incouple with their radii (119903)

12= 1199030 Equivalently the potential

energy arrives at its minimum Applying 120597119881eff120597119903 = 0 yields

(1199030)1=

ℎ2+ radicℎ4minus 6119869212058321198772

119890

2120583

(1199030)2=

ℎ2minus radicℎ4minus 6119869212058321198772

119890

2120583

(21)

which is in agreement with Humirsquos conclusion [20] and (1199030)2

is an extraneous root that should be discarded The angularvelocity and the orbital period of the spacecraft are given by

120596 =ℎ

1199032

0

119879 =21205871199032

0

ℎ (22)

Substituting 1199030into 1205972119881eff120597119903

2 yields

1205972119881eff1205971199032

100381610038161003816100381610038161003816100381610038161003816119903=1199030

= minus2120583

1199033

0

minus611986921205831198772

119890

1199035

0

+3ℎ2

1199034

0

gt 0 (23)

indicating that the 1198692-circular orbit is stable according to the

theorems of the classical mechanics [21]To detect the orbital evolution when small radial per-

turbations act on the particle the substitution 119906 = 1119903 isemployedThus the orbital dynamic equation is expressed as

1198892119906

1198891205792+ 119906 = 119869 (119906) (24)


where

119869 (119906) = minus1

ℎ2

119889

119889119906119881(

1

119906) =

1

ℎ2(3

211986921205831198772

1198901199062minus ℎ2119906 + 120583)

(25)

Since motion is stable 119906 is surely bounded and varieswithin a small region around 119906

0 To proceed we define

Δ119906 ≜ 119906 minus 1199060 denoting the deviation between perturbed and

unperturbed motion Expanding 119869(119906) in a Taylor series at 1199060

and neglecting all terms in 119869(119906) of second and higher ordersyield

119869 (119906) = 1199060+ (119906 minus 119906

0)119889119869

1198891199060

+ 119874 [(119906 minus 1199060)2

] (26)

then the equation of orbital motion around the 1198692-circular

orbit reduces to

1198892Δ119906

1198891205792+ Δ119906 = Δ119906

119889119869

1198891199060

(27)

For the sake of compactness denote

1205942= 1 minus

119889119869

1198891199060

= 2 minus311986921205831198772

1198901199060

ℎ2

(28)

then a simplified solution to (27) is obtained

Δ119906 = 119906 minus 1199060= 119860 cos (120594120579) (29)

where 119860 is related to the external perturbation imposed onthe particle Through transformation we have

119903 =1199030

1 + 1198601199030cos (120594120579)

(30)

Usually 120594 is an irrational number Hence the particle willcome to a pseudo-elliptic orbit and oscillate radially withina small region

5 Fourier Series Approximations

In the previous section the closed-form analytical solutionsfor 1198692-bounded equatorial orbits were derived in terms of

elliptic integrals and elliptic functions However due tothe lack of physical insight regarding these mathematicalfunctions instead of the elementary functions it is difficult toput them into practical use formission designHere a Fourierseries expansion is resorted for analytical approximations

Equation (5) shows that the radius is an even function of120579 with half radial period Δ120579 Hence the generalized Fourierseries evaluated on the interval of [0 Δ120579] can be written bymeans of cosines alone [3]

119903(infin)

app (120579) =119903min

1 + suminfin

119894=0119862119894cos (119894120587 (120579Δ120579))

(31)

where 119862119894are the Fourier coefficients expressed as

1198620=

1

2Δ120579int

Δ120579

0

[119903min119903 (120579)

minus 1] 119889120579

119862119894=

1

Δ120579int

Δ120579

0

[119903min119903 (120579)

minus 1] cos(119894120587 120579

Δ120579)119889120579

(32)

Consider a limiting case in which the first-order approx-imation is merely retained in (31) under the assumption that1198901198692≪ 1 As such the truncated expression of (31) is given by

119903(1)

app (120579) =119903min

1 + 1198610+ 1198611cos (120587 (120579Δ120579))

(33)

where 1198610and 119861

1are constants and different from the

Fourier coefficients119862119894in (31) Assume that the approximation

solution coincides with the exact solution at both endpointsof the half period [0 Δ120579] thus the maximum deviation willremain bounded

Suppose the particle starts at pericenter and ends atapocenter on a short timescale [0 Δ120579] then 119861

0and 119861

1are

calculated as follows

1198610= minus

119903max minus 119903min2119903max

1198611=119903max minus 119903min2119903max

(34)

To seek for a brief format of approximation denote

1198861198692=119903max + 119903min

2 119890

1198692=119903max minus 119903min119903max + 119903min

1199011198692= 1198861198692(1 minus 119890

2

1198692)

(35)

Substituting 1198610 1198611 and (35) into (33) yields

119903(1)

app (120579) =1199011198692

1 + 1198901198692cos (120587 (120579Δ120579))

(36)

Accordingly the first-order approximation of flight time asa function of polar angle can be computed by substitutingequation (36) into equation 119889120579119889119905 = ℎ1199032

119905(1)

app (120579) asymp int120579

0

[119903(1)

app (120579)]2

ℎ119889120579 =

Δ1205791199012

1198692

120587ℎ(1 minus 1198902

1198692)32

times

2 arctan[radic1 minus 1198901198692

1 + 1198901198692

tan( 120587120579

2Δ120579)]

minus

1198901198692radic1 minus 119890

2

1198692sin (120587120579Δ120579)

1 + 1198901198692cos (120587120579Δ120579)

(37)

To evaluate the accuracy of approximation it is necessaryto introduce the distance deviation

120576(1)

Δ=

10038161003816100381610038161003816119903(1)

app (120579) minus 11990310038161003816100381610038161003816

119903 (38)


Table 1 Least-squares fitting coefficients 1198610 1198611 1198612vary with 119890(2)

1198692

119890(2)

11986921198610

1198611

1198612

min 119869

01111 minus999953119890 minus 2 999966119890 minus 2 minus125461119890 minus 6 347383119890 minus 8





0 1 2 3 4 50

1

2

3

4

5

6

7

8times10

minus6

120576(1)

Δ

t

Figure 3 Variations of the deviation 120576(1)Δ

for 1198901198692= 4times 10

minus6≪ 1 and

isin [0 5]

And the flight time deviation is also constructed as

120576(1)

Δ119905=

10038161003816100381610038161003816119905(1)

app (120579) minus 11990510038161003816100381610038161003816

119905 (39)

Figure 3 shows that for ℎ = 11205 1198692= 108263 times 10

minus3 and1198901198692= 4times10

minus6≪ 1 120576(1)

Δvaries periodically and themagnitude

of 120576(1)Δ

is less than 8 times 10minus6 implying the effectiveness of thefirst-order Fourier series approximation for small 119890

1198692

As 1198901198692increases gradually even if 119890

1198692gt 01 the first-order

Fourier series approximation of flight time seems to be stilleffective the magnitude of 120576(1)

Δ119905is less than 3 times 10minus5 as shown

in Figure 4 Nevertheless the radius approximation becomesto be inadequate especially for 119890

1198692gt 01 as displayed in

Figure 5 Naturally it necessitates us to retain high-orderterms in the proceedings of approximating

To that end we retain119898 minus 1 orders (119898 gt 1) of the serieswritten by

119903(119898)

app (120579) =119903min

1 + sum119898

119894=0119861119894cos (119894120587 (120579Δ120579))

(40)

Once again assume that the approximation solution coin-cides with the exact solution at both endpoints of the half-period [0 Δ120579] Sequentially the coefficients in (40) satisfy thefollowing constraints

119898

sum

119894=0

119861119894= 0

119898

sum

119894=0

119861119894(minus1)119894=119903min119903max

minus 1 (41)

It is noteworthy that these constraints enable119898minus1degree-of-freedom One rational approach is to apply least-squaresfitting The objective function is constructed as

min 119869 (1198610 1198611 119861119898) =

119899

sum

119895=1

119898

sum

119894=0

[119903(119898)

app (119861119894 120579119895) minus 119903 (120579119895)]2

(42)

where 119899 denotes the quantity of total discrete points in theprocess of integration

Table 1 shows the coefficients with large 1198901198692 obtained

by Fourier series approximation of order 2 and the corre-sponding objective functions are also givenThe fixed angularmomentum ℎ = 11205 One should not neglect the factthat due to the constraint 119903(2)app(120579) gt 119877

119890 1198901198692lt 0254 must be

satisfiedFigure 6 displays the Fourier series approximation of

order 2 via least-squares fitting In contrast to the resultby first-order Fourier series approximations the maximumdeviation is less than 0025 of the orbital radius 119903(120579) In viewof this it is suitable for the high-accuracy required missions

Figure 7 illustrates the particlersquos actual orbit (solid line)and orbit by Fourier series approximations (dashed line) inpolar reference frame (120579 119903) with 119890

1198692= 025 In Figure 8 the

left plane shows the evolution of the actual radius and radiusof Fourier series approximation of order 2 via least-squaresfitting The right plane presents a magnified view of a localsegment

6 Conclusions

The main contribution of this paper is that a frameworkfor approximating 119869

2-bounded equatorial orbits is estab-

lished with arbitrary high-order Fourier series expansions


0 05 1 15 2 25 30

05

1

15

2

25

3

120579

times10minus5

eJ2 = 0250

eJ2 = 0212

eJ2 = 0177

eJ2 = 0143

120576(1)

Δt

Figure 4 Variations of the deviation 120576(1)Δ119905

for 1198901198692gt 01 and 120579 isin [0 3]

0 1 2 3 4 50

001

002

003

004

005

006

007

120576(1)

Δ

eJ2 = 0111

eJ2 = 0081

eJ2 = 0053

eJ2 = 0026

t

Figure 5 Variations of the deviation 120576(1)Δ and isin [0 5]

Since the distance and time vary periodically as a functionof polar angle the solutions are expressed in terms ofelementary trigonometric functions rather than Jacobianelliptic function and elliptic integrals that lack physicalinsight into the problem For Fourier series expansion ofsecond order or higher the coefficients can be selectedvia least-squares fitting and the deviations are guaranteedto be still bounded by imposing a constraint that theapproximation solutions coincide with the exact solutionsat both endpoints of the half period Also the approxi-mation of closed-form relationship for the flight time asa function of the polar angle is given using an analyticalapproach

0 1 2 3 4 50

05

1

15

2

25

0111

0143

0177

0212

0250

120576(2)

Δ

times10minus5

eJ2

t


for 1198901198692gt 01 and isin [0 5]

05

1

15

2

30

210

60

240

90

270

120

300

150

330

180 0

Figure 7 Actual orbit and orbit by Fourier series approximation for1198901198692gt 01 in polar reference frame (120579 119903)

The presented approximation method has a potentialfor space missions such as novel orbits designand compu-tational efficiency improvement for long-term low-altitudeorbits

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper


0 5 10 15 20 25 30 35 40 45 501

1112131415161718

465

17

465

175

465

18

465

185

465

19

1353713537135371353813538135381353813538135391353913539

t

rr(120579)

r(2)app (120579)

Figure 8 Evolution of actual radius and radius by Fourier series approximation for 1198901198692= 025

Acknowledgment

This work was supported by the Open Research Foundationof Science and Technology in Aerospace Flight DynamicsLaboratory of China (Grant no 2012afdl021)

References

[1] V I Arnold V V Kozlov and A I Neishtadt MathematicalAspects of Classical and Celestial Mechanics vol 3 SpringerBerlin Germany Third edition 2006

[2] J Binney and S Tremaine Galactic Dynamics Princeton Uni-versity Press Princeton NJ USA 2005

[3] A A Quarta and G Mengali ldquoNew look to the constantradial acceleration problemrdquo Journal of Guidance Control andDynamics vol 35 no 3 pp 919ndash929 2012

[4] M R Akella and R A Broucke ldquoAnatomy of the constant radialthrust problemrdquo Journal of Guidance Control and Dynamicsvol 25 no 3 pp 563ndash570 2002

[5] J E Prussing andVCoverstone-Carroll ldquoConstant radial thrustacceleration reduxrdquo Journal of Guidance Control andDynamicsvol 21 no 3 pp 516ndash518 1998

[6] G Mengali and A A Quarta ldquoEscape from elliptic orbitusing constant radial thrustrdquo Journal of Guidance Control andDynamics vol 32 no 3 pp 1018ndash1022 2009

[7] J F San-Juan L M Lopez and M Lara ldquoOn boundedsatellite motion under constant radial propulsive accelerationrdquoMathematical Problems in Engineering vol 2012 Article ID680394 12 pages 2012

[8] G Mengali A A Quarta and G Aliasi ldquoA graphical approachto electric sailmission designwith radial thrustrdquoActaAstronau-tica vol 82 no 2 pp 197ndash208 2012

[9] A J Trask W J Mason and V L Coverstone ldquoOptimalinterplanetary trajectories using constant radial thrust andgravitational assistsrdquo Journal of Guidance Control and Dynam-ics vol 27 no 3 pp 503ndash506 2004

[10] H Yamakawa ldquoOptimal radially accelerated interplanetarytrajectoriesrdquo Journal of Spacecraft and Rockets vol 43 no 1 pp116ndash120 2006

[11] A A Quarta and G Mengali ldquoOptimal switching strategyfor radially accelerated trajectoriesrdquo Celestial Mechanics andDynamical Astronomy vol 105 no 4 pp 361ndash377 2009

[12] F Topputo A H Owis and F Bernelli-Zazzera ldquoAnalyticalsolution of optimal feedback control for radially accelerated

orbitsrdquo Journal of Guidance Control and Dynamics vol 31 no5 pp 1352ndash1359 2008

[13] A A Quarta and G Mengali ldquoAnalytical results for solarsail optimal missions with modulated radial thrustrdquo CelestialMechanics amp Dynamical Astronomy vol 109 no 2 pp 147ndash1662011

[14] C R McInnes ldquoOrbits in a generalized two-body problemrdquoJournal of Guidance Control and Dynamics vol 26 no 5 pp743ndash749 2003

[15] VMartinusi and P Gurfil ldquoSolutions and periodicity of satelliterelative motion under even zonal harmonics perturbationsrdquoCelestial Mechanics amp Dynamical Astronomy vol 111 no 4 pp387ndash414 2011

[16] VMartinusi andPGurfil ldquoAnalytical solutions for 1198692-perturbed

unbounded equatorial orbitsrdquoCelestial Mechanics ampDynamicalAstronomy vol 115 no 1 pp 35ndash57 2013

[17] V Martinusi and P Gurfil ldquoKeplerization of motion in anycentral force fieldrdquo in Proceedings of the 1st IAA Conference onDynamics amp Control of Space Systems Porto Portugal March2013 IAA-AAS-DyCoSS1-08-01

[18] WWang J P Yuan J J Luo and J Cao ldquoAnatomy of J2bounded

orbits in equatorial planerdquo Scientia Sinica Physica Mechanica ampAstronomica vol 43 no 3 pp 309ndash317 2013

[19] D A Vallado Fundamentals of Astrodynamics and ApplicationsMicrocosm Press Hawthorne Calif USA 3rd edition 2001

[20] M Humi and T Carter ldquoOrbits and relative motion in thegravitational field of an oblate bodyrdquo Journal of GuidanceControl and Dynamics vol 31 no 3 pp 522ndash532 2008

[21] A H Nayfeh and B Balachandran Applied Nonlinear Dynam-ics Wiley-VCH Stuttgart Germany 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


The dynamical analysis of the 1198692-bounded equatorial

orbits which may be of deep insight to the problem isessential but scarce in the literature Besides although theelliptic integrals and elliptic functions may present compactclosed-form solutions according to the previous studiesthesemathematical functions complicate the practical use forpreliminarymission design due to the lack of physical insight

In view of this the present paper first offers a compre-hensive study on dynamical characteristics and initial condi-tions of 119869

2-perturbed equatorial orbits Then we establish a

methodology to obtain arbitrary high-order approximationsof the 119869

2-bounded equatorial orbits by means of Fourier

series expansions such as trigonometric functions since thedistance varies periodically as a function of time or polarangle In this paper we utilize the Fourier series of order 1 andorder 2 via least-squares fitting to the coefficients The devi-ations are quantifiable and negligible and are guaranteed tobe bounded by imposing a constraint that the approximationsolutions coincide with the exact solutions at both endpointsof the half period

As a result the solutions are expressed in terms ofelementary functions such as cosines rather than complexmathematical functions and the proposed methods aredemonstrated to be effective in simulation

2 Problem Formulationand Mathematic Model

The equatorial geopotential comprising 1198692

perturbativepotential and Kepler potential can be expressed as [19]

119881 (119903) = minus120583

119903[1 +

1198692

2(119877119890

119903)

2

] (2)

where 119877119890represents the mean equatorial radius of the

planet 1198692is the second zonal harmonics coefficient 120583 is

the gravitational parameter and 119903 denotes the modulus ofposition vector r

Since we are dealing only with gravitational forces inthis paper the orbit of a particle in a given field does notdepend on its mass Hence we examine the dynamics of aparticle of unit mass so the quantities such as momentumangular momentum and energy and functions such asthe Lagrangian and Hamiltonian are normally written perunit mass The Lagrangian 119871 of the system is formed bysubtracting the potential energy119881(119903) from the kinetic energy119879 where polar coordinates (119903 120579) are mostly convenientlyused as follows

119871 = 119879 minus 119881 (119903) =1

2[ 1199032+ (119903 120579)

2

] +120583

119903[1 +

1198692

2(119877119890

119903)

2

] (3)

Obviously 120579 is an ignorable coordinate from (3) thecanonical momentum also referred to as the first integralconstant yields from

120597119871

120597 120579

= 1199032 120579 = ℎ = const (4)

TheEuler-Lagrange equation for 119903 therefore becomes theradial force equation

119903 minus 119903 1205792= 119865 (119903) (5)

where119865(119903) = minus120583(1+311986921198772

11989021199032)1199032 denoting the acceleration

per mass that resulted from the center force or imposed byKepler and 119869

2perturbative forces along r Also the second

integral constant is easily obtained from the Hamiltonian

120576 = 119879 + 119881 =1

21199032+1

2(119903 120579)2

minus120583

119903minus11986921205831198772

119890

21199033

= const (6)

as the energy is conservedDenote the effective potential energy of the radial motion

as

119881eff =1

2(119903 120579)2

+ 119881 (119903) =ℎ2

21199032minus120583

119903minus11986921205831198772

119890

21199033 (7)

Substituting (6) into (7) yields

119903 = plusmnradic2 (120576 minus 119881eff) (8)

3 Initial Conditions of 1198692-Perturbed

Equatorial Orbits

The particle motion in equatorial plane subject to 1198692pertur-

bation is therefore established in Section 2 Itmay be checkedthat if we set 119903 = 0 a unified equation is obtained accordingto (8)

1205761199033+ 1205831199032minusℎ2

2119903 +

12058311986921198772

119890

2= 0 (9)

comprising three main cases that need to be discussed Notethat (9) is a cubic equation if 120576 = 0 To have three real rootsthe discriminant of (9) satisfies

Δ = (minus120583ℎ2

121205762minus

1205833

271205763minus11986921205831198772

119890

4120576)

2

+ (minusℎ2

6120576minus1205832

91205762)

3

le 0

(10)

where the equality sign indicates the existence of a couple ofidentical roots



1199091= minus

120583

3120576+3radicminus

120583ℎ2

121205762minus

1205833

271205763minus11986921205831198772

119890

4120576+ radicΔ

+3radicminus

120583ℎ2

121205762minus

1205833

271205763minus11986921205831198772

119890


1199092= minus

120583

3120576+minus1 + radic3119894

2

3radicminus

120583ℎ2

121205762minus

1205833

271205763minus11986921205831198772

119890

4120576+ radicΔ


2

3radicminus

120583ℎ2

121205762minus

1205833

271205763minus11986921205831198772

119890


1199093= minus

120583


2

3radicminus

120583ℎ2

121205762minus

1205833

271205763minus11986921205831198772

119890

4120576+ radicΔ


2

3radicminus

120583ℎ2

121205762minus

1205833

271205763minus11986921205831198772

119890


(11)


1199032minusℎ2

2120583119903 +

11986921198772

119890

2= 0 (12)


Δ = (ℎ2

2120583)

2

minus 211986921198772

119890gt 0 (13)

The two roots are

11990912=


441205832minus 211986921198772

119890

2

(14)


2= 108263times




3

119890120583 ℎ = ℎradic119877e120583







05 06 07 08 09 1 11 12 13 14 15

0

05

1

15

2

25

3

09222

1120513189

15172

minus1

minus05

r0

h



120576 lt 0

r

r

120576 = 0

120576 = 0

120576 gt 0

120576 gt 0

120576 lt 0

Veff

r





Equatorial Orbits






120576119903=1

21199032 120576

120579=1

2(119903 120579)2

(15)



119879119903= radic

2

minus120576[1205821119865(

120587

2 119896) + 120582

2119864(

120587

2 119896) + 120582

3Π(

120587

2 1205722 119896)]

Δ120579 =2ℎ

1205822

radic2

minus120576119865(

120587

2 119896)

119879120579=2120587

Δ120579119879119903= (120587120582

2[1205821119865(

120587

2 119896) + 120582

2119864(

120587

2 119896)

+1205823Π(

120587

2 1205722 119896)])

times (ℎ119865(120587

2 119896))

minus1

119903 (120579) = (119903min119903max)


times [119865(120587

2 119896) minus


2ℎ120579 119896])

minus1

(16)







(17)




119903max119903min

(18)




119905 = int

119903

119903min

119889119905

119889119903119889119903 = int

119903

119903min

1

radic2 (120576 minus 119881eff)119889119903 =

1

radicminus2120576


+ 1205821119865 (120579 119896)

+1205822119864 (120579 119896) + 120582

3Π(120579 120572

2 119896) ]

(19)


119905 =

119899119879119903+ int

119903

119903min

119889119905

119889119903119889119903 119905 isin [119899119879

119903 (119899 +

1

2)119879]

(119899 + 1) 119879119903minus int

119903max

119903

119889119905

119889119903119889119903 119905 isin [(119899 +

1

2)119879119903 (119899 + 1) 119879]

120579 =

119899Δ120579 + int

119903

119903min

119889120579

119889119903119889119903 119905isin[119899119879

119903 (119899 +

1

2)119879]

(119899 + 1) Δ120579 minus int

119903max

119903

119889120579

119889119903119889119903 119905 isin [(119899 +

1

2)119879119903(119899 + 1) 119879]

(20)

where 119899 isin N




(1199030)1=

ℎ2+ radicℎ4minus 6119869212058321198772

119890

2120583

(1199030)2=


119890

2120583

(21)



120596 =ℎ

1199032

0

119879 =21205871199032

0

ℎ (22)


2 yields

1205972119881eff1205971199032

100381610038161003816100381610038161003816100381610038161003816119903=1199030

= minus2120583

1199033

0

minus611986921205831198772

119890

1199035

0

+3ℎ2

1199034

0

gt 0 (23)




1198892119906

1198891205792+ 119906 = 119869 (119906) (24)


where

119869 (119906) = minus1

ℎ2

119889

119889119906119881(

1

119906) =

1

ℎ2(3

211986921205831198772

1198901199062minus ℎ2119906 + 120583)

(25)






119869 (119906) = 1199060+ (119906 minus 119906

0)119889119869

1198891199060

+ 119874 [(119906 minus 1199060)2

] (26)


orbit reduces to

1198892Δ119906

1198891205792+ Δ119906 = Δ119906

119889119869

1198891199060

(27)


1205942= 1 minus

119889119869

1198891199060

= 2 minus311986921205831198772

1198901199060

ℎ2

(28)


Δ119906 = 119906 minus 1199060= 119860 cos (120594120579) (29)


119903 =1199030

1 + 1198601199030cos (120594120579)

(30)






119903(infin)

app (120579) =119903min

1 + suminfin

119894=0119862119894cos (119894120587 (120579Δ120579))

(31)


1198620=

1

2Δ120579int

Δ120579

0

[119903min119903 (120579)

minus 1] 119889120579

119862119894=

1

Δ120579int

Δ120579

0

[119903min119903 (120579)

minus 1] cos(119894120587 120579

Δ120579)119889120579

(32)


119903(1)

app (120579) =119903min

1 + 1198610+ 1198611cos (120587 (120579Δ120579))

(33)

where 1198610and 119861





0and 119861

1are


1198610= minus


1198611=119903max minus 119903min2119903max

(34)


1198861198692=119903max + 119903min

2 119890


1199011198692= 1198861198692(1 minus 119890

2

1198692)

(35)


119903(1)

app (120579) =1199011198692

1 + 1198901198692cos (120587 (120579Δ120579))

(36)


119905(1)

app (120579) asymp int120579

0

[119903(1)

app (120579)]2

ℎ119889120579 =

Δ1205791199012

1198692

120587ℎ(1 minus 1198902

1198692)32

times


1 + 1198901198692

tan( 120587120579

2Δ120579)]

minus

1198901198692radic1 minus 119890

2

1198692sin (120587120579Δ120579)

1 + 1198901198692cos (120587120579Δ120579)

(37)


120576(1)

Δ=

10038161003816100381610038161003816119903(1)

app (120579) minus 11990310038161003816100381610038161003816

119903 (38)



1198692

119890(2)

11986921198610

1198611

1198612

min 119869






0 1 2 3 4 50

1

2

3

4

5

6

7

8times10

minus6

120576(1)

Δ

t


for 1198901198692= 4times 10

minus6≪ 1 and

isin [0 5]


120576(1)

Δ119905=

10038161003816100381610038161003816119905(1)

app (120579) minus 11990510038161003816100381610038161003816

119905 (39)


minus3 and1198901198692= 4times10

minus6≪ 1 120576(1)


of 120576(1)Δ


1198692









119903(119898)

app (120579) =119903min

1 + sum119898

119894=0119861119894cos (119894120587 (120579Δ120579))

(40)


119898

sum

119894=0

119861119894= 0

119898

sum

119894=0

119861119894(minus1)119894=119903min119903max

minus 1 (41)


min 119869 (1198610 1198611 119861119898) =

119899

sum

119895=1

119898

sum

119894=0

[119903(119898)

app (119861119894 120579119895) minus 119903 (120579119895)]2

(42)




119890 1198901198692lt 0254 must be




1198692= 025 In Figure 8 the


6 Conclusions





0 05 1 15 2 25 30

05

1

15

2

25

3

120579

times10minus5

eJ2 = 0250

eJ2 = 0212

eJ2 = 0177

eJ2 = 0143

120576(1)

Δt


for 1198901198692gt 01 and 120579 isin [0 3]

0 1 2 3 4 50

001

002

003

004

005

006

007

120576(1)

Δ

eJ2 = 0111

eJ2 = 0081

eJ2 = 0053

eJ2 = 0026

t



0 1 2 3 4 50

05

1

15

2

25

0111

0143

0177

0212

0250

120576(2)

Δ

times10minus5

eJ2

t


for 1198901198692gt 01 and isin [0 5]

05

1

15

2

30

210

60

240

90

270

120

300

150

330

180 0






0 5 10 15 20 25 30 35 40 45 501

1112131415161718

465

17

465

175

465

18

465

185

465

19

1353713537135371353813538135381353813538135391353913539

t

rr(120579)

r(2)app (120579)


Acknowledgment


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1199091= minus

120583

3120576+3radicminus

120583ℎ2

121205762minus

1205833

271205763minus11986921205831198772

119890

4120576+ radicΔ

+3radicminus

120583ℎ2

121205762minus

1205833

271205763minus11986921205831198772

119890


1199092= minus

120583

3120576+minus1 + radic3119894

2

3radicminus

120583ℎ2

121205762minus

1205833

271205763minus11986921205831198772

119890

4120576+ radicΔ


2

3radicminus

120583ℎ2

121205762minus

1205833

271205763minus11986921205831198772

119890


1199093= minus

120583


2

3radicminus

120583ℎ2

121205762minus

1205833

271205763minus11986921205831198772

119890

4120576+ radicΔ


2

3radicminus

120583ℎ2

121205762minus

1205833

271205763minus11986921205831198772

119890


(11)


1199032minusℎ2

2120583119903 +

11986921198772

119890

2= 0 (12)


Δ = (ℎ2

2120583)

2

minus 211986921198772

119890gt 0 (13)

The two roots are

11990912=


441205832minus 211986921198772

119890

2

(14)


2= 108263times




3

119890120583 ℎ = ℎradic119877e120583







05 06 07 08 09 1 11 12 13 14 15

0

05

1

15

2

25

3

09222

1120513189

15172

minus1

minus05

r0

h



120576 lt 0

r

r

120576 = 0

120576 = 0

120576 gt 0

120576 gt 0

120576 lt 0

Veff

r





Equatorial Orbits






120576119903=1

21199032 120576

120579=1

2(119903 120579)2

(15)



119879119903= radic

2

minus120576[1205821119865(

120587

2 119896) + 120582

2119864(

120587

2 119896) + 120582

3Π(

120587

2 1205722 119896)]

Δ120579 =2ℎ

1205822

radic2

minus120576119865(

120587

2 119896)

119879120579=2120587

Δ120579119879119903= (120587120582

2[1205821119865(

120587

2 119896) + 120582

2119864(

120587

2 119896)

+1205823Π(

120587

2 1205722 119896)])

times (ℎ119865(120587

2 119896))

minus1

119903 (120579) = (119903min119903max)


times [119865(120587

2 119896) minus


2ℎ120579 119896])

minus1

(16)







(17)




119903max119903min

(18)




119905 = int

119903

119903min

119889119905

119889119903119889119903 = int

119903

119903min

1

radic2 (120576 minus 119881eff)119889119903 =

1

radicminus2120576


+ 1205821119865 (120579 119896)

+1205822119864 (120579 119896) + 120582

3Π(120579 120572

2 119896) ]

(19)


119905 =

119899119879119903+ int

119903

119903min

119889119905

119889119903119889119903 119905 isin [119899119879

119903 (119899 +

1

2)119879]

(119899 + 1) 119879119903minus int

119903max

119903

119889119905

119889119903119889119903 119905 isin [(119899 +

1

2)119879119903 (119899 + 1) 119879]

120579 =

119899Δ120579 + int

119903

119903min

119889120579

119889119903119889119903 119905isin[119899119879

119903 (119899 +

1

2)119879]

(119899 + 1) Δ120579 minus int

119903max

119903

119889120579

119889119903119889119903 119905 isin [(119899 +

1

2)119879119903(119899 + 1) 119879]

(20)

where 119899 isin N




(1199030)1=

ℎ2+ radicℎ4minus 6119869212058321198772

119890

2120583

(1199030)2=


119890

2120583

(21)



120596 =ℎ

1199032

0

119879 =21205871199032

0

ℎ (22)


2 yields

1205972119881eff1205971199032

100381610038161003816100381610038161003816100381610038161003816119903=1199030

= minus2120583

1199033

0

minus611986921205831198772

119890

1199035

0

+3ℎ2

1199034

0

gt 0 (23)




1198892119906

1198891205792+ 119906 = 119869 (119906) (24)


where

119869 (119906) = minus1

ℎ2

119889

119889119906119881(

1

119906) =

1

ℎ2(3

211986921205831198772

1198901199062minus ℎ2119906 + 120583)

(25)






119869 (119906) = 1199060+ (119906 minus 119906

0)119889119869

1198891199060

+ 119874 [(119906 minus 1199060)2

] (26)


orbit reduces to

1198892Δ119906

1198891205792+ Δ119906 = Δ119906

119889119869

1198891199060

(27)


1205942= 1 minus

119889119869

1198891199060

= 2 minus311986921205831198772

1198901199060

ℎ2

(28)


Δ119906 = 119906 minus 1199060= 119860 cos (120594120579) (29)


119903 =1199030

1 + 1198601199030cos (120594120579)

(30)






119903(infin)

app (120579) =119903min

1 + suminfin

119894=0119862119894cos (119894120587 (120579Δ120579))

(31)


1198620=

1

2Δ120579int

Δ120579

0

[119903min119903 (120579)

minus 1] 119889120579

119862119894=

1

Δ120579int

Δ120579

0

[119903min119903 (120579)

minus 1] cos(119894120587 120579

Δ120579)119889120579

(32)


119903(1)

app (120579) =119903min

1 + 1198610+ 1198611cos (120587 (120579Δ120579))

(33)

where 1198610and 119861





0and 119861

1are


1198610= minus


1198611=119903max minus 119903min2119903max

(34)


1198861198692=119903max + 119903min

2 119890


1199011198692= 1198861198692(1 minus 119890

2

1198692)

(35)


119903(1)

app (120579) =1199011198692

1 + 1198901198692cos (120587 (120579Δ120579))

(36)


119905(1)

app (120579) asymp int120579

0

[119903(1)

app (120579)]2

ℎ119889120579 =

Δ1205791199012

1198692

120587ℎ(1 minus 1198902

1198692)32

times


1 + 1198901198692

tan( 120587120579

2Δ120579)]

minus

1198901198692radic1 minus 119890

2

1198692sin (120587120579Δ120579)

1 + 1198901198692cos (120587120579Δ120579)

(37)


120576(1)

Δ=

10038161003816100381610038161003816119903(1)

app (120579) minus 11990310038161003816100381610038161003816

119903 (38)



1198692

119890(2)

11986921198610

1198611

1198612

min 119869






0 1 2 3 4 50

1

2

3

4

5

6

7

8times10

minus6

120576(1)

Δ

t


for 1198901198692= 4times 10

minus6≪ 1 and

isin [0 5]


120576(1)

Δ119905=

10038161003816100381610038161003816119905(1)

app (120579) minus 11990510038161003816100381610038161003816

119905 (39)


minus3 and1198901198692= 4times10

minus6≪ 1 120576(1)


of 120576(1)Δ


1198692









119903(119898)

app (120579) =119903min

1 + sum119898

119894=0119861119894cos (119894120587 (120579Δ120579))

(40)


119898

sum

119894=0

119861119894= 0

119898

sum

119894=0

119861119894(minus1)119894=119903min119903max

minus 1 (41)


min 119869 (1198610 1198611 119861119898) =

119899

sum

119895=1

119898

sum

119894=0

[119903(119898)

app (119861119894 120579119895) minus 119903 (120579119895)]2

(42)




119890 1198901198692lt 0254 must be




1198692= 025 In Figure 8 the


6 Conclusions





0 05 1 15 2 25 30

05

1

15

2

25

3

120579

times10minus5

eJ2 = 0250

eJ2 = 0212

eJ2 = 0177

eJ2 = 0143

120576(1)

Δt


for 1198901198692gt 01 and 120579 isin [0 3]

0 1 2 3 4 50

001

002

003

004

005

006

007

120576(1)

Δ

eJ2 = 0111

eJ2 = 0081

eJ2 = 0053

eJ2 = 0026

t



0 1 2 3 4 50

05

1

15

2

25

0111

0143

0177

0212

0250

120576(2)

Δ

times10minus5

eJ2

t


for 1198901198692gt 01 and isin [0 5]

05

1

15

2

30

210

60

240

90

270

120

300

150

330

180 0






0 5 10 15 20 25 30 35 40 45 501

1112131415161718

465

17

465

175

465

18

465

185

465

19

1353713537135371353813538135381353813538135391353913539

t

rr(120579)

r(2)app (120579)


Acknowledgment


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











120576119903=1

21199032 120576

120579=1

2(119903 120579)2

(15)



119879119903= radic

2

minus120576[1205821119865(

120587

2 119896) + 120582

2119864(

120587

2 119896) + 120582

3Π(

120587

2 1205722 119896)]

Δ120579 =2ℎ

1205822

radic2

minus120576119865(

120587

2 119896)

119879120579=2120587

Δ120579119879119903= (120587120582

2[1205821119865(

120587

2 119896) + 120582

2119864(

120587

2 119896)

+1205823Π(

120587

2 1205722 119896)])

times (ℎ119865(120587

2 119896))

minus1

119903 (120579) = (119903min119903max)


times [119865(120587

2 119896) minus


2ℎ120579 119896])

minus1

(16)







(17)




119903max119903min

(18)




119905 = int

119903

119903min

119889119905

119889119903119889119903 = int

119903

119903min

1

radic2 (120576 minus 119881eff)119889119903 =

1

radicminus2120576


+ 1205821119865 (120579 119896)

+1205822119864 (120579 119896) + 120582

3Π(120579 120572

2 119896) ]

(19)


119905 =

119899119879119903+ int

119903

119903min

119889119905

119889119903119889119903 119905 isin [119899119879

119903 (119899 +

1

2)119879]

(119899 + 1) 119879119903minus int

119903max

119903

119889119905

119889119903119889119903 119905 isin [(119899 +

1

2)119879119903 (119899 + 1) 119879]

120579 =

119899Δ120579 + int

119903

119903min

119889120579

119889119903119889119903 119905isin[119899119879

119903 (119899 +

1

2)119879]

(119899 + 1) Δ120579 minus int

119903max

119903

119889120579

119889119903119889119903 119905 isin [(119899 +

1

2)119879119903(119899 + 1) 119879]

(20)

where 119899 isin N




(1199030)1=

ℎ2+ radicℎ4minus 6119869212058321198772

119890

2120583

(1199030)2=


119890

2120583

(21)



120596 =ℎ

1199032

0

119879 =21205871199032

0

ℎ (22)


2 yields

1205972119881eff1205971199032

100381610038161003816100381610038161003816100381610038161003816119903=1199030

= minus2120583

1199033

0

minus611986921205831198772

119890

1199035

0

+3ℎ2

1199034

0

gt 0 (23)




1198892119906

1198891205792+ 119906 = 119869 (119906) (24)


where

119869 (119906) = minus1

ℎ2

119889

119889119906119881(

1

119906) =

1

ℎ2(3

211986921205831198772

1198901199062minus ℎ2119906 + 120583)

(25)






119869 (119906) = 1199060+ (119906 minus 119906

0)119889119869

1198891199060

+ 119874 [(119906 minus 1199060)2

] (26)


orbit reduces to

1198892Δ119906

1198891205792+ Δ119906 = Δ119906

119889119869

1198891199060

(27)


1205942= 1 minus

119889119869

1198891199060

= 2 minus311986921205831198772

1198901199060

ℎ2

(28)


Δ119906 = 119906 minus 1199060= 119860 cos (120594120579) (29)


119903 =1199030

1 + 1198601199030cos (120594120579)

(30)






119903(infin)

app (120579) =119903min

1 + suminfin

119894=0119862119894cos (119894120587 (120579Δ120579))

(31)


1198620=

1

2Δ120579int

Δ120579

0

[119903min119903 (120579)

minus 1] 119889120579

119862119894=

1

Δ120579int

Δ120579

0

[119903min119903 (120579)

minus 1] cos(119894120587 120579

Δ120579)119889120579

(32)


119903(1)

app (120579) =119903min

1 + 1198610+ 1198611cos (120587 (120579Δ120579))

(33)

where 1198610and 119861





0and 119861

1are


1198610= minus


1198611=119903max minus 119903min2119903max

(34)


1198861198692=119903max + 119903min

2 119890


1199011198692= 1198861198692(1 minus 119890

2

1198692)

(35)


119903(1)

app (120579) =1199011198692

1 + 1198901198692cos (120587 (120579Δ120579))

(36)


119905(1)

app (120579) asymp int120579

0

[119903(1)

app (120579)]2

ℎ119889120579 =

Δ1205791199012

1198692

120587ℎ(1 minus 1198902

1198692)32

times


1 + 1198901198692

tan( 120587120579

2Δ120579)]

minus

1198901198692radic1 minus 119890

2

1198692sin (120587120579Δ120579)

1 + 1198901198692cos (120587120579Δ120579)

(37)


120576(1)

Δ=

10038161003816100381610038161003816119903(1)

app (120579) minus 11990310038161003816100381610038161003816

119903 (38)



1198692

119890(2)

11986921198610

1198611

1198612

min 119869






0 1 2 3 4 50

1

2

3

4

5

6

7

8times10

minus6

120576(1)

Δ

t


for 1198901198692= 4times 10

minus6≪ 1 and

isin [0 5]


120576(1)

Δ119905=

10038161003816100381610038161003816119905(1)

app (120579) minus 11990510038161003816100381610038161003816

119905 (39)


minus3 and1198901198692= 4times10

minus6≪ 1 120576(1)


of 120576(1)Δ


1198692









119903(119898)

app (120579) =119903min

1 + sum119898

119894=0119861119894cos (119894120587 (120579Δ120579))

(40)


119898

sum

119894=0

119861119894= 0

119898

sum

119894=0

119861119894(minus1)119894=119903min119903max

minus 1 (41)


min 119869 (1198610 1198611 119861119898) =

119899

sum

119895=1

119898

sum

119894=0

[119903(119898)

app (119861119894 120579119895) minus 119903 (120579119895)]2

(42)




119890 1198901198692lt 0254 must be




1198692= 025 In Figure 8 the


6 Conclusions





0 05 1 15 2 25 30

05

1

15

2

25

3

120579

times10minus5

eJ2 = 0250

eJ2 = 0212

eJ2 = 0177

eJ2 = 0143

120576(1)

Δt


for 1198901198692gt 01 and 120579 isin [0 3]

0 1 2 3 4 50

001

002

003

004

005

006

007

120576(1)

Δ

eJ2 = 0111

eJ2 = 0081

eJ2 = 0053

eJ2 = 0026

t



0 1 2 3 4 50

05

1

15

2

25

0111

0143

0177

0212

0250

120576(2)

Δ

times10minus5

eJ2

t


for 1198901198692gt 01 and isin [0 5]

05

1

15

2

30

210

60

240

90

270

120

300

150

330

180 0






0 5 10 15 20 25 30 35 40 45 501

1112131415161718

465

17

465

175

465

18

465

185

465

19

1353713537135371353813538135381353813538135391353913539

t

rr(120579)

r(2)app (120579)


Acknowledgment


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










where

119869 (119906) = minus1

ℎ2

119889

119889119906119881(

1

119906) =

1

ℎ2(3

211986921205831198772

1198901199062minus ℎ2119906 + 120583)

(25)






119869 (119906) = 1199060+ (119906 minus 119906

0)119889119869

1198891199060

+ 119874 [(119906 minus 1199060)2

] (26)


orbit reduces to

1198892Δ119906

1198891205792+ Δ119906 = Δ119906

119889119869

1198891199060

(27)


1205942= 1 minus

119889119869

1198891199060

= 2 minus311986921205831198772

1198901199060

ℎ2

(28)


Δ119906 = 119906 minus 1199060= 119860 cos (120594120579) (29)


119903 =1199030

1 + 1198601199030cos (120594120579)

(30)






119903(infin)

app (120579) =119903min

1 + suminfin

119894=0119862119894cos (119894120587 (120579Δ120579))

(31)


1198620=

1

2Δ120579int

Δ120579

0

[119903min119903 (120579)

minus 1] 119889120579

119862119894=

1

Δ120579int

Δ120579

0

[119903min119903 (120579)

minus 1] cos(119894120587 120579

Δ120579)119889120579

(32)


119903(1)

app (120579) =119903min

1 + 1198610+ 1198611cos (120587 (120579Δ120579))

(33)

where 1198610and 119861





0and 119861

1are


1198610= minus


1198611=119903max minus 119903min2119903max

(34)


1198861198692=119903max + 119903min

2 119890


1199011198692= 1198861198692(1 minus 119890

2

1198692)

(35)


119903(1)

app (120579) =1199011198692

1 + 1198901198692cos (120587 (120579Δ120579))

(36)


119905(1)

app (120579) asymp int120579

0

[119903(1)

app (120579)]2

ℎ119889120579 =

Δ1205791199012

1198692

120587ℎ(1 minus 1198902

1198692)32

times


1 + 1198901198692

tan( 120587120579

2Δ120579)]

minus

1198901198692radic1 minus 119890

2

1198692sin (120587120579Δ120579)

1 + 1198901198692cos (120587120579Δ120579)

(37)


120576(1)

Δ=

10038161003816100381610038161003816119903(1)

app (120579) minus 11990310038161003816100381610038161003816

119903 (38)



1198692

119890(2)

11986921198610

1198611

1198612

min 119869






0 1 2 3 4 50

1

2

3

4

5

6

7

8times10

minus6

120576(1)

Δ

t


for 1198901198692= 4times 10

minus6≪ 1 and

isin [0 5]


120576(1)

Δ119905=

10038161003816100381610038161003816119905(1)

app (120579) minus 11990510038161003816100381610038161003816

119905 (39)


minus3 and1198901198692= 4times10

minus6≪ 1 120576(1)


of 120576(1)Δ


1198692









119903(119898)

app (120579) =119903min

1 + sum119898

119894=0119861119894cos (119894120587 (120579Δ120579))

(40)


119898

sum

119894=0

119861119894= 0

119898

sum

119894=0

119861119894(minus1)119894=119903min119903max

minus 1 (41)


min 119869 (1198610 1198611 119861119898) =

119899

sum

119895=1

119898

sum

119894=0

[119903(119898)

app (119861119894 120579119895) minus 119903 (120579119895)]2

(42)




119890 1198901198692lt 0254 must be




1198692= 025 In Figure 8 the


6 Conclusions





0 05 1 15 2 25 30

05

1

15

2

25

3

120579

times10minus5

eJ2 = 0250

eJ2 = 0212

eJ2 = 0177

eJ2 = 0143

120576(1)

Δt


for 1198901198692gt 01 and 120579 isin [0 3]

0 1 2 3 4 50

001

002

003

004

005

006

007

120576(1)

Δ

eJ2 = 0111

eJ2 = 0081

eJ2 = 0053

eJ2 = 0026

t



0 1 2 3 4 50

05

1

15

2

25

0111

0143

0177

0212

0250

120576(2)

Δ

times10minus5

eJ2

t


for 1198901198692gt 01 and isin [0 5]

05

1

15

2

30

210

60

240

90

270

120

300

150

330

180 0






0 5 10 15 20 25 30 35 40 45 501

1112131415161718

465

17

465

175

465

18

465

185

465

19

1353713537135371353813538135381353813538135391353913539

t

rr(120579)

r(2)app (120579)


Acknowledgment


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1198692

119890(2)

11986921198610

1198611

1198612

min 119869






0 1 2 3 4 50

1

2

3

4

5

6

7

8times10

minus6

120576(1)

Δ

t


for 1198901198692= 4times 10

minus6≪ 1 and

isin [0 5]


120576(1)

Δ119905=

10038161003816100381610038161003816119905(1)

app (120579) minus 11990510038161003816100381610038161003816

119905 (39)


minus3 and1198901198692= 4times10

minus6≪ 1 120576(1)


of 120576(1)Δ


1198692









119903(119898)

app (120579) =119903min

1 + sum119898

119894=0119861119894cos (119894120587 (120579Δ120579))

(40)


119898

sum

119894=0

119861119894= 0

119898

sum

119894=0

119861119894(minus1)119894=119903min119903max

minus 1 (41)


min 119869 (1198610 1198611 119861119898) =

119899

sum

119895=1

119898

sum

119894=0

[119903(119898)

app (119861119894 120579119895) minus 119903 (120579119895)]2

(42)




119890 1198901198692lt 0254 must be




1198692= 025 In Figure 8 the


6 Conclusions





0 05 1 15 2 25 30

05

1

15

2

25

3

120579

times10minus5

eJ2 = 0250

eJ2 = 0212

eJ2 = 0177

eJ2 = 0143

120576(1)

Δt


for 1198901198692gt 01 and 120579 isin [0 3]

0 1 2 3 4 50

001

002

003

004

005

006

007

120576(1)

Δ

eJ2 = 0111

eJ2 = 0081

eJ2 = 0053

eJ2 = 0026

t



0 1 2 3 4 50

05

1

15

2

25

0111

0143

0177

0212

0250

120576(2)

Δ

times10minus5

eJ2

t


for 1198901198692gt 01 and isin [0 5]

05

1

15

2

30

210

60

240

90

270

120

300

150

330

180 0






0 5 10 15 20 25 30 35 40 45 501

1112131415161718

465

17

465

175

465

18

465

185

465

19

1353713537135371353813538135381353813538135391353913539

t

rr(120579)

r(2)app (120579)


Acknowledgment


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 05 1 15 2 25 30

05

1

15

2

25

3

120579

times10minus5

eJ2 = 0250

eJ2 = 0212

eJ2 = 0177

eJ2 = 0143

120576(1)

Δt


for 1198901198692gt 01 and 120579 isin [0 3]

0 1 2 3 4 50

001

002

003

004

005

006

007

120576(1)

Δ

eJ2 = 0111

eJ2 = 0081

eJ2 = 0053

eJ2 = 0026

t



0 1 2 3 4 50

05

1

15

2

25

0111

0143

0177

0212

0250

120576(2)

Δ

times10minus5

eJ2

t


for 1198901198692gt 01 and isin [0 5]

05

1

15

2

30

210

60

240

90

270

120

300

150

330

180 0






0 5 10 15 20 25 30 35 40 45 501

1112131415161718

465

17

465

175

465

18

465

185

465

19

1353713537135371353813538135381353813538135391353913539

t

rr(120579)

r(2)app (120579)


Acknowledgment


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 5 10 15 20 25 30 35 40 45 501

1112131415161718

465

17

465

175

465

18

465

185

465

19

1353713537135371353813538135381353813538135391353913539

t

rr(120579)

r(2)app (120579)


Acknowledgment


References
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article Fourier Series Approximations to -Bounded ...downloads.hindawi.com/journals/mpe/2014/568318.pdf · Such pseudo-elliptic bounded orbit displays a rosette shape in