The angle w-n$ will attain a maximum o r minimum only if its derivative vanishes... In the derivation of eqs..[r]

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ACKNOWLEDGMENTS The author would like to thank Dr.[r]

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Gelf’gat [5] examined the **restricted** **three**-**body** prob- lem of **variable** masses in which the primary bodies move within the framework of the GMP and established the existence of five libration points (collinear & triangular) analogous to the classical libration points. Bekov [6] found **two** additional equilibrium points, called the co- **planar** points. A few recent characterizations of the GMP were examined by Gurfil and Belyanin [7] and Singh and Leke [8]. The majority of the authors have been inter- ested in the stellar applications of this **problem** than the solar system.

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master and slave system and they demonstrated that chaotic synchronization could be achieved by driving or replacing one of the variables of a chaotic system with a **variable** of another similar chaotic device. Many methods for chaos synchronization of various chaotic systems have been developed, such as non linear feedback control [2], OGY approach [3], sliding mode control [4], anti synchronization method [5], adaptive synchronization [6], active control [7] and so on. The active control methods for synchronizing the chaotic systems has been applied to many practical systems such as spatiotemporal dynamical systems (Codreanu [8]), the Rikitake **two**-disc dynamo-a geographical systems (Vincent 9]), Complex dynamos (Mahmoud [10]) and Hyper-chaotic and time delay systems (Israr Ahmad et al. [11]) etc. Shihua Chen and Jinhu [12] proposed a new adaptive control method for adaptive synchronization of **two** uncertain chaotic systems, using a speci_c uncertain uni_ed chaotic model.

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Many mathematicians have made the huge contributions to the analytical, qualitative and numerical studies of the **restricted** **three**-**body** **problem** when the primaries are moving in the circular motion around there centre of mass in the uniform motion. A detailed analysis of this **problem** is illustrated in the work of American mathematician Szebehely [15]. In (1975) Sharma, R. K. and Subbarao[16] have discussed the collinear equilibria and their characteristic exponents in the **restricted** **three** **body** **problem** when the primaries are oblate spheroids. The Lagrangian triangular equilibria in the **planar** **restricted** **three** **body** **problem** where the primaries are oblate homogeneous spheroids discussed by Arredondo, J.A. et al [17]. Khan and Shahzad [18] investigated the synchronization behavior of the **two** identical circular **restricted** **three** **body** **problem** influenced by radiation evolving from different initial conditions via the active control. In an another paper the Complete synchronization, anti-synchronization and hybrid synchronization of **two** identical parabolic **restricted** **three** **body** **problem** have been studied by Khan and Rimpi pal [19]. Arif [20] studied the complete synchronization, anti-synchronization and hybrid synchronization in the **planar** **restricted** **three** **problem** by taking into consideration the small primary is ellipsoid and bigger primary an oblate spheroid via active control technique.

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In the present work, the collinear equilibrium points of the **restricted** **three**-**body** **problem** are studied under the effect of oblateness of the bigger primary using an analytical and numerical approach. The periodic orbits around these points are investigated for the Earth-Moon system. The Lissaj- ous orbits and the phase spaces are obtained under the effect of oblateness.

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The PN approximation may be described as a method for obtaining the motions of the system to one higher power of the small parameters GM r and v 2 than given by Newtonian mechanics. It is sometimes referred to as an **expansion** in inverse powers of the speed of light. We prefer to say that our **expansion** parameter is c 2 , note that geometric units will not be used, so that G 1 , and c 1 . We now proceed to find the equations of mo- tion of the relativistic **three** **body** **problem** in the PN for- malism, or more precisely the equation of the RTBP.

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The linear stability of the triangular points was studied for the Robes **restricted** **three**-**body** **problem** when the bigger primary (rigid shell) is oblate spheroid and the second primary is radiating. The critical mass obtained depends on the oblateness of the rigid shell and radiation of the second primary as well as the density parameter . The stability of the triangular points depends largely on the values of . The destabilizing tendencies of the oblateness and radiation fac- tors were enhanced when and weakened for

In this paper we study the elliptical **restricted** **three** **body** **problem** which has not been studied much in comparison to the circular RTBP further we have both major bodies radiating and oblate. We obtain the equations of motion of the infinitesimal **body** moving under the influence of **two** major bodies which are both oblate and radiating and the major bodies move in elliptical orbits around their common centre of mass, the equations (1) & (2) represent the equations of motion of the infinitesimal **body**. The locations of triangular librational points L 4 & L 5 are obtained and equations (9)

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The linear solutions given in the previous section will only closely approx- imate the motion of the sail given in (1) for small amplitudes. For larger amplitude periodic orbits, we compute high order approximations using the method of Linstedt-Poincar´e. 5 This **procedure** is well known and is

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The circular **restricted** **three**-**body** **problem** (CRTBP) is a special case of the **restricted** **three**-**body** **problem** where bigger and smaller primaries move in cir- cular motion around their common center of mass. The five equilibrium points of CRTBP are known as the Lagrangian points where the gravitational forces due to **two** primaries and the centrifugal force on a spacecraft are balanced.

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In this paper, new families of displaced non-Keplerian orbits are generated using impulsive, rather than continuous thrust. The use of impulse thrust was considered by Nock (1984), Yashko and Hastings (1996), McInnes (1998), Hope and Trask (2003) and Spilker (2003) to generate small displacements away from a circular Keplerian orbit at linear order in the **two**-**body** **problem**. In this paper, the more general **problem** is considered with arbitrarily large displacements. The displaced non-Keplerian orbits are approximated by a sequence of individual Keplerian arcs connected by impulses. The sequence of arcs is therefore continuous in position, with discontinuities in velocity which are removed using the impulses. For frequent impulses the families of orbits approximate the continuous thrust non-Keplerian orbits found in previous studies. The conditions for displaced non-Keplerian orbits using impulse thrust can be found by linearising the dynamics of the **two** and **three**-**body** problems in the vicinity of an arbitrary reference point in a rotating frame of reference. While the motion of the spacecraft relative to the reference point is considered at linear order, the reference point can be selected arbitrarily so that, when viewed from an inertial frame, the resulting orbit has arbitrarily large displacements away from a Keplerian orbit. Finally, some applications are considered for displaced geostationary orbits.

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Therefore, there exist one and only one value of x in each of the above interval such that f x ( ) 0 . Further, f 2 0 , f (0) 0 and f 1 0 . Therefore, there are only **three** real roots of equation (12), one lying in each of the intervals

In the present paper we have studied the location of libration points triangular as well as libration points in the generalized photogravitational elliptic **restricted** **three** **body** **problem**. We have taken the bigger primary as an oblate spheroid and the smaller primary libration points contain eccentricity, oblateness and radiation factor which are different from those in classical case.

last forever. In Figure 6(b), the left parts, the projections on ξ and η change like those in Case A; the projection on axis ζ changes as a sine curve and the amplitude remains about the same. The right parts are the variations of **three** angles, where α changes in a small scale about from 89.8 ˚ to 90˚, −90˚ ≤ γ , φ ≤ 90˚. Figure 8(a) displays a direct perturbation trajectory of the sail derived from L 1 point. The maximum acceleration of solar radiation

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This paper has investigated equilibria in the solar sail elliptical **restricted** **three**-**body** **problem**, and the following results obtained: There are no equilibrium surfaces in the **three**-dimensional, elliptical **restricted** **three**-**body** **problem**, but **two**-dimensional **planar** equilibrium curves do exist in the plane of the system, with the curve shapes in the non-uniformly rotating, pulsating system the same as the circular case; These equilibrium points are unstable; Although out-of-plane equilibrium points do not exist, when the sail can be controlled around approximated equilibrium points. Therefore, solar sails can still be utilized for practical applications at out-of-plane equilibria of the Sun-Earth elliptical **restricted** **three**-**body** **problem**.

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Being motivated by the above discussion, in this article we have discussed the complete synchronization behavior of the **planar** **restricted** **three** **body** **problem** when the bigger primary is an uniform circular disc evolving from deferent initial conditions using active control technique based on the Lyapunov-stability theory and Routh-Hurwitz criteria. It has been observed that the system is chaotic for some values of parameter. Hence the slave chaotic system completely traces the dynamics of the master system in the course of time. The paper is organized as follows.

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Therefore, the interest is to find a model that can maintain dynamical as- tronomy within the context of classical mechanics, as well as proffering justifica- tions for the observed phenomena as offered by the relativity theory. Such a model meets the theoretical needs of celestial mechanics (by preserving the sim- plicity and advantages of Newtonian mechanics), and can also describe accu- rately the orbits coming close to collisions. By using physical principles, the Bulgarian Physicist George Manev obtained a similar model in the twenties, and proposed an alternative substitute for the relativity theory [2] [3] [4] [5] [6]. In the corresponding central force **problem** with unit mass for the satellite, Manev’s potential gives A = µ and 2 2

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