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1.
《New Astronomy》2020
The objective of this paper is to find periodic solutions of the circular Sitnikov problem by the multiple scales method which is used to remove the secular terms and find the periodic approximated solutions in closed forms. Comparisons among a numerical solution (NS), the first approximated solution (FA) and the second approximated solution (SA) via multiple scales method are investigated graphically under different initial conditions. We observe that the initial conditions play a vital role in the numerical and approximated solutions behaviour. The obtained motion is periodic, but the difference of its amplitude is directly proportional with the initial conditions. We prove that the obtained motion by the numerical or the second approximated solutions is a regular and periodic, when the infinitesimal body starts its motion from a nearer position to the common center of primaries. Otherwise when the start point distance of motion is far from this center, the numerical solution may not be represent a periodic motion for along time, while the second approximated solution may present a chaotic motion, however it is always periodic all time. But the obtained motion by the first approximated solution is periodic and has regularity in its periodicity all time. Finally we remark that the provided solutions by multiple scales methods reflect the true motion of the Sitnikov restricted three–body problem, and the second approximation has more accuracy than the first approximation. Moreover the solutions of multiple scales technique are more realistic than the numerical solution because there is always a warranty that the motion is periodic all time. 相似文献
2.
This paper presents the procedure of a computational scheme leading to approximate general solution of the axi-symmetric,2-degrees
of freedom dynamical systems. Also the results of application of this scheme in two such systems of the non-linear double
oscillator with third and fifth order potentials in position variables. Their approximate general solution is constructed
by computing a dense set of families of periodic solutions and their presentation is made through plots of initial conditions.
The accuracy of the approximate general solution is defined by two error parameters, one giving a measure of the accuracy
of the integration and calculation of periodic solutions procedure, and the second the density in the initial conditions space
of the periodic solutions calculated. Due to the need to compute families of periodic solutions of large periods the numerical
integrations were carried out using the eighth order, variable step, R-K algorithm, which secured for almost all results presented
here conservation of the energy constant between 10-9 and 10-12 for single runs of any and all solutions. The accuracy of the approximate general solution is controlled by increasing the
number of family curves and also by `zooming' into parts of the space of initial conditions. All families of periodic solutions
were checked for their stability. The computation of such families within areas of `deterministic chaos' did not encounter
any difficulty other than poorer precision. Furthermore, on the basis of the stability study of the computed families, the
boundaries of areas of `order' and `chaos' were approximately defined. On the basis of these results it is concluded that
investigations in thePoincaré sections have to disclose 3 distinct types of areas of `order' and 2 distinct types of areas
of `chaos'. Verification of the `order'/`chaos' boundary calculation was made by working out several Poincaré surfaces of
sections.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
3.
C. N. Frangakis 《Astrophysics and Space Science》1973,23(1):17-42
The long period problem provides the initial conditions for numerical computation of close periodic solutions separated in three categories. For each type of commensurability a number of periodic solutions are computed and their stability is studied by computing the characteristic exponents of the matrizant. The Runge-Kutta method for the solution of differential equations of motion was used in all cases. The results obtained are presented for a four cases of commensurability. 相似文献
4.
考虑周期解的数值延拓问题并提出基于Broyden拟牛顿法来延拓周期解的一种有效算法,先后以布鲁塞尔振子、平面圆型限制性三体问题(Planar Circular Restricted Three-Body Problem, PCRTBP)的周期解为例进行了验证.这里的Broyden方法包含线性搜索、正交三角分解求线性方程组的步骤.对一般的周期解,周期性条件方程组中含有周期作为待延拓参数,可用周期来决定积分时长,将解代入周期性条件得到积分型的非线性方程组,利用Broyden方法迭代延拓直至初值收敛.根据两次垂直通过一个超平面的轨道是对称周期轨道的性质,可采用插值的方法求得再次抵达超平面的解分量,得到周期性条件方程组,再用Broyden方法求解.结合哈密顿系统的对称性和PCRTBP周期轨道的一些分类,对2/1、3/1的内共振周期解族进行了数值研究.最后,对算法和计算结果做了总结和讨论. 相似文献
5.
Using the 2D numerical simulation we have studied the nonlinear evolution of kinetic Alfvén wave (KAW) in intermediate β plasmas (β?m e /m i ?1). The coupled equations of kinetic Alfvén wave (KAW) and ion acoustic wave (IAW) have been studied with different initial conditions using (1) periodic perturbation, (2) Gaussian perturbation and (3) random perturbation. We have studied the effect of initial conditions on the filament formation and on the turbulent scaling laws. The scale size of the localized structures is also obtained under different conditions. 相似文献
6.
Gang Zhang Di Zhou Zhaowei Sun Xibin Cao 《Celestial Mechanics and Dynamical Astronomy》2013,117(2):137-148
The problem of two-body linearized periodic relative orbits with eccentric reference orbits is studied in this paper. The periodic relative orbit in the target-orbital coordinate system can be used in fly-around and formation-flying orbit design. Based on the closed-form solutions to the Tschauner–Hempel equations, the initial condition for periodic relative orbits is obtained. Then the minimum-fuel periodic-orbit condition with a single impulse is analytically derived for given initial position and velocity vectors. When considering the initial coasting time, the impulse position of the global minimum-fuel periodic orbit is proved to be near to the perigee of the target and can be obtained by numerical optimization algorithms. Moreover, the condition for a special periodic orbit, i.e., the rectilinear relative orbit in the target-orbital frame, is obtained. Numerical simulations are used to demonstrate the efficacy of the method, and show the geometry of the periodic relative orbit and the rectilinear relative orbit. 相似文献
7.
We consider the general spatial three body problem and study the dynamics of planetary systems consisting of a star and two planets which evolve into 2/1 mean motion resonance and into inclined orbits. Our study is focused on the periodic orbits of the system given in a suitable rotating frame. The stability of periodic orbits characterize the evolution of any planetary system with initial conditions in their vicinity. Stable periodic orbits are associated with long term regular evolution, while unstable periodic orbits are surrounded by regions of chaotic motion. We compute many families of symmetric periodic orbits by applying two schemes of analytical continuation. In the first scheme, we start from the 2/1 (or 1/2) resonant periodic orbits of the restricted problem and in the second scheme, we start from vertical critical periodic orbits of the general planar problem. Most of the periodic orbits are unstable, but many stable periodic orbits have been, also, found with mutual inclination up to 50?–60?, which may be related with the existence of real planetary systems. 相似文献
8.
One- and two-dimensional sections of the region of initial conditions in the vicinity of a periodic Ducati orbit have been studied in detail in the plane equal-mass three-body problem. A continuous stability region generated by the periodic Ducati orbit has been revealed. In addition, a number of other stability regions that are probably related to stable hierarchical triple systems have been found. Several specific trajectories from the stability regions and in the boundary zones are analyzed. 相似文献
9.
The surfaces of section in a harmonic oscillator potential, perturbed by quartic terms, are obtained analytically. A succession
of action‐angle, Lissajous and Lie transformations near the 1:1 commensurability, reduces the three‐dimensional motion to
a one‐dimensional one. The latter is solved in terms of Jacobi's elliptic functions. Existence conditions for periodic orbits
are found and two general families of such solutions are introduced. Two examples of regular motions in oblate and prolate
spheroids are discussed.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
10.
Victor Vladimirovich Orlov Anna V. Petrova Kiyotaka Tanikawa Masaya M. Saito Alija I. Martynova 《Celestial Mechanics and Dynamical Astronomy》2008,100(2):93-120
The rectilinear equal-mass and unequal-mass three-body problems are considered. The first part of the paper is a review that
covers the following items: regularization of the equations of motion, integrable cases, triple collisions and their vicinities,
escapes, periodic orbits and their stability, chaos and regularity of motions. The second part contains the results of our
numerical simulations in this problem. A classification of orbits in correspondence with the following evolution scenarios
is suggested: ejections, escapes, conditional escapes (long ejections), periodic orbits, quasi-stable long-lived systems in
the vicinity of stable periodic orbits, and triple collisions. Homothetic solutions ending by triple collisions and their
dependence on initial parameters are found. We study how the ejection length changes in response to the variation of the triple
approach parameters. Regions of initial conditions are outlined in which escapes occur after a definite number of triple approaches
or a definite time. In the vicinity of a stable Schubart periodic orbit, we reveal a region of initial parameters that corresponds
to trajectories with finite motions. The regular and chaotic structure of the manifold of orbits is mostly defined by this
periodic orbit. We have studied the phase space structure via Poincaré sections. Using these sections and symbolic dynamics,
we study the fine structure of the region of initial conditions, in particular the chaotic scattering region. 相似文献
11.
Using specialized codes for the search of periodic and linear components we show that direct solar radiation leads to short-period
variations of all the orbital elements of geosynchronous satellites. The variation period of the semimajor axis a, orbit inclination i and the longitude of the ascending node Ω is 1 day. Eccentricity e, the argument of perigee ω and the mean anomaly M vary with a period of 0.5 days. Direct solar radiation also leads to long-period variations in e, ω and M with a period of 1 year. The elements a, i and Ω undergo variations only in the amplitude of diurnal variations with a period of 1 or 0.5 years. Secular variability
(linear components) are not detected. To obtain the initial value array of the orbital elements we used the Lagrange equations
of perturbed motion in the form of a Gaussian with their subsequent integration via a special method of harmonics: the values
of the derived orbital elements, obtained from the Lagrange equations, were presented through the periodic functions that
are easy to integrate. 相似文献
12.
The planetary dynamics of 4/3, 3/2, 5/2, 3/1 and 4/1 mean motion resonances is studied by using the model of the general three body problem in a rotating frame and by determining families of periodic orbits for each resonance. Both planar and spatial cases are examined. In the spatial problem, families of periodic orbits are obtained after analytical continuation of vertical critical orbits. The linear stability of orbits is also examined. Concerning initial conditions nearby stable periodic orbits, we obtain long-term planetary stability, while unstable orbits are associated with chaotic evolution that destabilizes the planetary system. Stable periodic orbits are of particular importance in planetary dynamics, since they can host real planetary systems. We found stable orbits up to 60° of mutual planetary inclination, but in most families, the stability does not exceed 20°–30°, depending on the planetary mass ratio. Most of these orbits are very eccentric. Stable inclined circular orbits or orbits of low eccentricity were found in the 4/3 and 5/2 resonance, respectively. 相似文献
13.
We study the simple periodic orbits of a particle that is subject to the gravitational action of the much bigger primary bodies
which form a regular polygonal configuration of (ν+1) bodies when ν=8. We investigate the distribution of the characteristic curves of the families and their evolution in the phase space of
the initial conditions, we describe various types of simple periodic orbits and we study their linear stability. Plots and
tables illustrate the obtained material and reveal many interesting aspects regarding particle dynamics in such a multi-body
system. 相似文献
14.
The proposed method connects two unstable periodic orbits by employing trajectories of their associated invariant manifolds that are perturbed in two levels. A first level of velocity perturbations is applied on the trajectories of the discretized manifolds at the points where they approach the nominal unstable periodic orbit in order to accelerate them. A second level of structured velocity perturbations is applied to trajectories that have already been subjected to first level perturbations in order to approximately meet the necessary conditions for a low \(\varDelta \text {V}\) transfer. Due to this two-level perturbation approach, the number of the trajectories obtained is significantly larger compared with approaches that employ traditional invariant manifolds. For this reason, the problem of connecting two unstable periodic orbits through perturbed trajectories of their manifolds is transformed into an equivalent discrete optimization problem that is solved with a very low computational complexity algorithm that is proposed in this paper. Finally, the method is applied to a lunar observation mission of practical interest and is found to perform considerably better in terms of \(\varDelta \text {V}\) cost and time of flight when compared with previous techniques applied to the same project. 相似文献
15.
The solar radiation effects upon the orbital behaviour of an arbitrarily shaped spacecraft (or a solar sail in particular) in a general fixed orientation with respect to the local coordinate frame are investigated. Through introduction of a quasi-angle in the osculating plane, the motion of the orbital plane becomes uncoupled from the in-plane perturbations. Exact solutions in the form of conic sections and logarithmic spirals can readily be formulated for certain specific initial conditions. An effective out-of-plane spiral transfer trajectory is obtained by reversing the force component normal to the orbital plane at specified positions in the orbit. By choosing the appropriate control angles for the sail orientation, any point in space can be reached eventually. In the case of general initial conditions, the long-term orbital behaviour is assessed asymptotically by means of the two-variable expansion procedure. An implicit expression for the eccentricity is derived and explicit results are established by an iteration scheme. The other orbital elements can be expressed in terms of the eccentricity and their asymptotic series for near-circular initial orbits are also obtained. While equations for the higher-order contributions as well as the periodic parts of their solutions can be formulated readily, their secular terms are determined only for a circular initial orbit. 相似文献
16.
Tilemahos J. Kalvouridis 《Celestial Mechanics and Dynamical Astronomy》2008,102(1-3):191-206
The focus of this contribution is an effort to review and report the main results obtained so far, concerning the periodic motions of a small body in the combined gravitational field created by a regular ν-gon arrangement of ν big bodies with equal masses, where ν > 7, and another central primary with different mass. Various types of planar periodic motions are presented and networks of characteristic curves of families are depicted, in order to show their distribution in the space of the initial conditions, as well as the evolution of their members that are also examined under the variation of the parameters of the system. Furthermore, the regions of the allowed three-dimensional motions, as well as their variation, are illustrated by means of the zero-velocity surfaces. All this new material is added to the already existing data, and completes thus the profile of the dynamical behavior of the system. 相似文献
17.
18.
V.S. Kalantonis E.A. Perdios A.E. Perdiou O. Ragos M.N. Vrahatis 《Astrophysics and Space Science》2003,288(4):479-488
The techniques used for the numerical computation of families of periodic orbits of dynamical systems rely on predictor-corrector
algorithms. These algorithms usually depend on the solution of systems of approximate equations constructed from the periodicity
conditions of these orbits. In this contribution we transform the root finding procedure to an optimization one which is applied
on an objective function based on the exact periodicity conditions. Thus, the determination of periodic solutions and families
of such orbits can be accomplished through unconstrained optimization. In this paper we apply and compare some well-known
minimization methods for the solution of this problem. The obtained results are promising.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
19.
G. A. Tsirogiannis E. A. Perdios V. V. Markellos 《Celestial Mechanics and Dynamical Astronomy》2009,103(1):49-78
We present an improved grid search method for the global computation of periodic orbits in model problems of Dynamics, and
the classification of these orbits into families. The method concerns symmetric periodic orbits in problems of two degrees
of freedom with a conserved quantity, and is applied here to problems of Celestial Mechanics. It consists of two main phases;
a global sampling technique in a two-dimensional space of initial conditions and a data processing procedure for the classification
(clustering) of the periodic orbits into families characterized by continuous evolution of the orbital parameters of member
orbits. The method is tested by using it to recompute known results. It is then applied with advantage to the determination
of the branch families of the family f of retrograde satellites in Hill’s Lunar problem, and to the determination of irregular families of periodic orbits in a
perturbed Hill problem, a species of families which are difficult to find by continuation methods.
相似文献
20.
A. M. Elnaggar 《Astrophysics and Space Science》1992,190(2):177-190
The object of the present paper is to investigate the influence of initial stress on the waves propagation in a generalized thermoelastic granular medium subjected to the boundary conditions that the outer surface is traction free. In addition, it is subjected to temperature boundary conditions. The wave velocity equation for the generalized thermoelastic granular medium Rayleigh wave under the influence of initial stress has been obtained. The classical result has been derived as a limiting case similar to one which was obtained by Ewinget al. (1957). 相似文献