共查询到20条相似文献，搜索用时 500 毫秒
1.
Wei-Duo Hu Daniel J. Scheeres 《天体物理学报》2008,8(1):108-118
Periodic orbits in an arbitrary 2nd degree and order uniformly rotating gravity field are studied. We investigate the four equilibrium points in this gravity field. We see that close relation exists between the stability of these equilibria and the existence and stability of their nearby periodic orbits. We check the periodic orbits with non-zero periods. In our searching procedure for these periodic orbits, we remove the two unity eigenvalues from the state transition matrix to find a robust, non-singular linear map to solve for the periodic orbits. The algorithm converges well, especially for stable periodic orbits. Using the searching procedure, which is relatively automatic, we find five basic families of periodic orbits in the rotating second degree and order gravity field for planar motion, and discuss their existence and stability at different central body rotation rates. 相似文献
2.
Periodic orbits in an arbitrary 2nd degree and order uniformly rotating gravity field are studied. We investigate the four equilibrium points in this gravity field. We see that close relation exists between the stability of these equilibria and the existence and stability of their nearby periodic orbits. We check the periodic orbits with non-zero periods. In our searching procedure for these periodic orbits, we remove the two unity eigenvalues from the state transition matrix to find a robust, non-singular linear map to solve for the periodic orbits. The algorithm converges well, especially for stable periodic orbits. Using the searching procedure, which is relatively automatic, we find five basic families of periodic orbits in the rotating second degree and order gravity field for planar motion, and discuss their existence and stability at different central body rotation rates. 相似文献
3.
B. Barbanis 《Celestial Mechanics and Dynamical Astronomy》1990,48(1):57-77
We investigate the escape regions of a quartic potential and the main types of irregular periodic orbits. Because of the symmetry of the model the zero velocity curve consists of four summetric arcs forming four open channels around the lines y = ± x through which an orbit can escape. Four unstable Lyapunov periodic orbits bridge these openings.We have found an infinite sequence of families of periodic orbits which is the outer boundary of one of the escape regions and several infinite sequences of periodic orbits inside this region that tend to homoclinic and heteroclinic orbits. Some of these sequences of periodic orbits tend to homoclinic orbits starting perpendicularly and ending asymptotically at the x-axis. The other sequences tend to heteroclinic orbits which intersect the x-axis perpendicularly for x > 0 and make infinite oscillations almost parallel to each of the two Lyapunov orbits which correspond to x > 0 or x < 0. 相似文献
4.
A systematic numerical exploration of the families of asymmetric periodic orbits of the restricted three-body problem when
a) the primary bodies are equal and b) for the Earth-Moon mass ratio, is presented. Decades families of asymmetric periodic
solutions were found and three of the simplest ones, in the first case, and ten of the second one are illustrated. All of
these families consist of periodic orbits which are asymmetric with respect to x-axis while are simple symmetric periodic orbits with respect to y-axis (i.e. the orbit has only one perpendicular intersection at half period with y-axis). Many asymmetric periodic orbits, members of these families, are calculated and plotted. We studied the stability of
all the asymmetric periodic orbits we found. These families consist, mainly, of unstable periodic solutions but there exist
very small, with respect to x, intervals where these families have stable periodic orbits. We also found, using appropriate Poincaré surface of sections,
that a relatively large region of phase space extended around all these stable asymmetric periodic orbits shows chaotic motion. 相似文献
5.
We have extend Stormer’s problem considering four magnetic dipoles in motion trying to justify the phenomena of extreme “orderlines”
such as the ones observed in the rings of Saturn; the aim is to account the strength of the Lorentz forces estimating that
the Lorentz field, co-acting with the gravity field of the planet, will limit the motion of all charged particles and small
size grains with surface charges inside a layer of about 200 m thickness as that which is observed in the rings of Saturn.
For this purpose our interest feast in the motion of charged particles with neglected mass where only electromagnetic forces
accounted in comparison to the weakness of the Newtonian fields. This study is particularly difficult because in the regions
we investigate these motions there is enormous three dimensional instability. Following the Poincare’s hypothesis that periodic
solutions are ‘dense’ in the set of all solutions in Hamiltonian systems we try to calculate many families of periodic solutions
and to study their stability. In this work we prove that in this environment charged particles can trace planar symmetric
periodic orbits. We discuss these orbits in details and we give their symplectic relations using the Hamiltonian formulation
which is related to the symplectic matrix. We apply numerical procedures to find families of these orbits and to study their
stability. Moreover we give the bifurcations of these families with families of planar asymmetric periodic orbits and families
of three dimensional symmetric periodic orbits. 相似文献
6.
Masaya Masayoshi Saito Kiyotaka Tanikawa 《Celestial Mechanics and Dynamical Astronomy》2009,103(3):191-207
We study the change of phase space structure of the rectilinear three-body problem when the mass combination is changed. Generally,
periodic orbits bifurcate from the stable Schubart periodic orbit and move radially outward. Among these periodic orbits there
are dominant periodic orbits having rotation number (n − 2)/n with n ≥ 3. We find that the number of dominant periodic orbits is two when n is odd and four when n is even. Dominant periodic orbits have large stable regions in and out of the stability region of the Schubart orbit (Schubart
region), and so they determine the size of the Schubart region and influence the structure of the Poincaré section out of
the Schubart region. Indeed, with the movement of the dominant periodic orbits, part of complicated structure of the Poincaré
section follows these orbits. We find stable periodic orbits which do not bifurcate from the Schubart orbit. 相似文献
7.
P. S. Soulis K. E. Papadakis T. Bountis 《Celestial Mechanics and Dynamical Astronomy》2008,100(4):251-266
We study the existence, linear stability and bifurcations of what we call the Sitnikov family of straight line periodic orbits
in the case of the restricted four-body problem, where the three equal mass primary bodies are rotating on a circle and the
fourth (small body) is moving in the direction vertical to the center mass of the other three. In contrast to the restricted
three-body Sitnikov problem, where the Sitnikov family has infinitely many stability intervals (hence infinitely many Sitnikov critical orbits), as the “family parameter” ż0 varies within a finite interval (while z
0 tends to infinity), in the four-body problem this family has only one stability interval and only twelve 3-dimensional (3D) families of symmetric periodic orbits exist which bifurcate from twelve
corresponding critical Sitnikov periodic orbits. We also calculate the evolution of the characteristic curves of these 3D
branch-families and determine their stability. More importantly, we study the phase space dynamics in the vicinity of these
orbits in two ways: First, we use the SALI index to investigate the extent of bounded motion of the small particle off the
z-axis along its interval of stable Sitnikov orbits, and secondly, through suitably chosen Poincaré maps, we chart the motion
near one of the 3D families of plane-symmetric periodic orbits. Our study reveals in both cases a fascinating structure of
ordered motion surrounded by “sticky” and chaotic orbits as well as orbits which rapidly escape to infinity. 相似文献
8.
《New Astronomy》2019
In the current study, we use the polyhedral model to compute the potential of the asteroid. There are five equilibrium points in the gravitational field of the asteroid 283 Emma. We concluded that the zero-velocity surfaces and the equilibrium points change with the suppositive variation of the rotational speed of the asteroid. It is found that if the rotational speed equals a half as it is in present, the number of equilibrium points is also five. However, if the rotational speed equals twice as it is in present, there are only three equilibrium points left. Four different periodic orbits are calculated using the hierarchical grid searching method. We calculated characteristic multipliers of periodic orbits to investigate the stability of these periodic orbits. The orbit near the primary's equatorial plane is more likely to be stable when the separation/ primary-radius is a large number. 相似文献
9.
We distinguish between regular orbits, that bifurcate from the main families of periodic orbits (those that exist also in the unperturbed case) and irregular periodic orbits, that are independent of the above. The genuine irregular families cannot be made to join the regular families by changing some parameters. We present evidence that all irregular families appear inside lobes formed by the asymptotic curves of the unstable periodic orbits. We study in particular a dynamical system of two degrees of freedom, that is symmetric with respect to the x-axis, and has also a triple resonance in its unperturbed form. The distribution of the periodic orbits (points on a Poincaré surface of section) shows some conspicuous lines composed of points of different multiplicities. The regular periodic orbits along these lines belong to Farey trees. But there are also lines composed mainly of irregular orbits. These are images of the x-axis in the map defined on the Poincaré surface of section. Higher order iterations of this map , close to the unstable triple periodic orbit, produce lines that are close to the asymptotic curves of this unstable orbit. The homoclinic tangle, formed by these asymptotic curves, contains many regular orbits, that were generated by bifurcation from the central orbit, but were trapped inside the tangle as the perturbation increased. We found some stable periodic orbits inside the homoclinic tangle, both regular and irregular. This proves that the homoclinic tangle is not completely chaotic, but contains gaps (islands of stability) filled with KAM curves. 相似文献
10.
M. Hénon 《Celestial Mechanics and Dynamical Astronomy》1974,10(3):375-388
We show by a general argument that periodic solutions of the planar problem of three bodies (with given masses) form one-parameter families. This result is confirmed by numerical investigations: two orbits found earlier by Standish and Szebehely are shown to belong to continuous one-parameter families of periodic orbits. In general these orbits have a non-zero angular momentum, and the configuration after one period is rotated with respect to the initial configuration. Similar general arguments whow that in the three-dimensional problem, periodic orbits form also one-parameter families; in the one-dimensional problem, periodic orbits are isolated. 相似文献
11.
We study symmetric relative periodic orbits in the isosceles three-body problem using theoretical and numerical approaches.
We first prove that another family of symmetric relative periodic orbits is born from the circular Euler solution besides
the elliptic Euler solutions. Previous studies also showed that there exist infinitely many families of symmetric relative
periodic orbits which are born from heteroclinic connections between triple collisions as well as planar periodic orbits with
binary collisions. We carry out numerical continuation analyses of symmetric relative periodic orbits, and observe abundant
families of symmetric relative periodic orbits bifurcating from the two families born from the circular Euler solution. As
the angular momentum tends to zero, many of the numerically observed families converge to heteroclinic connections between
triple collisions or planar periodic orbits with binary collisions described in the previous results, while some of them converge
to “previously unknown” periodic orbits in the planar problem. 相似文献
12.
R. C. Calleja E. J. Doedel A. R. Humphries A. Lemus-Rodríguez E. B. Oldeman 《Celestial Mechanics and Dynamical Astronomy》2012,114(1-2):77-106
We demonstrate the remarkable effectiveness of boundary value formulations coupled to numerical continuation for the computation of stable and unstable manifolds in systems of ordinary differential equations. Specifically, we consider the circular restricted three-body problem (CR3BP), which models the motion of a satellite in an Earth–Moon-like system. The CR3BP has many well-known families of periodic orbits, such as the planar Lyapunov orbits and the non-planar vertical and halo orbits. We compute the unstable manifolds of selected vertical and halo orbits, which in several cases leads to the detection of heteroclinic connections from such a periodic orbit to invariant tori. Subsequent continuation of these connecting orbits with a suitable end point condition and allowing the energy level to vary leads to the further detection of apparent homoclinic connections from the base periodic orbit to itself, or the detection of heteroclinic connections from the base periodic orbit to other periodic orbits. Some of these connecting orbits are of potential interest in space mission design. 相似文献
13.
We consider the bifurcation of 3D periodic orbits from the plane of motion of the primaries in the restricted three-body problem with oblateness. The simplest 3D periodic orbits branch-off at the plane periodic orbits of indifferent vertical stability. We describe briefly suitable numerical techniques and apply them to produce the first few such vertical-critical orbits of the basic families of periodic orbits of the problem, for varying mass parameter and fixed oblateness coefficent A1 = 0.005, as well as for varying A1 and fixed = 1/2. The horizontal stability of these orbits is also determined leading to predictions about the stability of the branching 3D orbits. 相似文献
14.
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
15.
Roger A. Broucke 《Celestial Mechanics and Dynamical Astronomy》2001,81(4):321-341
The present research was motivated by the recent discovery of planets around binary stars. Our initial intention was thus to investigate the 3-dimensional nearly circular periodic orbits of the circular restricted problem of three bodies; more precisely Stromgren's class L, (direct) and class m, (retrograde). We started by extending several of Hénon's vertical critical orbits of these 2 classes to three dimensions, looking especially for orbits which are near circular and have stable characteristic exponents.We discovered early on that the periodic orbits with the above two qualifications are fairly rare and we decided thus to undertake a systematic exploration, limiting ourselves to symmetric periodic orbits. However, we examined all 16 possible symmetry cases, trying 10000 sets of initial values for periodicity in each case, thus 160000 integrations, all with z
o or o equal to 0.1 This gave us a preliminary collection of 171 periodic orbits, all fairly near the xy-plane, thus with rather low inclinations. Next, we integrated a second similar set of 160000 cases with z
o or o equal to 0.5, in order to get a better representation of the large inclinations. This time, we found 167 periodic orbits, but it was later discovered that at least 152 of them belong to the same families as the first set with 0.1Our paper quickly describes the definition of the problem, with special emphasis on the symmetry properties, especially for the case of masses with equal primaries. We also allow a section to describe our approach to stability and characteristic exponents, following our paper on this subject, (Broucke, 1969). Then we describe our numerical results, as much as space permits in the present paper.We found basically only about a dozen families with sizeable segments of simple stable periodic orbits. Some of them are around one of the two stars only but we do not describe them here because of a lack of space. We extended about 170 periodic orbits to families of up to 500 members, (by steps of 0.005 in the parameter), although, in many cases, we do not know the real end of the families. We also give an overview of the different types of periodic orbits that are most often encountered. We describe some of the rather strange orbits, (some of which are actually stable). 相似文献
16.
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
17.
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
18.
We consider the main problem in satellite theory restricted to the polar plane. For suitable values of the energy the system has two unstable periodic orbits. We classify the trajectories in terms of their ultimate behavior with respect these periodic orbits in: oscillating, asymptotic and capture orbits. We study the energy level set and the existence and properties of the mentioned types of motion. 相似文献
19.
N. Caranicolas 《Celestial Mechanics and Dynamical Astronomy》1984,33(3):209-215
We study the periodic orbits in a two dimensional dynamical system, symmetric with respect to both axes, with two equal or nearly equal frequencies. It is shown that the periodic orbits can be found directly from the equations of motion. The form of these orbits depends on the value of the coupling parameter . We verify the theoretical results by numerical calculations. 相似文献
20.
G. Voyatzis T. Kotoulas J.D. Hadjidemetriou 《Celestial Mechanics and Dynamical Astronomy》2005,91(1-2):191-202
We present families of periodic orbits and their stability for the exterior mean motion resonances 1:2, 1:3 and 1:4 with Neptune
in the framework of the planar circular restricted three-body problem. We found that in each resonance there exist two branches
of symmetric elliptic periodic orbits with stable and unstable segments. Asymmetric periodic orbits bifurcate from the corresponding
symmetric ones. Asymmetric periodic orbits are stable and the motion in their neighbourhood is a libration with respect to
the resonant angle variable. In all the families of asymmetric periodic orbits the eccentricity extends to high values. Poincaré
sections reveal the changes of the topology in phase space. 相似文献