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1.
In this paper the effect of the Galactic tidal field on a Sun–comet pair will be considered when the comet is situated in the Oort cloud and planetary perturbations can be neglected. First, two averaged models were created, one of which can be solved analytically in terms of Jacobi elliptic functions. In the latter system, switching between libration and circulation of the argument of perihelion is prohibited. The non-averaged equations of motion are integrated numerically in order to determine the regions of the ( e ,  i ) phase space in which chaotic orbits can be found, and an effort is made to explain why the chaotic orbits manifest in these regions only. It is evident that for moderate values of semimajor axis, a ∼50 000 au , chaotic orbits are found for ( e <0.15 , 40°≤ i ≤140°) as determined by integrating the evolution of the comet over a period of 104 orbits. These regions of chaos increase in size with increasing semimajor axis. The typical e-folding times for these orbits range from around 600 Myr to 1 Gyr and thus are of little practical interest, as the time-scales for chaos arising from passing stars are much shorter. As a result of Galactic rotation, the chaotic regions in ( e ,  i ) phase space are not symmetric for prograde and retrograde orbits.  相似文献   

2.
In the present paper we apply a new method of distinction between ordered and chaotic motion in galactic potentials. The method uses the Fourier Transform of a series of time intervals each one representing the time that elapsed between two successive points on the Poincaré surface of section. Examples of the methods ability to achieve an early and clear detection of an orbit's behavior are provided using two galactic potentials. The new method can also be applied in order to have an early distinction between ordered and sticky orbits. The method is generalized in order to be used in models with more than two dimensions. Finally we have tried to find an one‐number index to give us the nature of the orbit instead of checking by eye the whole spectrum. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)  相似文献   

3.
We consider the possibility of particles being injected at the interior of a reconnecting current sheet (RCS), and study their orbits by dynamical systems methods. As an example we consider orbits in a 3D Harris type RCS. We find that, despite the presence of a strong electric field, a 'mirror' trapping effect persists, to a certain extent, for orbits with appropriate initial conditions within the sheet. The mirror effect is stronger for electrons than for protons. In summary, three types of orbits are distinguished: (i) chaotic orbits leading to escape by stochastic acceleration, (ii) regular orbits leading to escape along the field lines of the reconnecting magnetic component, and (iii) mirror-type regular orbits that are trapped in the sheet, making mirror oscillations. Dynamically, the latter orbits lie on a set of invariant KAM tori that occupy a considerable amount of the phase space of the motion of the particles. We also observe the phenomenon of 'stickiness', namely chaotic orbits that remain trapped in the sheet for a considerable time. A trapping domain, related to the boundary of mirror motions in velocity space, is calculated analytically. Analytical formulae are derived for the kinetic energy gain in regular or chaotic escaping orbits. The analytical results are compared with numerical simulations.  相似文献   

4.
We investigate the regular or chaotic nature of star orbits moving in the meridional plane of an axially symmetric galactic model with a disk and a spherical nucleus. We study the influence of some important parameters of the dynamical system, such as the mass and the scale length of the nucleus, the angular momentum or the energy, by computing in each case the percentage of chaotic orbits, as well as the percentages of orbits of the main regular resonant families. Some heuristic arguments to explain and justify the numerically derived outcomes are also given. Furthermore, we present a new method to find the threshold between chaos and regularity for both Lyapunov Characteristic Numbers and SALI, by using them simultaneously.  相似文献   

5.
We study the orbital structure in a series of self-consistent N -body configurations simulating rotating barred galaxies with spiral and ring structures. We perform frequency analysis in order to measure the angular and the radial frequencies of the orbits at two different time snapshots during the evolution of each N -body system. The analysis is done separately for the regular and the chaotic orbits. We thereby identify the various types of orbits, determine the shape and percentages of the orbits supporting the bar and the ring/spiral structures, and study how the latter quantities change during the secular evolution of each system. Although the frequency maps of the chaotic orbits are scattered, we can still identify concentrations around resonances. We give the distributions of frequencies of the most important populations of orbits. We explore the phase-space structure of each system using projections of the 4D surfaces of section. These are obtained via the numerical integration not only of the orbits of test particles, but also of the real N -body particles. We thus identify which domains of the phase space are preferred and which are avoided by the real particles. The chaotic orbits are found to play a major role in supporting the shape of the outer envelope of the bar as well as the rings and the spiral arms formed outside corotation.  相似文献   

6.
The behavior of the orbits in a galaxy model composed of an harmonic core and a strong bar potential is studied. Numerical calculations show that a large number of orbits display chaotic motion. These orbits are low angular momentun orbits. The percentage of chaotic orbits increases as the angular velocity of the system increases or the strength of the harmonic term decreases. A new dynamical parameter, the S(c) spectrum, is introduced and used to detect the island motion and the evolution of the sticky regions. Comparison to previously obtained results reveals the leading role of the new spectrum. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)  相似文献   

7.
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.
It is already known (Froeschlé et al., 1997a) that the fast Lyapunov indicator (hereafter FLI), i.e. the computation on a relatively short time of a quantity related to the largest Lyapunov indicator, allows us to discriminate between ordered and weak chaotic motion. Using the FLI many results have been obtained on the standard map taken as a model problem. On this model we are not only able to discriminate between a short time weak chaotic motion and an ordered one, but also among regular motion between non resonant and resonant orbits. Moreover, periodic orbits are characterised by constant FLI values which appear to be related to the order of periodic orbits (Lega and Froeschlé, 2001). In the present paper we extend all these results to the case of continuous dynamical systems (the Hénon and Heiles system and the restricted three-body problem). Especially for the periodic orbits we need to introduce a new value: the orthogonal FLI in order to fully recover the results obtained for mappings.  相似文献   

9.
A galaxy model with a satellite companion is used to study the character of motion for stars moving in the x ‐y plane. It is observed that a large part of the phase plane is covered by chaotic orbits. The percentage of chaotic orbits increases when the galaxy has a dense nucleus of massMn. The presence of the dense nucleus also increases the stellar velocities near the center of the galaxy. For small values of the distance R between the two bodies, low energy stars display a chaotic region near the centre of the galaxy, when the dense nucleus is present, while for larger values of R the motion in active galaxies is regular for low energy stars. Our results suggest that in galaxies with a satellite companion, the chaotic character of motion is not only a result of galactic interaction but also a result caused by the dense nucleus. Theoretical arguments are used to support the numerical outcomes. We follow the evolution of the galaxy, as mass is transported adiabatically from the disk to the nucleus. Our numerical results are in satisfactory agreement with observational data from M51‐type binary galaxies (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)  相似文献   

10.
The spectra of ‘stretching numbers’ (or ‘local Lyapunov characteristic numbers’) are different in the ordered and in the chaotic domain. We follow the variation of the spectrum as we move from the centre of an island outwards until we reach the chaotic domain. As we move outwards the number of abrupt maxima in the spectrum increases. These maxima correspond to maxima or minima in the curve a(θ), where a is the stretching number, and θ the azimuthal angle. We explain the appearance of new maxima in the spectra of ordered orbits. The orbits just outside the last KAM curve are confined close to this curve for a long time (stickiness time) because of the existence of cantori surrounding the island, but eventually escape to the large chaotic domain further outside. The spectra of sticky orbits resemble those of the ordered orbits just inside the last KAM curve, but later these spectra tend to the invariant spectrum of the chaotic domain. The sticky spectra are invariant during the stickiness time. The stickiness time increases exponentially as we approach an island of stability, but very close to an island the increase is super exponential. The stickiness time varies substantially for nearby orbits; thus we define a probability of escape Pn(x) at time n for every point x. Only the average escape time in a not very small interval Δx around each x is reliable. Then we study the convergence of the spectra to the final, invariant spectrum. We define the number of iterations, N, needed to approach the final spectrum within a given accuracy. In the regular domain N is small, while in the chaotic domain it is large. In some ordered cases the convergence is anomalously slow. In these cases the maximum value of ak in the continued fraction expansion of the rotation number a = [a0,a1,... ak,...] is large. The ordered domain contains small higher order chaotic domains and higher order islands. These can be located by calculating orbits starting at various points along a line parallel to the q-axis. A monotonic variation of the sup {q}as a function of the initial condition q0 indicates ordered motions, a jump indicates the crossing of a localized chaotic domain, and a V-shaped structure indicates the crossing of an island. But sometimes the V-shaped structure disappears if the orbit is calculated over longer times. This is due to a near resonance of the rotation number, that is not followed by stable islands. This revised version was published online in July 2006 with corrections to the Cover Date.  相似文献   

11.
Spiral galaxies contain both ordered and chaotic orbits. In normal spirals the perturbations are weak (of order 2–10%) and most orbits are ordered. The density wave theory refers mainly to linear perturbations. Nonlinear effects appear in the outer parts of the open spirals (S_b, S_c) and produce the termination of these spirals near the 4/1 resonance. On the other hand in barred spirals the perturbations are relatively strong (of order 100%). Then the outer spirals and the envelope of the bar are composed mainly of chaotic orbits, while the main body of the bar is composed of ordered orbits. The chaotic orbits of the spiral arms of strong barred galaxies are sticky, i.e. they do not escape from the galaxy for at least a Hubble time. The forms of these spirals are delineated by the unstable manifolds of the unstable periodic orbits L_1, L_2 near the ends of the bar and of other unstable periodic orbits inside and outside corotation.  相似文献   

12.
The correlation dimension, that is the dimension obtained by computing the correlation function of pairs of points of a trajectory in phase space, is a numerical technique introduced in the field of non-linear dynamics in order to compute the dimension of the manifold in which an orbit moves, without the need of knowing the actual equations of motion that give rise to the trajectory. This technique has been proposed in the past as a method to measure the dimension of stellar orbits in astronomical potentials, that is the number of isolating integrals of motion the orbits obey. Although the algorithm can in principle yield that number, some care has to be taken in order to obtain good results. We studied the relevant parameters of the technique, found their optimal values, and tested the validity of the method on a number of potentials previously studied in the literature, using the Smaller Alignment Index (SALI), Lyapunov exponents and spectral dynamics as gauges.  相似文献   

13.
Chaos appears in various problems of Relativity and Cosmology. Here we discuss (a) the Mixmaster Universe model, and (b) the motions around two fixed black holes. (a) The Mixmaster equations have a general solution (i.e. a solution depending on 6 arbitrary constants) of Painlevé type, but there is a second general solution which is not Painlevé. Thus the system does not pass the Painlevé test, and cannot be integrable. The Mixmaster model is not ergodic and does not have any periodic orbits. This is due to the fact that the sum of the three variables of the system (α + β + γ) has only one maximum for τ = τm and decreases continuously for larger and for smaller τ. The various Kasner periods increase exponentially for large τ. Thus the Lyapunov Characteristic Number (LCN) is zero. The "finite time LCN" is positive for finite τ and tends to zero when τ → ∞. Chaos is introduced mainly near the maximum of (α + β + γ). No appreciable chaos is introduced at the successive Kasner periods, or eras. We conclude that in the Belinskii-Khalatnikov time, τ, the Mixmaster model has the basic characteristics of a chaotic scattering problem. (b) In the case of two fixed black holes M1 and M2 the orbits of photons are separated into three types: orbits falling into M1 (type I), or M2 (type II), or escaping to infinity (type III). Chaos appears because between any two orbits of different types there are orbits of the third type. This is a typical chaotic scattering problem. The various types of orbits are separated by orbits asymptotic to 3 simple unstable orbits. In the case of particles of nonzero rest mass we have intervals where some periodic orbits are stable. Near such orbits we have order. The transition from order to chaos is made through an infinite sequence of period doubling bifurcations. The bifurcation ratio is the same as in classical conservative systems. This revised version was published online in July 2006 with corrections to the Cover Date.  相似文献   

14.
The dynamics of the two Jupiter triangular libration points perturbed by Saturn is studied in this paper. Unlike some previous works that studied the same problem via the pure numerical approach, this study is done in a semianalytic way. Using a literal solution, we are able to explain the asymmetry of two orbits around the two libration points with symmetric initial conditions. The literal solution consists of many frequencies. The amplitudes of each frequency are the same for both libration points, but the initial phase angles are different. This difference causes a temporary spatial asymmetry in the motions around the two points, but this asymmetry gradually disappears when the time goes to infinity. The results show that the two Jupiter triangular libration points should have symmetric spatial stable regions in the present status of Jupiter and Saturn. As a test of the literal solution, we study the resonances that have been extensively studied in Robutel and Gabern (Mon Not R Astron Soc 372:1463–1482, 2006). The resonance structures predicted by our analytic theory agree well with those found in Robutel and Gabern (Mon Not R Astron Soc 372:1463–1482, 2006) via a numerical approach. Two kinds of chaotic orbits are discussed. They have different behaviors in the frequency map. The first kind of chaotic orbits (inner chaotic orbits) is of small to moderate amplitudes, while the second kind of chaotic orbits (outer chaotic orbits) is of relatively larger amplitudes. Using analytical theory, we qualitatively explain the transition process from the inner chaotic orbits to the outer chaotic orbits with increasing amplitudes. A critical value of the diffusion rate is given to separate them in the frequency map. In a forthcoming paper, we will study the same problem but keep the planets in migration. The time asymmetry, which is unimportant in this paper, may cause an observable difference in the two Jupiter Trojan groups during a very fast planet migration process.  相似文献   

15.
In the present paper, an efficient iterative method of arbitrary integer order of convergent ≥2 based on the homotopy continuation techniques for the solution of the initial value problem of space dynamics using the universal Y functions is presented. The method is of dynamic nature in the sense that, ongoing from one iterative scheme to the subsequent one, only additional instruction is needed. Most importantly, the method does not need any prior knowledge of the initial guess. This is a property which avoids the critical situations between divergent to very slow convergent solutions that may exist in other numerical methods which depend on initial guesses. A computational package for digital implementation of the method is given, together with numerical applications for elliptic, hyperbolic, and parabolic orbits. The accuracy of the results for all orbits is O(10–16). (© 2016 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)  相似文献   

16.
We use probability density functions (pdfs) of sums of orbit coordinates, over time intervals of the order of one Hubble time, to distinguish weakly from strongly chaotic orbits in a barred galaxy model. We find that, in the weakly chaotic case, quasi-stationary states arise, whose pdfs are well approximated by q-Gaussian functions (with 1 <?q < 3), while strong chaos is identified by pdfs which quickly tend to Gaussians (q =?1). Typical examples of weakly chaotic orbits are those that ??stick?? to islands of ordered motion. Their presence in rotating galaxy models has been investigated thoroughly in recent years due to their ability to support galaxy structures for relatively long time scales. In this paper, we demonstrate, on specific orbits of 2 and 3 degree of freedom barred galaxy models, that the proposed statistical approach can distinguish weakly from strongly chaotic motion accurately and efficiently, especially in cases where Lyapunov exponents and other local dynamic indicators appear to be inconclusive.  相似文献   

17.
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18.
We discuss the issue of ordered and chaotic trajectories in the Bohmian approach of Quantum Mechanics from points of view relevant to the methods of Celestial Mechanics. The Bohmian approach gives the same results as the orthodox (Copenhagen) approach, but it considers also underlying trajectories guided by the wave. The Bohmian trajectories are rather different from the corresponding classical trajectories. We give examples of a classical chaotic system that is ordered quantum-mechanically and of a classically ordered system that is mostly chaotic quantum mechanically. Then we consider quantum periodic orbits and ordered orbits, that can be represented by formal series of the “third integral” type, and we study their asymptotic properties leading to estimates of exponential stability. Such orbits do not approach the “nodal points” where the wavefunction ψ vanishes. On the other hand, when an orbit comes close to a nodal point, chaos is generated in the neighborhood of a hyperbolic point (called X-point). The generation of chaos is maximum when the X-point is close to the nodal point. Finally we remark that high order periodic orbits may behave as “effectively ordered” or “effectively chaotic” for long times before reaching the period.  相似文献   

19.
We compare two different N-body models simulating elliptical galaxies. Namely, the first model is a non-rotating triaxial N-body equilibrium model with smooth center, called SC model. The second model, called CM model, is derived from the SC by inserting a central mass in it, so that all possible differences between the two models are due to the effect of the central mass. The central mass is assumed to be mainly due to a massive central black hole of mass about 1% of the total mass of the galaxy. By using the fundamental frequency analysis, the two systems are thoroughly investigated as regards the types of orbits described either by test particles, or by the real particles of the systems at all the energy levels. A comparison between the orbits of test particles and the orbits of real particles at various energy levels is made on the rotation number plane. We find that extensive stable regions of phase space, detected by test particles remain empty, i.e. these regions are not occupied by real particles, while many real particles move in unstable regions of phase space describing chaotic orbits. We run self-consistently the two models for more than a Hubble time. During this run, in spite of the noise due to small variations of the potential, the SC model maintains (within a small uncertainly) the number of particles moving on orbits of each particular type. In contrast, the CM model is unstable, due to the large amount of mass in chaotic motion caused by the central mass. This system undergoes a secular evolution towards an equilibrium state. During this evolution it is gradually self-organized by converting chaotic orbits to ordered orbits mainly of the short axis tube type approaching an oblate spheroidal equilibrium. This is clearly demonstrated in terms of the fundamental frequencies of the orbits on the rotation number plane and the time evolution of the triaxiality index.  相似文献   

20.
In a previous paper (Voglis et al., Paper I), we demonstrated that, in a rotating galaxy with a strong bar, the unstable asymptotic manifolds of the short-period family of unstable periodic orbits around the Lagrangian points L 1 or L 2 create correlations among the apocentric positions of many chaotic orbits, thus supporting a spiral structure beyond the bar. In this paper, we present evidence that the unstable manifolds of all the families of unstable periodic orbits near and beyond corotation contribute to the same phenomenon. Our results refer to a N -body simulation, a number of drawbacks of which, as well as the reasons why these do not significantly affect the main results, are discussed. We explain the dynamical importance of the invariant manifolds as due to the fact that they produce a phenomenon of 'stickiness' slowing down the rate of chaotic escape in an otherwise non-compact region of the phase space. We find a stickiness time of the order of 100 dynamical periods, which is sufficient to support a long-living spiral structure. Manifolds of different families become important at different ranges of values of the Jacobi constant. The projections of the manifolds of all the different families in the configuration space produce a pattern due to the 'coalescence' of the invariant manifolds. This follows closely the maxima of the observed   m = 2  component near and beyond corotation. Thus, the manifolds support both the outer edge of the bar and the spiral arms.  相似文献   

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