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1.
Wavelet Analysis: the effect of varying basic wavelet parameters 总被引：4，自引：1，他引：3
The most commonly used methods to analyse (observed) quasi-periodic signals are standard techniques such as Fourier and wavelet analysis. Whereas a Fourier transform provides information on the dominant frequencies, wavelet analysis has the added advantage of providing the time localisation of the various frequency components. The usefulness and robustness of wavelet analysis is investigated by varying the different parameters which characterise the `mother' wavelet. We examine the effect of varying these parameters on the temporal and frequency resolution and the damping profile, which can be obtained from the wavelet transform. Additionally, the effect of a changing periodicity on the wavelet transform is investigated. Both simple harmonic functions and intensity oscillations observed by TRACE are used to demonstrate the various advantages and disadvantages of the different methods. In general, using the Paul wavelet or a smaller value of the wavelet parameter k provides a better time resolution, whereas the Morlet wavelet or a larger value of k improves the frequency resolution. Overall, our results indicate that great care is needed when using a wavelet analysis and that all the possible factors that could affect the transform should be taken into consideration. 相似文献
2.
We present results of a study of the so-called “stickiness” regions where orbits in mappings and dynamical systems stay for very long times near an island and then escape to the surrounding chaotic region. First we investigated the standard map in the form xi+1 = xi+yi+1 and yi+1 = yi+K/2π · sin(2πxi) with a stochasticity parameter K = 5, where only two islands of regular motion survive. We checked now many consecutive points—for special initial conditions of the mapping—stay within a certain region around the island. For an orbit on an invariant curve all the points remain forever inside this region, but outside the “last invariant curve” this number changes significantly even for very small changes in the initial conditions. In our study we found out that there exist two regions of “sticky” orbits around the invariant curves: A small region I confined by Cantori with small holes and an extended region II is outside these cantori which has an interesting fractal character. Investigating also the Sitnikov-Problem where two equally massive primary bodies move on elliptical Keplerian orbits, and a third massless body oscillates through the barycentre of the two primaries perpendicularly to the plane of the primaries—a similar behaviour of the stickiness region was found. Although no clearly defined border between the two stickiness regions was found in the latter problem the fractal character of the outer region was confirmed. 相似文献
3.
In this work, we have simulated orbits of a particle moving in gravitational field of the Sun-Jupiter system. The effect of
solar radiation pressure, including Poynting Robertson drag, on the evolution of particle orbits in phase space have been
studied for different values of the parameter β
1 (the ratio of radiation to gravitational force) and initial conditions. Characteristics of various computed trajectories
have been studied using wavelet transform (WT), Fourier transform (FT) and Poincare surface of section method. We use wavelet
analysis to identify transitions of a trajectory in time-frequency plane and further apply it to classify it as regular or
chaotic in phase space. Unlike the Fourier transform method (FT), we observe that the wavelet transform (WT) also provides
a basis to identify ‘sticky’ trajectories in the present dynamical system. 相似文献
4.
The dynamics of space debris with very high A/m near the geostationary orbit is dominated by the gravitational coefficient C
22 and the solar radiation pressure. An analysis of the stability of the orbits by the chaos indicator MEGNO and frequency analysis map FAM shows chaotic layers around the separatrix and reveals a web of sub-structures associated to resonances with the annual period
of the Sun. This succession of stable thin islands and chaotic layers can be reproduced and explained by a quite simple toy
model, based on a pendulum approach, perturbed, through the eccentricity, by the external (Sun) frequency. The use of suitable
action-angle variables in the circulation and libration regions of the pendulum allows to point out new resonances between
the geostationary libration angle and the Sun’s longitude. They correspond very well (positions, shape, width) to the structures
visible on the FAM representations. 相似文献
5.
James H. Bartlett 《Celestial Mechanics and Dynamical Astronomy》1982,28(3):295-317
The Henon-Heiles mappingx=x+a(y–y
3), y=y–a(x–x3) has been studied, with the aim of finding where the unstable regions of the (x, y) plane are. When this mapping is put into the normal form, it is found to be a typical twist mapping. The criteria of Moser (1971) are used to obtain an upper limit to the size of a stable region around the origin, and this limit decreases to zero as the value of the parameter a increases toward 2.0. However, direct calculation fora=1.99 shows that there is a fairly large region insidex=0.412,y=0, from which escape from near the outer boundary requires at least 160 mappings. The region of high stability thus appears to be much larger than any region of absolute stability predicted by the KAM theorem.A general survey has been made of instability regions for the parameter valuea=1.0, this survey having been carried out to the extent which is allowed by a computer with 18-decimal-place accuracy. First, for all thex-axis fixed points (of the above mapping) deemed to be representative and significant, both the locations and variational matrix traces have been calculated. (The latter show whether the fixed point is elliptic or hyperbolic.) Ifn is the number of mappings andk is the number of circuits around the origin, then the listing (Table IV) is for fractionsk/n between 1/6 and 1/22, inclusive. (This covers the range 0x<0.96, withx=0 the fixed point forn=6,k=1).Escape toward infinity can be rapid, with less than 200 mappings necessary to reach the vicinity of then=1 fixed points (atx=±1,y=0 andx=0,y=±1) from outer regions of the (x, y) plane, such as for |x|>0.93,y=0. In this case, the unstable regions may be tongues encircling the origin. However, as the distance from the origin is decreased, the tongues can be replaced by exceedingly fine threads rapidly becoming less than say 10–16 in thickness. Such a thread issues fromx=0.905468199,y=0 and requires of the order of 40 000 mappings to escape. It does so by spiralling about the origin and penetrating through several series of loops associated with various fixed points at successively greater (absolute) values ofx(y=0). The region between this thread and the origin is therefore highly stable.
Practical stability of a region may be regarded as attained when the region is interior to a series of loops for which the trace of the variational matrix is close to 2.0. This occurs forn=53,k=4, with fixed point atx=0.819786,y=0 and Trace=2.0000 0004.If an invariant curve does in fact exist, then one must be able to show that the outward spiralling from a given series of loops is brought to a halt at some stage. This does not occur in the region where direct computation is possible, as we show in this article, and it remains to be seen under what conditions it can take place. 相似文献
6.
G. Contopoulos N. Voglis C. Efthymiopoulos C. Froeschlé R. Gonczi E. Lega R. Dvorak E. Lohinger 《Celestial Mechanics and Dynamical Astronomy》1997,67(4):293-317
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
7.
8.
Using the standard map as a model problem and in the spirit of cluster analysis we have studied the invariance of the distributions of different indicators introduced to detect and measure weak chaos. We show that the problem is less straightforward than expected and that, except for very strong chaotic dynamical systems, all the complexities (islands, sticking phenomena, cantori) of mixed Hamiltonian systems are reflected into the indicators of convergence towards invariant distributions. 相似文献
9.
Kleomenis Tsiganis Harry Varvoglis Rudolf Dvorak 《Celestial Mechanics and Dynamical Astronomy》2005,92(1-3):71-87
It has recently been shown that Jupiter Trojans may exhibit chaotic behavior, a fact that has put in question their presumed long term stability. Previous numerical results suggest a slow dispersion of the Trojan swarms, but the extent of the ‘effective’ stability region in orbital elements space is still an open problem. In this paper, we tackle this problem by means of extensive numerical integrations. First, a set of 3,200 fictitious objects and 667 numbered Trojans is integrated for 4 Myrs and their Lyapunov time, TL, is estimated. The ones following chaotic orbits are then integrated for 1 Gyr, or until they escape from the Trojan region. The results of these experiments are presented in the form of maps of TLand the escape time, TE, in the space of proper elements. An effective stability region for 1 Gyr is defined on these maps, in which chaotic orbits also exist. The distribution of the numbered Trojans follows closely the TE=1 Gyr level curve, with 86% of the bodies lying inside and 14% outside the stability region. This result is confirmed by a 4.5 Gyr integration of the 246 chaotic numbered Trojans, which showed that 17% of the numbered Trojans are unstable over the age of the solar system. We show that the size distributions of the stable and unstable populations are nearly identical. Thus, the existence of unstable bodies should not be the result of a size-dependent transport mechanism but, rather, the result of chaotic diffusion. Finally, in the large chaotic region that surrounds the stability zone, a statistical correlation between TLandTE is found. 相似文献
10.
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
11.
12.
Xiyun Hou Daniel J. Scheeres Lin Liu 《Celestial Mechanics and Dynamical Astronomy》2014,119(2):119-142
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
13.
Collisions in the Solar System play an important role in its history. Impact processes depend essentially on the velocity distribution of meteoroids colliding with a chosen planet. According to Carleman's theorem it is sufficient to find the set of M
k
= mathematical expectation of v
k
, v being the collisional velocity. We suppose that M
k
for meteoroids of asteroidal nature differs slightly from that for asteroids themselves. So among all numbered minor planets we select those which may potentially collide with the chosen major planet. Then we calculate v at intersection points and count the average over all such points and all selected asteroids. The gravitation of a body-target may be taken into account or not. Numerical results are collected in four Tables.St.Petersburg University 相似文献
14.
This paper investigates the triangular libration points in the photogravitational restricted three-body problem of variable
mass, in which both the attracting bodies are radiating as well and the infinitesimal body vary its mass with time according
to Jeans’ law. Firstly, applying the space-time transformation of Meshcherskii in the special case when q=1/2, k=0, n=1, the differential equations of motion of the problem are given. Secondly, in analogy to corresponding problem with constant
mass, the positions of analogous triangular libration points are obtained, and the fact that these triangular libration points
cease to be classical ones when α≠0, but turn to classical L
4 and L
5 naturally when α=0 is pointed out. Lastly, introducing the space-time inverse transformation of Meshcherskii, the linear stability of triangular
libration points is tested when α>0. It is seen that the motion around the triangular libration points become unstable in general when the problem with constant
mass evolves into the problem with decreasing mass. 相似文献
15.
S. Ferraz-Mello T. A. Michtchenko D. Nesvorný F. Roig A. Simula 《Planetary and Space Science》1998,46(11-12)
This paper presents a comparative analysis of the 2/1 and 3/2 asteroidal resonances based on several analytical and numerical tools. The frequency map analysis was used to obtain a refined estimation of the chaotic transport. Fourier and wavelet analyses were used to construct the web of inner resonances and showed that they are the seat of the strongly unstable motion observed in the numerical simulations. The most regular regions in both resonances were classified. A fast symplectic mapping allowed a number of direct runs over 108 years of the orbits initially in these regions. The stability of orbits over the age of the solar system was discussed and compared to the distribution of the observed asteroids in both resonances. 相似文献
16.
This paper points out the errors in the solutions of a research work by N. Nanousis under the same title published in this journal, volume 199, 1993. The correct solutions of the problem for the velocity field and the drag on the plate, by the Laplace transform technique, are presented. The results are discussed for two cases of an arbitrary time-dependent forcing effect. It is shown that the viscoelastic parameterk > 0 influences the velocity and introduces reverse flow. For a suddenly accelerated plate,k > 0 increases the velocity forz <
and decreases it forz >
. In the case of the ramp-type boundary condition,k > 0 tends to decrease the velocity. 相似文献
17.
N. D. Caranicolas 《Journal of Astrophysics and Astronomy》2001,22(4):309-319
We present a map for the study of resonant motion in a potential made up of two harmonic oscillators with quartic perturbing
terms. This potential can be considered to describe motion in the central parts of non-rotating elliptical galaxies. The map
is based on the averaged Hamiltonian. Adding on a semi-empirical basis suitable terms in the unperturbed averaged Hamiltonian,
corresponding to the 1:1 resonant case, we are able to construct a map describing motion in several resonant cases. The map
is used in order to find thex − p
x
Poincare phase plane for each resonance. Comparing the results of the map, with those obtained by numerical integration of
the equation of motion, we observe, that the map describes satisfactorily the broad features of orbits in all studied cases
for regular motion. There are cases where the map describes satisfactorily the properties of the chaotic orbits as well. 相似文献
18.
19.
The Brans-Dicke field equations for a viscous distribution representing slowly rotating fluid spheres are investigated. Exact solutions are obtained for differential rotation by imposing physical restrictions on the matter rotation (r,t). The physical properties are discussed fork=±1. 相似文献
20.
Exact solution for a homogeneous cosmological model in 5D space-time-mass gravity theory proposed by Wesson (Astron. Astrophys.
119:145, 1983) is obtained by assuming the time-dependent equation of state. The behavior of the solution is discussed for the two cases
k<0 and k=0. It is found that the observed constancy of the rest mass of an isolated particle in the present era may be interpreted
as a consequence of the decreasing rate of change of rest mass with time. Moreover, a spontaneous compactification-like phenomenon
of an extra dimension takes place in the case of k=0. It is also found that with decrease in extra space the observable three-dimensional space entropy increases, thus accounting
for the large value of entropy observable at present. 相似文献