Generalized Lyapunov exponents indicators in Hamiltonian dynamics: An application to a double star system

The Lyapunov characteristic numbers (LCNs) which are defined as the mean value of the distribution of the local variations of the tangent vectors to the flow (=ln _k ⁱ ) (see Froeschlé, 1984) have been found to be sensitive indicators of stochasticity. So we computed the distribution of these local variations and determined the moments of higher order for the integrable and stochastic regions in a binary star system with =0.5.     

Generalized Lyapunov characteristic indicators and corresponding Kolmogorov like entropy of the standard mapping

Lyapunov characteristic indicators are currently defined as the mean, i.e. the first moment, of the distribution of the local variations of the tangent vector to the flow. Higher moments of the distribution give further informations about the fluctuations around the average.     

The use of symplectic algorithm in the computation of Lyapunov characteristic numbers

It is found that, in the computation of Lyapunov characteristic numbers, symplectic algorithms are clearly more advantageous than non-symplectic ones.     

Velocity curve analysis of the spectroscopic binary stars by the non-linear least squares

We introduce a new method to derive the orbital parameters of spectroscopic binary stars by a non-linear least squares analysis of residuals. Using measured radial velocity data of four double lined spectroscopic binary systems RZ Cas, CC Cas, V1130 Tau and Y Cygni, we find corresponding orbital and spectroscopic elements. Our numerical results are in good agreement with those obtained by others using more traditional methods.     

Photometric analysis of EL Aqr

We present an analysis of the Hipparcos and ASAS photometry of the W UMa star EL Aqr. The Wilson-Devinney code is employed in conjunction with recent spectroscopic mass ratio of 0.203 to model the V band light curve. We confirm that the system is an A type contact binary however our analysis suggests that it is over massive for its spectral class and has a relatively high fillout of 44%. This revised version was published online in August 2006 with corrections to the Cover Date.     

on the relationship between fast lyapunov indicator and periodic orbits for symplectic mappings

The computation on a relatively short time of a quantity, related to the largest Lyapunov Characteristic Exponent, called Fast Lyapunov Indicator allows to discriminate between ordered and weak chaotic motion and also, under certain conditions, between resonant and non resonant regular orbits. The aim of this paper is to study numerically the relationship between the Fast Lyapunov Indicator values and the order of periodic orbits. Using the two-dimensional standard map as a model problem we have found that the Fast Lyapunov Indicator increases as the logarithm of the order of periodic orbits up to a given order. For higher order the Fast Lyapunov Indicator grows linearly with the order of the periodic orbits. We provide a simple model to explain the relationship that we have found between the values of the Fast Lyapunov Indicator, the order of the periodic orbits and also the minimum number of iterations needed to obtain the Fast Lyapunov Indicator values.     

Contact Binaries: A Study of the Proximity Effects and Gravity Darkening Based on Kopal’s Fourier Method

A new method for the determination of the proximity effects and gravity darkening exponents in contact binaries of W UMa type is presented. The method is based on Kopal’s method of Fourier analysis of the light changes of eclipsing variables in the Frequency Domain. The method was applied to 36 W UMa systems for which geometric and photometric elements have been derived by the most powerful techniques. The derived values are very close to those predicted by the existing theory of radiative transfer or convective equilibrium.     

Stability of S-type Orbits in Binaries

This numerical investigation is concerned with the stability of planets moving around one component of a double star system. Since the discovery of four extra solar planets moving in such orbits, there is a growing interest of stability studies thereto. We determined the stable regions in the elliptic restricted three body problem, for the whole range of mass-ratios from 0.1 to 0.9, by means of the Fast Lyapunov Indicators. The computations have been carried out for eccentricities of the binary and of the planet in the range 0–0.5. Therefore, we present for the first time the variation of the stable regions when the initial eccentricity of the planet is increased. We have found a correlation between the reduction of the stable zones if the eccentricity of the planet or of the binary is increased — of course the latter one is more effective.This revised version was published online in October 2005 with corrections to the Cover Date.     

Spectra of stretching numbers and helicity angles in dynamical systems

We define a stretching number (or Lyapunov characteristic number for one period) (or stretching number) a = In % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dc9vqaqpepu0xbbG8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqWaaeaada% Wcaaqaaiabe67a4jaadshacqGHRaWkcaaIXaaabaGaeqOVdGNaamiD% aaaaaiaawEa7caGLiWoaaaa!3F1E!\[\left| {\frac{{\xi t + 1}}{{\xi t}}} \right|\]as the logarithm of the ratio of deviations from a given orbit at times t and t + 1. Similarly we define a helicity angle as the angle between the deviation % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dc9vqaqpepu0xbbG8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOVdGNaam% iDaaaa!3793!\[\xi t\]and a fixed direction. The distributions of the stretching numbers and helicity angles (spectra) are invariant with respect to initial conditions in a connected chaotic domain. We study such spectra in conservative and dissipative mappings of 2 degrees of freedom and in conservative mappings of 3-degrees of freedom. In 2-D conservative systems we found that the lines of constant stretching number have a fractal form.     

On the Stochasticity of the Asteroid Belt

The study of the stochasticity of the asteroid belt requires the analysis of a large number of orbits. We detect the dynamical character of a set of 5 400 asteroids using the Fast Lyapunov Indicator, a method of analysis closely related to the computation of the Lyapunov Characteristic Exponents, but cheaper in computational time. For both regular and chaotic orbits we try to associate the motion to the underlying resonances network. For it we consider different methods of classification of rational numbers proposed by number theory, and we choose the one which seems to be strictly related to the dynamical behaviour of a system. This revised version was published online in July 2006 with corrections to the Cover Date.     

Frequency analysis of a dynamical system

Frequency analysis is a new method for analyzing the stability of orbits in a conservative dynamical system. It was first devised in order to study the stability of the solar system (Laskar, Icarus, 88, 1990). It is a powerful method for analyzing weakly chaotic motion in hamiltonian systems or symplectic maps. For regular motions, it yields an analytical representation of the solutions. In cases of 2 degrees of freedom system with monotonous torsion, precise numerical criterions for the destruction of KAM tori can be found. For a 4D symplectic map, plotting the frequency map in the frequency plane provides a clear representation of the global dynamics and describes the actual Arnold web of the system.     

Comparison of convergence towards invariant distributions for rotation angles, twist angles and local Lyapunov characteristic numbers

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.     

A new interpretation of zero Lyapunov exponents in BKL time for Mixmaster cosmology

A global relation ship between cosmological time and Belinskii-Khalatnikov-Lifshitz(BKL)time during the entire evolution of the Mixmaster Bianchi Ⅸ universe is used to explain why all the Lyapunov exponents are zero at the BKL time.The actual reason is that the domain of the cosmological time is finite as the BKL time runs from minus infinity to infinity.     

Wavelet analysis of the standard map: Structure and scaling

Wavelet analysis is applied to distributions of points generated by iterating the standard map. The initial condition is chosen so that the points fill the largest chaotic region. When the standard map parameterk=1.3, the distribution of points contains many voids corresponding to islands in the chaotic region. The wavelet transform is dominated by contributions from these islands. Fork=10 the chaos fills phase space and no structure is apparent; the wavelet transform reveals statistical fluctuations in the distribution of points.     

Stochastic data in astronomy: Fourier analysis of highly nonuniform time series

The features of the CLEAN algorithm for Fourier analysis of time series with data separated by long pauses are analyzed in detail. Estimates are obtained for the limits of variability of the parameters of harmonic components that can be determined on a specified time grid. This analysis are used to search for harmonic components in the spectral line profile variations of the star Ori A (O8III) obtained in 2001 with the 1 m telescope at the Special Astrophysical Observatory of the Russian Academy of Sciences.Translated from Astrofizika, Vol. 48, No. 1, pp. 87–100 (February 2005).     

Photometric detection of high proper motions in dense stellar fields using difference image analysis

The difference image analysis (DIA) of the images obtained by the Optical Gravitational Lensing Experiment (OGLE-II) revealed a peculiar artefact in the sample of stars proposed as variable by Woźniak in one of the Galactic bulge fields: the occurrence of pairs of candidate variables showing anti-correlated light curves monotonic over a period of 3 yr. This effect can be understood, quantified and related to the stellar proper motions. DIA photometry supplemented with a simple model offers an effective and easy way to detect high proper motion stars in very dense stellar fields, where conventional astrometric searches are extremely inefficient.     

A mapping for the study of the 1/1 resonance in a galactic type Hamiltonian

We derive an algebraic mapping for an autonomous, two-dimensional galactic type Hamiltonian in the 1/1 resonance case. We use the mapping to study the stability of the periodic orbits. Using the x — p _x Poincaré surface section, we compare the results of the mapping with those found by the numerical integration of the full equations of motion. For small values of the perturbation the results of the two methods are in very good agreement while satisfactory agreement is obtained for larger perturbations.     

A non-adiabatic analysis for axisymmetric pulsations of magnetic stars

This paper presents the results of a non-adiabatic analysis for axisymmetric non-radial pulsations including the effect of a dipole magnetic field. Convection is assumed to be suppressed in the stellar envelope, and the diffusion approximation is used to radiative transport. As in a previous adiabatic analysis, the eigenfunctions are expanded in a series of spherical harmonics. The analysis is applied to a  1.9-M_⊙  , main-sequence model  (log  T _eff= 3.913)  . The presence of a magnetic field always stabilizes low-order acoustic modes. All the low-order modes of the model that are excited by the κ-mechanism in the He  ii ionization zone in the absence of a magnetic field are found to be stabilized if the polar strength of the dipole magnetic field is larger than about 1 kG. For high-order p modes, on the other hand, distorted dipole and quadrupole modes excited by the κ-mechanism in the H ionization zone remain overstable, even in the presence of a strong magnetic field. It is found, however, that all the distorted radial high-order modes are stabilized by the effect of the magnetic field. Thus, our non-adiabatic analysis suggests that distorted dipole modes and distorted quadrupole modes are most likely excited in rapidly oscillating Ap stars. The latitudinal amplitude dependence is found to be in reasonable agreement with the observationally determined one for HR 3831. Finally, the expected amplitude of magnetic perturbations at the surface is found to be very small.     

Evidence for Activity of Capella

Results of observations of Capella in the infrared He I 10830 line obtained with the Shajn 2.6 m telescope of the Crimean Astrophysical Observatory over 10 years are analyzed. The He I equivalent width varies with the 104-day orbital period, and the amplitude of these variations in September 1993-March 1994 was significantly lower than during our observations in 1985-1988. A comparison of these observations with those obtained by us since 1985 suggests the presence of an activity cycle with a period of at least 5 years. The main He I absorption forms in the chromosphere of the quiescent G6 giant, but near the maximum of the activity cycle, helium absorption forming on the more active, hotter F9 giant becomes observable. These helium data, together with the EUV results for the high-temperature Fe XX-Fe XXIII lines can be explained using a model in which an enhanced magnetized stellar wind originating from the hot corona on the active F9 giant reaches the Lagrangian point and then forms a shock wave in the corona of the quiescent G6 giant.     

A Symplectic Mapping Model for the Study of 2:3 Resonant Trans-Neptunian Motion

A symplectic mapping is constructed for the study of the dynamical evolution of Edgeworth-Kuiper belt objects near the 2:3 mean motion resonance with Neptune. The mapping is six-dimensional and is a good model for the Poincaré map of the real system, that is, the spatial elliptic restricted three-body problem at the 2:3 resonance, with the Sun and Neptune as primaries. The mapping model is based on the averaged Hamiltonian, corrected by a semianalytic method so that it has the basic topological properties of the phase space of the real system both qualitatively and quantitatively. We start with two dimensional motion and then we extend it to three dimensions. Both chaotic and regular motion is observed, depending on the objects' initial inclination and phase. For zero inclination, objects that are phase-protected from close encounters with Neptune show ordered motion even at eccentricities as large as 0.4 and despite being Neptune-crossers. On the other hand, not-phase-protected objects with eccentricities greater than 0.15 follow chaotic motion that leads to sudden jumps in their eccentricity and are removed from the 2:3 resonance, thus becoming short period comets. As inclination increases, chaotic motion becomes more widespread, but phase-protection still exists and, as a result, stable motion appears for eccentricities up to e = 0.3 and inclinations as high as i = 15°, a region where plutinos exist.