arXiv subjects

Ivan I. Shevchenko

Publications and source records attributed to Ivan I. Shevchenko.

29 records · Page 2Linked to original sources

arXiv2016

On the recurrence and Lyapunov time scales of the motion near the chaos border

Conditions for the emergence of a statistical relationship between $T_r$, the chaotic transport (recurrence) time, and $T_L$, the local Lyapunov time (the inverse of the numerically measured largest Lyapunov characteristic exponent), are considered for the motion inside the chaotic layer around the separatrix of a nonlinear resonance. When numerical values of the Lyapunov exponents are measured on a time interval not greater than $T_r$, the relationship is shown to resemble the quadratic one. This tentatively explains numerical results presented in the literature.

nlin.CD↗

arXiv2016

On the maximum Lyapunov exponent of the motion in a chaotic layer

The maximum Lyapunov exponent (referred to the mean half-period of phase libration) of the motion in the chaotic layer of a nonlinear resonance subject to symmetric periodic perturbation, in the limit of infinitely high frequency of the perturbation, has been numerically estimated by two independent methods. The newly derived value of this constant is 0.80, with precision presumably better than 0.01.

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arXiv2015

Spiral patterns in planetesimal circumbinary disks

Planet formation scenarios and the observed planetary dynamics in binaries pose a number of theoretical challenges, especially in what concerns circumbinary planetary systems. We explore the dynamical stirring of a planetesimal circumbinary disk in the epoch when the gas component disappears. For this purpose, following theoretical approaches by Heppenheimer (1978) and Moriwaki and Nakagawa (2004), we develop a secular theory for the dynamics of planetesimals in circumbinary disks. If the binary is eccentric and its components have unequal masses, a spiral density wave is generated, engulfing the disk on the secular timescale, which may exceed 10^7 yr, depending on the problem parameters. The spiral pattern is transient; thus, its observed presence may betray system's young age. We explore the pattern both analytically and in numerical experiments. The derived analytical spiral is a modified lituus; it matches the numerical density wave in the gas-free case perfectly. Using the SPH scheme, we explore the effect of residual gas on the wave propagation.

astro-ph.EP↗

arXiv2015

Chaotic zones around gravitating binaries

The extent of the continuous zone of chaotic orbits of a small-mass tertiary around a system of two gravitationally bound primaries (a double star, a double black hole, a binary asteroid, etc.) is estimated analytically, in function of the tertiary's orbital eccentricity. The separatrix map theory is used to demonstrate that the central continuous chaos zone emerges (above a threshold in the primaries mass ratio) due to overlapping of the orbital resonances corresponding to the integer ratios p:1 between the tertiary and the central binary periods. In this zone, the unlimited chaotic orbital diffusion of the tertiary takes place, up to its ejection from the system. The primaries mass ratio, above which such a chaotic zone is universally present at all initial eccentricities of the tertiary, is estimated. The diversity of the observed orbital configurations of biplanetary and circumbinary exosystems is shown to be in accord with the existence of the primaries mass parameter threshold.

astro-ph.EP↗

arXiv2013

The Kepler map in the three-body problem

The Kepler map was derived by Petrosky (1986) and Chirikov and Vecheslavov (1986) as a tool for description of the long-term chaotic orbital behaviour of the comets in nearly parabolic motion. It is a two-dimensional area-preserving map, describing the motion of a comet in terms of energy and time. Its second equation is based on Kepler's third law, hence the title of the map. Since 1980s the Kepler map has become paradigmatic in a number of applications in celestial mechanics and atomic physics. It represents an important kind of general separatrix maps. Petrosky and Broucke (1988) used refined methods of mathematical physics to derive analytical expressions for its single parameter. These methods became available only in the second half of the 20th century, and it may seem that the map is inherently a very modern mathematical tool. With the help of the Jacobi integral I show that the Kepler map, including analytical formulae for its parameter, can be derived by quite elementary methods. The prehistory and applications of the Kepler map are considered and discussed.

nlin.CD↗

arXiv2013

The width of a chaotic layer

A model of nonlinear resonance as a periodically perturbed pendulum is considered, and a new method of analytical estimating the width of a chaotic layer near the separatrices of the resonance is derived for the case of slow perturbation (the case of adiabatic chaos). The method turns out to be successful not only in the case of adiabatic chaos, but in the case of intermediate perturbation frequencies as well.

nlin.CD↗

arXiv2013

Width of the chaotic layer: maxima due to marginal resonances

Modern theoretical methods for estimating the width of the chaotic layer in presence of prominent marginal resonances are considered in the perturbed pendulum model of nonlinear resonance. The fields of applicability of these methods are explicitly and precisely formulated. The comparative accuracy is investigated in massive and long-run numerical experiments. It is shown that the methods are naturally subdivided in classes applicable for adiabatic and non-adiabatic cases of perturbation. It is explicitly shown that the pendulum approximation of marginal resonance works good in the non-adiabatic case. In this case, the role of marginal resonances in determining the total layer width is demonstrated to diminish with increasing the main parameter λ(equal to the ratio of the perturbation frequency to the frequency of small-amplitude phase oscillations on the resonance). Solely the "bending effect" is important in determining the total amplitude of the energy deviations of the near-separatrix motion at λ> 7. In the adiabatic case, it is demonstrated that the geometrical form of the separatrix cell can be described analytically quite easily by means of using a specific representation of the separatrix map. It is shown that the non-adiabatic (and, to some extent, intermediary) case is most actual, in comparison with the adiabatic one, for the physical or technical applications that concern the energy jumps in the near-separatrix chaotic motion.

nlin.CD↗

arXiv2011

Lyapunov and diffusion timescales in the solar neighborhood

We estimate the Lyapunov times (characteristic times of predictability of motion) in Quillen's models for the dynamics in the solar neighborhood. These models take into account perturbations due to the Galactic bar and spiral arms. For estimating the Lyapunov times, an approach based on the separatrix map theory is used. The Lyapunov times turn out to be typically of the order of 10 Galactic years. We show that only in a narrow range of possible values of the problem parameters the Galactic chaos is adiabatic; usually it is not slow. We also estimate the characteristic diffusion times in the chaotic domain. In a number of models, the diffusion times turn out to be small enough to permit migration of the Sun from the inner regions of the Milky Way to its current location. Moreover, due to the possibility of ballistic flights inside the chaotic layer, the chaotic mixing might be even far more effective and quicker than in the case of normal diffusion. This confirms the dynamical possibility of Minchev and Famaey's migration concept.

astro-ph.GA↗

arXiv2010

Hamiltonian intermittency and Lévy flights in the three-body problem

We consider statistics of the disruption and Lyapunov times in an hierarchical restricted three-body problem. We show that at the edge of disruption the orbital periods and the size of the orbit of the escaping body exhibit Lévy flights. Due to them, the time decay of the survival probability is heavy-tailed with the power-law index equal to -2/3, while the relation between the Lyapunov and disruption times is quasilinear. Applicability of these results in an "hierarchical resonant scattering" setting for a three-body interaction is discussed.

astro-ph.EP↗

arXiv2010

How do the small planetary satellites rotate?

We investigate the problem of the typical rotation states of the small planetary satellites from the viewpoint of the dynamical stability of their rotation. We show that the majority of the discovered satellites with unknown rotation periods cannot rotate synchronously, because no stable synchronous 1:1 spin-orbit state exists for them. They rotate either much faster than synchronously (those tidally unevolved) or, what is much less probable, chaotically (tidally evolved objects or captured slow rotators).

astro-ph.EP↗

arXiv2008

On reverberation and cross-correlation estimates of the size of the broad-line region in active galactic nuclei

It is known that the dependence of the emission-line luminosity of a typical cloud in the active galactic nuclei broad-line regions (BLRs) upon the incident flux of ionizing continuum can be nonlinear. We study how this nonlinearity can be taken into account in estimating the size of the BLR by means of the "reverberation" methods. We show that the BLR size estimates obtained by cross-correlation of emission-line and continuum light curves can be much (up to an order of magnitude) less than the values obtained by reverberation modelling. This is demonstrated by means of numerical cross-correlation and reverberation experiments with model continuum flares and emission-line transfer functions and by means of practical reverberation modelling of the observed optical spectral variability of NGC 4151. The time behaviour of NGC 4151 in the H_alpha and H_beta lines is modelled on the basis of the observational data by Kaspi et al. (1996, ApJ, 470, 336) and the theoretical BLR model by Shevchenko (1984, Sov. Astron. Lett., 10, 377; 1985, Sov. Astron. Lett., 11, 35). The values of the BLR parameters are estimated that allow to judge on the size and physical characteristics of the BLR. The small size of the BLR, as determined by the cross-correlation method from the data of Kaspi et al. (1996, ApJ, 470, 336), is shown to be an artifact of this method. So, the hypothesis that the BLR size varies in time is not necessitated by the observational data.

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