arXiv subjects

Ivan I. Shevchenko

Publications and source records attributed to Ivan I. Shevchenko.

At least 19 recordsLinked to original sources

arXiv2026

Melnikov-Arnold integrals and optimal normal forms

The Melnikov-Arnold integrals (MA-integrals) is a well-known instrument used to measure the splitting of separatrices in Hamiltonian systems. In this article, we explore how calculation of MA-integrals can be used as well to estimate sizes of secondary resonances. Within the standard map model, we show how the newly developed MA-based procedure allows one to estimate the sizes of secondary resonances of any order (up to the order of the optimal normal form), without relying on the cumbersome traditional normalization procedure.

nlin.CD↗

arXiv2025

Dynamical entropy of the separatrix map

We calculate the maximum Lyapunov exponent of the motion in the separatrix map's chaotic layer, along with calculation of its width, as functions of the adiabaticity parameter $λ$. The separatrix map is set in natural variables; and the case of the layer's least perturbed border is considered, i.~e., the winding number of the layer's border (the last invariant curve) is the golden mean. Although these two dependences (for the Lyapunov exponent and the layer width) are strongly non-monotonous and evade any simple analytical description, the calculated dynamical entropy $h$ turns out to be a close-to-linear function of $λ$. In other words, if normalized by $λ$, the entropy is a quasi-constant. We discuss whether the function $h(λ)$ can be in fact exactly linear, $h \propto λ$. The function $h(λ)$ forms a basis for calculating the dynamical entropy for any perturbed nonlinear resonance in the first fundamental model, as soon as the corresponding Melnikov--Arnold integral is estimated.

nlin.CD↗

arXiv2024

Lyapunov exponents in fundamental models of nonlinear resonance

The problem of analytical estimation of the Lyapunov exponents and Lyapunov timescales of the motion in multiplets of interacting nonlinear resonances is considered. To this end, we elaborate a unified framework, based on the separatrix map theory, which incorporates both an earlier approach for the first fundamental model of perturbed resonance (given by the perturbed pendulum Hamiltonian) and a new one for its second fundamental model (given by the perturbed Andoyer Hamiltonian). Within this framework, new accurate estimates for the Lyapunov timescales of the inner and outer subsystems of the Solar planetary system are presented and discussed.

astro-ph.EP↗

arXiv2024

Evolution of Planetary Chaotic Zones in Planetesimal Disks

Extensive numerical experiments on the long-term dynamics of planetesimal disks with planets in systems of single stars have been carried out. The planetary chaotic zone clearing timescales $T_\mathrm{cl}$ as a function of mass parameter $μ$ (planet-star mass ratio) have been determined numerically with a high accuracy separately for the outer and inner parts of the chaotic zone. Diffusional components $\propto μ^{-6/7}$ and $\propto μ^{-2}$ have been revealed in the dependence $T_\mathrm{cl}(μ)$. The results obtained are discussed and interpreted in light of existing analytical theories based on the mean motion resonance overlap criterion and in comparison with previous numerical approaches to the problem.

astro-ph.EP↗

arXiv2024

Rendez-vous with massive interstellar objects, as triggers of destabilisation

We study how close passages of interstellar objects of planetary and substellar masses may affect the immediate and long-term dynamics of the Solar system. We consider two nominal approach orbits, namely, the orbits of actual interstellar objects 1I/'Oumuamua and 2I/Borisov, assuming them to be typical or representative for interstellar swarms of matter. Thus, the nominal orbits of the interloper in our models cross the inner part of the Solar system. Series of massive numerical experiments are performed, in which the interloper's mass is varied with a small step over a broad range. We find that, even if a Jovian-mass interloper does not experience close encounters with the Solar system planets (and this holds for our nominal orbits), our planetary system can be destabilised on timescales as short as several million years. In what concerns substellar-mass interlopers (free-floating brown dwarfs), an immediate (on a timescale of $\sim 10 - 100$ yr) consequence of such a MISO flyby is a sharp increase in the orbital eccentricities and inclinations of the outer planets. On an intermediate timescale ($\sim 10^3 - 10^5$ yr after the MISO flyby), Uranus or Neptune can be ejected from the system, as a result of their mutual close encounters and encounters with Saturn. On a secular timescale ($\sim 10^6 - 10^7$ yr after the MISO flyby), the perturbation wave formed by secular planetary interactions propagates from the outer Solar system to its inner zone.

astro-ph.EP↗

arXiv2023

Azimuthal patterns in planetesimal circumstellar disks

Ways of formation of azimuthal resonant patterns in circumstellar planetesimal disks with planets are considered. Our analytical estimates and massive numerical experiments show that the disk particles that initially reside in zones of low-order mean-motion resonances with the planet may eventually concentrate into potentially observable azimuthal patterns. The structuring process is rapid, usually taking ~100 orbital periods of the planet. It is found that the relative number of particles that retain their resonant position increases with decreasing the mass parameter $μ$ (the ratio of masses of the perturbing planet and the parent star), but a significant fraction of the particle population is always removed from the disk due to accretion of the particles onto the star and planet, as well as due to their transition to highly elongated and hyperbolic orbits. Expected radio images of azimuthally structured disks are constructed. In the considered models, azimuthal patterns associated with the 2:1 and 3:2 resonances are most clearly manifested; observational manifestations of the 1:2 and 2:3 resonances are also possible.

astro-ph.EP↗

arXiv2023

Dynamical environments of (486958) Arrokoth: prior evolution and present state

We consider dynamical environments of (486958) Arrokoth, focusing on both their present state and their long-term evolution, starting from the KBO's formation. Both analytical (based on an upgraded Kepler-map formalism) and numerical (based on massive simulations and construction of stability diagrams in the 3D setting of the problem) approaches to the problem are used. The debris removal is due to either absorption by the KBO or by leaving the Hill sphere; the interplay of these processes is considered. The clearing mechanisms are explored, and the debris removal timescales are estimated. We assess survival opportunities for any debris orbiting around Arrokoth. The generic chaotization of Arrokoth's circumbinary debris disk's inner zone and generic cloudization of the disk's periphery, which is shown to be essential in the general 3D case, naturally explains the current absence of any debris in its vicinities.

astro-ph.EP↗

arXiv2022

Extended planetary chaotic zones

We consider the chaotic motion of low-mass bodies in two-body high-order mean-motion resonances with planets in model planetary systems, and analytically estimate the Lyapunov and diffusion timescales of the motion in multiplets of interacting subresonances corresponding to the mean-motion resonances. We show that the densely distributed (though not overlapping) high-order mean-motion resonances, when certain conditions on the planetary system parameters are satisfied, may produce extended planetary chaotic zones -- "zones of weak chaotization," -- much broader than the well-known planetary connected chaotic zone, the Wisdom gap. This extended planetary chaotic zone covers the orbital range between the 2/1 and 1/1 resonances with the planet. On the other hand, the orbital space inner (closer to the host star) with respect to the 2/1 resonance location is essentially long-term stable. This difference arises because the adiabaticity parameter of subresonance multiplets specifically depends on the particle's orbit size. The revealed effect may control the structure of planetesimal disks in planetary systems: the orbital zone between the 2/1 and 1/1 resonances with a planet should be normally free from low-mass material (only that occasionally captured in the first-order 3/2 or 4/3 resonances may survive); whereas any low-mass population inner to the 2/1 resonance location should be normally long-lived (if not perturbed by secular resonances, which we do not consider in this study).

astro-ph.EP↗

arXiv2020

Dynamical environments of MU69 and similar objects

We explore properties of the long-term dynamics of particles (moonlets, fragments, debris etc.) around KBO 2014 MU69 (Arrokoth), as well as around similar contact-binary objects potentially present in the Kuiper belt. The chaotic diffusion of particles inside the Hill sphere of MU69 (or, generally, a similar object) is studied by means of construction of appropriate stability diagrams and by application of analytical approaches generally based on the Kepler map theory. The formation and evolution of the particle clouds, due to the chaotic diffusion inside the Hill sphere, are studied and the cloud lifetimes are estimated.

astro-ph.EP↗

arXiv2019

Dynamical environments of relativistic binaries: The phenomenon of resonance shifting

In this article, we explore both numerically and analytically how the dynamical environments of mildly relativistic binaries evolve with increasing the general relativity factor $γ$ (the normalized inverse of the binary size measured in the units of the gravitational radius corresponding to the total mass of the system). Analytically, we reveal a phenomenon of the relativistic shifting of mean-motion resonances: on increasing $γ$, the resonances between the test particle and the central binary shift, due to the relativistic variation of the mean motions of the primary and secondary binaries and the relativistic advance of the tertiary's pericenter. To exhibit the circumbinary dynamics globally, we numerically integrate equations of the circumbinary motion of a test particle, and construct relevant scans of the maximum Lyapunov exponents and stability diagrams in the "pericentric distance -- eccentricity" plane of initial conditions. In these scans and diagrams, regular and chaotic domains are identified straightforwardly. Our analytical and numerical estimates of the shift size are in a good agreement. Prospects for identification of the revealed effect in astronomical observations are discussed.

astro-ph.EP↗

arXiv2019

Massive evaluation and analysis of Poincaré recurrences on grids of initial data: a tool to map chaotic diffusion

We present a novel numerical method aimed to characterize global behaviour, in particular chaotic diffusion, in dynamical systems. It is based on an analysis of the Poincaré recurrence statistics on massive grids of initial data or values of parameters. We concentrate on Hamiltonian systems, featuring the method separately for the cases of bounded and non-bounded phase spaces. The embodiments of the method in each of the cases are specific. We compare the performances of the proposed Poincaré recurrence method (PRM) and the custom Lyapunov exponent (LE) methods and show that they expose the global dynamics almost identically. However, a major advantage of the new method over the known global numerical tools, such as LE, FLI, MEGNO, and FA, is that it allows one to construct, in some approximation, charts of local diffusion timescales. Moreover, it is algorithmically simple and straightforward to apply.

nlin.CD↗

arXiv2018

Tidal decay of circumbinary planetary systems

It is shown that circumbinary planetary systems are subject to universal tidal decay (shrinkage of orbits), caused by the forced orbital eccentricity inherent to them. Circumbinary planets (CBP) are liberated from parent systems, when, owing to the shrinkage, they enter the circumbinary chaotic zone. On shorter timescales (less than the current age of the Universe), the effect may explain, at least partially, the observed lack of CBP of close-enough (with periods < 5 days) stellar binaries; on longer timescales (greater than the age of the Universe but well within stellar lifetimes), it may provide massive liberation of chemically evolved CBP. Observational signatures of the effect may comprise (1) a prevalence of large rocky planets (super-Earths) in the whole population of rogue planets (if this mechanism were the only source of rogue planets); (2) a mass-dependent paucity of CBP in systems of low-mass binaries: the lower the stellar mass, the greater the paucity.

astro-ph.EP↗

arXiv2017

Chaotic dynamics around cometary nuclei

We apply a generalized Kepler map theory to describe the qualitative chaotic dynamics around cometary nuclei, based on accessible observational data for five comets whose nuclei are well-documented to resemble dumb-bells. The sizes of chaotic zones around the nuclei and the Lyapunov times of the motion inside these zones are estimated. In the case of Comet 1P/Halley, the circumnuclear chaotic zone seems to engulf an essential part of the Hill sphere, at least for orbits of moderate to high eccentricity.

astro-ph.EP↗

arXiv2017

Habitability properties of circumbinary planets

It is shown that several habitability conditions (in fact, at least seven such conditions) appear to be fulfilled automatically by circumbinary planets of main-sequence stars (CBP-MS), whereas on Earth these conditions are fulfilled only by chance. Therefore, it looks natural that most of the production of replicating biopolymers in the Galaxy is concentrated on particular classes of CBP-MS, and life on Earth is an outlier, in this sense. In this scenario, Lathe's mechanism for the tidal "chain reaction" abiogenesis on Earth is favored as generic for CBP-MS, due to photo-tidal synchronization inherent to them. Problems with this scenario are discussed in detail.

astro-ph.EP↗

arXiv2017

Chaotic zones around rotating small bodies

Small bodies of the Solar system, like asteroids, trans-Neptunian objects, cometary nuclei, planetary satellites, with diameters smaller than one thousand kilometers usually have irregular shapes, often resembling dumb-bells, or contact binaries. The spinning of such a gravitating dumb-bell creates around it a zone of chaotic orbits. We determine its extent analytically and numerically. We find that the chaotic zone swells significantly if the rotation rate is decreased, in particular, the zone swells more than twice if the rotation rate is decreased ten times with respect to the "centrifugal breakup" threshold. We illustrate the properties of the chaotic orbital zones in examples of the global orbital dynamics about asteroid 243 Ida (which has a moon, Dactyl, orbiting near the edge of the chaotic zone) and asteroid 25143 Itokawa.

astro-ph.EP↗

arXiv2016

Three-lane and multi-lane signatures of planets in planetesimal disks

In massive numerical experiments we show that a planet embedded in a planetesimal disk induces a characteristic multi-lane "planetosignature" representing a pattern of several stellar-centric rings. If the planet's mass is large enough, the multi-lane signature degenerates to a three-lane one: then it consists of three rings, one bright coorbital with the planet, and two dark gaps in the radial distribution of the particles. The gaps correspond to orbital resonances 2:1 and 1:2 with the planet. This theoretical prediction may explain recent ALMA observations of the disk of HL Tau.

astro-ph.EP↗

arXiv2016

Marginal resonances and intermittent behaviour in the motion in the vicinity of a separatrix

A condition upon which sporadic bursts (intermittent behaviour) of the relative energy become possible is derived for the motion in the chaotic layer around the separatrix of non-linear resonance. This is a condition for the existence of a marginal resonance, i.e. a resonance located at the border of the layer. A separatrix map in Chirikov's form [Chirikov, B. V., Phys. Reports 52, 263 (1979)] is used to describe the motion. In order to provide a straightforward comparison with numeric integrations, the separatrix map is synchronized to the surface of the section farthest from the saddle point. The condition of intermittency is applied to clear out the nature of the phenomenon of bursts of the eccentricity of chaotic asteroidal trajectories in the 3/1 mean motion commensurability with Jupiter. On the basis of the condition, a new intermittent regime of resonant asteroidal motion is predicted and then identified in numeric simulations.

nlin.CD↗

arXiv2016

Numeric Deduction in Symbolic Computation. Application to Normalizing Transformations

Algorithms of numeric (in exact arithmetic) deduction of analytical expressions, proposed and described by Shevchenko and Vasiliev (1993), are developed and implemented in a computer algebra code. This code is built as a superstructure for the computer algebra package by Shevchenko and Sokolsky (1993a) for normalization of Hamiltonian systems of ordinary differential equations, in order that high complexity problems of normalization could be solved. As an example, a resonant normal form of a Hamiltonian describing the hyperboloidal precession of a dynamically symmetric satellite is derived by means of the numeric deduction technique. The technique provides a considerable economy, about 30 times in this particular application, in computer memory consumption. It is naturally parallelizable. Thus the economy of memory consumption is convertible into a gain in computation speed.

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