arXiv subjects

Ivan Bizyaev

Publications and source records attributed to Ivan Bizyaev.

5 recordsLinked to original sources

arXiv2025

Dynamics of spinning test bodies in the Schwarzschild space-time: reduction and circular orbits

This paper investigates the motion of a rotating test body in the Schwarzschild space-time. Previously, it was shown that this problem reduces to investigating a two-dimensional Poincare map. The paper presents a detailed analysis of bifurcations of periodic solutions using this map. In the Poincare map, as the energy of the body increases, one can observe two pitchfork bifurcations that follow one after the other: a supercritical and a subcritical one. This gives rise to five fixed points in the Poincare map. In addition, new circular orbits are found for which the total angular momentum is not parallel to the angular momentum of the test body. For these circular orbits, the radial coordinate satisfies the condition r>3 (in units of the mass of a black hole). For values of the total angular momentum of the test body that corresponds to neutron stars or black holes, these asymmetric circular orbits turn out to be unstable.

math.DS↗

arXiv2025

Trajectories of light beams in a Kerr metric: the influence of the rotation of an observer on the shadow of a black hole

This paper investigates the trajectories of light beams in a Kerr metric, which describes the gravitational field in the neighborhood of a rotating black hole. After reduction by cyclic coordinates, this problem reduces to analysis of a Hamiltonian system with two degrees of freedom. A bifurcation diagram is constructed and a classification is made of the types of trajectories of the system according to the values of first integrals. Relations describing the boundary of the shadow of the black hole are obtained for a stationary observer who rotates with an arbitrary angular velocity about the axis of rotation of the black hole.

math.DS↗

arXiv2024

Classification of the trajectories of uncharged particles in the Schwarzschild-Melvin metric

This paper investigates the trajectories of neutral particles in the Schwarzschild-Melvin spacetime. After reduction by cyclic coordinates this problem reduces to investigating a two-degree-of-freedom Hamiltonian system that has no additional integral. A classification of regions of possible motion of a particle is performed according to the values of the momentum and energy integrals. Bifurcations of periodic solutions of the reduced system are analyzed using a Poincare map.

math.DS↗

arXiv2024

Dynamics of a multilink wheeled vehicle: partial solutions and unbounded speedup

A mathematical model featuring the motion of a multilink wheeled vehicle is developed using a nonholonomic model. A detailed analysis of the inertial motion is made. Fixed points of the reduced system are identified, their stability is analyzed, and invariant manifolds are found. For the case of three platforms (links), a phase portrait for motion on an invariant manifold is shown and trajectories of the attachment points of the wheel pairs of the three-link vehicle are presented. In addition, an analysis is made of motion in the case where the leading platform has a rotor whose angular velocity is a periodic function of time. The existence of trajectories for which one of the velocity components increases without bound is established, and the asymptotics for it is found.

math.DS↗

arXiv2018

The Chaplygin sleigh with parametric excitation: chaotic dynamics and nonholonomic acceleration

This paper is concerned with the Chaplygin sleigh with timevarying mass distribution (parametric excitation). The focus is on the case where excitation is induced by a material point that executes periodic oscillations in a direction transverse to the plane of the knife edge of the sleigh. In this case, the problem reduces to investigating a reduced system of two first-order equations with periodic coefficients, which is similar to various nonlinear parametric oscillators. Depending on the parameters in the reduced system, one can observe different types of motion, including those accompanied by strange attractors leading to a chaotic (diffusion) trajectory of the sleigh on the plane. The problem of unbounded acceleration (an analog of Fermi acceleration) of the sleigh is examined in detail. It is shown that such an acceleration arises due to the position of the moving point relative to the line of action of the nonholonomic constraint and the center of mass of the platform. Various special cases of existence of tensor invariants are found.

nlin.CD↗