Pavel Krasil'nikov
Publications:
Krasil'nikov P. S., Ismagilov A. R.
On the DumbBell Equilibria in the Generalized Sitnikov Problem
2022, Vol. 18, no. 4, pp. 577588
Abstract
This paper discusses and analyzes the dumb–bell equilibria in a generalized Sitnikov problem. This has been done by assuming that the dumb–bell is oriented along the normal to the plane of motion of two primaries. Assuming the orbits of primaries to be circles, we apply bifurcation theory to investigate the set of equilibria for both symmetrical and asymmetrical dumb–bells. We also investigate the linear stability of the trivial equilibrium of a symmetrical dumb–bell in the elliptic Sitnikov problem. In the case of the dumb–bell length $l\geqslant 0.983819$, an instability of the trivial equilibria for eccentricity $e \in (0,\,1)$ is proved. 
Krasil'nikov P. S., Maiorov A. Y.
On Ziegler Effect with Arbitrary Large Viscous Friction
2022, Vol. 18, no. 2, pp. 161170
Abstract
We study a mechanical system with two degrees of freedom simulating the motion of rotor
blades on an elastic bushing of a mediumsized helicopter. For small values of some problem
parameters, the destabilizing effect due to small linear viscous friction forces has been studied
earlier. Here we study the problem with arbitrary large friction forces for arbitrary values of
the problem parameters. In the plane of parameters, the regions of asymptotic stability and
instability are calculated. As a result, necessary and sufficient conditions for the existence of
a destabilizing effect under the action of potential, follower forces and arbitrary friction forces
have been obtained. It is shown that, if some critical friction coefficient $k_*$ tends to infinity, then
there exists a Ziegler area with arbitrarily large dissipative forces.

Krasil'nikov P. S.
Hill’s curves and libration points in the lowthrust restricted circular threebody problem
2017, Vol. 13, No. 4, pp. 543–556
Abstract
The plane circular restricted threebody problem is considered, where the massless body is a constant lowthrust spacecraft. It is assumed that the vector of lowthrust is directed along the $Ox$ axis connecting the main bodies. The problem of plotting a family of oneparameter Hill’s curves is investigated. The existence conditions of artificial triangulartype and collineartype libration points are obtained. The values of the effective force function at libration points are investigated also. Six different topological types of the family of oneparameter Hill’s curves are described. It is shown that these types differ in the number of critical values of the constant Jacobi integral and in the ordering of these values. For the Earth – Moon system, a family of oneparameter Hill’s curves is plotted for each of the six types.

Rodnikov A. V., Krasil'nikov P. S.
On spacial motions of an orbital tethered system
2017, Vol. 13, No. 4, pp. 505–518
Abstract
We study motions of a particle along a rope with ends fixed to an extended rigid body whose center of mass traces out a circular orbit in the central Newtonian force field. (Such a rope is called a tether.) We assume that the tether realizes an ideal unilateral constraint. We derive particle motion equations on the surface of the ellipsoid, which restricts the particle motion, and conditions that guarantee such motions. We also study the existence and stability of relative equilibria of the particle with respect to the orbital frame of reference. We prove stability of the integral manifold of the particle motions in the plane of the orbit. We note that smallamplitude librations near this manifold can be described by approximate equations that can be reduced to Riccati’s equation. We establish that generally the spacial motions of the particle are chaotic for initial conditions from some vicinity of the separatrix motion in the plane of the orbit and are regular in other cases. We also note that chaotic motions usually lead to a situation where the particle comes off the constraint, in other words, to motions inside the abovementioned ellipsoid.

Krasil'nikov P. S., Amelin R. N.
On the rotation of Mars around its center of mass under the action of gravity the sun, Jupiter and Earth.
2015, Vol. 11, No. 2, pp. 329342
Abstract
The Mars rotation under the action of gravity torque from the Sun, Jupiter, Earth is considered. It is assumed that Mars is the axially symmetric rigid body $(A = B)$, the orbits of Mars, Earth and Jupiter are Kepler ellipses. Elliptical mean motions of Earth and Jupiter are the independent small parameters. The averaged Hamiltonian of problem and integrals of evolution equations are obtained. By assumption that the equatorial plane of unit sphere parallel to the plane of Jupiter orbit, the set of trajectories for angular momentum vector of Mars ${\bf I}_2$ is drawn. It is well known that “classic” equilibriums of vector ${\bf I}_2$ belong to the normal to the Mars orbit plane. It is shown that they are saved by the action of gravitational torque of Jupiter and Earth. Besides that there are two new stationary points of ${\bf I}_2$ on the normal to the Jupiter orbit plane. These equilibriums are unstable, homoclinic trajectories pass through them. In addition, there are a pair of unstable equilibriums on the great circle that is parallel to the Mars orbit plane. Four heteroclinic curves pass through these equilibriums. There are two stable equilibriums of ${\bf I}_2$ between pairs of these curves. 
Kalas V. O., Krasil'nikov P. S.
On the investigation of stability of equilibrium in Sitnikov problem in nonlinear formulation
2015, Vol. 11, No. 1, pp. 117126
Abstract
With regard to nonlinear terms in the equations of motion, the stability of the trivial equilibrium in Sitnikov problem is investigated. For Hamilton’s equations of the problem, the mapping of phase space into itself in the time $t = 2\pi$ was constructed up to terms of third order. With the help of point mapping method, the stability of equilibrium is investigated for eccentricity from the interval $[0, 1)$. It is shown that Lyapunov stability takes place for $e \in [0, 1)$, if we exclude the
discrete sequence of values ${e_j}$ for which the trace of the monodromy matrix is equal to $\pm2$. The first and second values of the eccentricity of the specified sequence are investigated. The equilibrium is stable if $e = e_1$. Eccentricity value $e = e_2$ corresponds to degeneracy stability theorems, therefore the stability analysis requires the consideration of the terms of order higher than the third. The remaining values of eccentricity from discrete sequence have not been studied. 
Krasil'nikov P. S.
Small plane oscillations of satellite in an weakly elliptical orbit
2013, Vol. 9, No. 4, pp. 671696
Abstract
The equation of plane nonlinear oscillations of satellite in an weakly elliptical orbit is investigated. Suppose, that equation of motion contains two small parameters. Various kinds of procedure which reduce the equation to one small parameter case are investigated. Lacks of such procedure are described. New resonance effects of satellite’s rotation are described with the help of the generalized averaging method with independent small parameters.
