Vol. 14, no. 3
Vol. 14, no. 3, 2018
Vilke V. G., Shatina A. V., Osipova L. S.
Abstract
The classical $N$body problem in the case when one of the bodies (the Sun) has a much larger
mass than the rest of the mutually gravitating bodies is considered. The system of equations
in canonical Delaunay variables describing the motion of the system relative to the barycentric
coordinate system is derived via the methods of analitical dynamics. The procedure of averaging
over the fast angular variables (mean anomalies) leads to the equation describing the evolution
of a single Solar system planet’s perihelion as the sum of two terms. The first term corresponds
to the gravitational disturbances caused by the rest of the planets, as in the case of a motionless
Sun. The second appears because the problem is considered in the barycentric coordinate system
and the orbits’ inclinations are taken into account. This term vanishes if all planets are assumed
to be moving in one static plane. This term contributes substantially to the Mercury’s and
Venus’s perihelion evolutions. For the rest of the planet this term is small compared to the
first one. For example, for Mercury the values of the two terms in question were calculated
to be 528.67 and 39.64 angular seconds per century, respectively.

Kubyshkin E. P., Moriakova A. R.
Abstract
We study equilibrium states and bifurcations of periodic solutions from the equilibrium
state of the Ikeda delaydifferential equation well known in nonlinear optics. This equation was
proposed as a mathematical model of a passive optical resonator in a nonlinear environment.
The equation, written in a characteristic time scale, contains a small parameter at the derivative,
which makes it singular. It is shown that the behavior of solutions of the equation with initial
conditions from the fixed neighborhood of the equilibrium state in the phase space of the equation
is described by a countable system of nonlinear ordinary differential equations. This system
of equations has a minimal structure and is called the normal form of the equation in the
neighborhood of the equilibrium state. The system of equations allows us to pick out one
“fast” and a countable number of “slow” variables and apply the averaging method to the
system obtained. It is shown that the equilibrium states of a system of averaged equations with
“slow” variables correspond to periodic solutions of the same type of stability. The possibility of
simultaneous bifurcation of a large number of stable periodic solutions(multistability bifurcation)
is shown. With further increase in the bifurcation parameter each of the periodic solutions
becomes a chaotic attractor through a series of perioddoubling bifurcations. Thus, the behavior
of the solutions of the Ikeda equation is characterized by chaotic multistability.

Pochinka O. V., Loginova A. S., Nozdrinova E. V.
Abstract
This article presents a number of models that arise in physics, biology, chemistry, etc.,
described by a onedimensional reactiondiffusion equation. The local dynamics of such models
for various values of the parameters is described by a rough transformation of the circle. Accordingly,
the control of such dynamics reduces to the consideration of a continuous family of
maps of the circle. In this connection, the question of the possibility of joining two maps of the
circle by an arc without bifurcation points naturally arises. In this paper it is shown that any
orientationpreserving sourcesink diffeomorphism on a circle is joined by such an arc. Note that
such a result is not true for multidimensional spheres.

Nazarov V. E., Kiyashko S. B.
Abstract
On the basis of the elastic contact model of rough surfaces of solids, a quadraticallybimodular
equation of state for microinhomogeneous media containing cracks is derived.
A study is made of the propagation of elastic single unipolar pulse perturbations and bipolar
periodic waves in such media. Exact analytical solutions that describe the evolution of
initially triangular pulses and periodic sawtooth waves are obtained. A numerical and graphical
analysis of the solutions is also carried out.

Safronov A. A., Koroteev A. A., Filatov N. I., Grigoriev A. L.
Abstract
The influence of longrange interactions on the progress of heat waves in the radiationcooling
disperse flow is considered. It is shown that the system exhibits oscillations attendant
on the process of establishing an equilibrium temperature profile. The oscillation amplitude and
the rate of oscillation damping are determined. The conditions under which the radiation cooling
process can be unstable with respect to temperature field perturbations are revealed. The results
of theoretical analysis and numerical calculation of the actual droplet flow are compared.

Kevorkov S. S., Khamidullin R. K., Koroleva (Kikot) I. P., Smirnov V. V., Gusarova E. B., Manevitch L. I.
Abstract
The results of an experimental and numerical investigation of the dynamics of a string
with three uniformly distributed discrete masses are presented. This system can be used as
a resonant energy sink for protecting structural elements from the effects of undesirable dynamic
loads over a wide frequency range. Preliminary studies of the nonlinear dynamics of the system
under consideration showed its high energy capacity. In this paper, we present the results of
an experimental study in which a shaker’s table mounted cantilever beam was being protected.
As a result, the efficiency of the sink was confirmed, and data were also obtained to refine the
mathematical model. It was shown that the experimental data obtained are in good agreement
with the results of computer simulation.

Morozov A. D., Morozov K. E.
Abstract
We study the role of quasiperiodic perturbations in systems close to twodimensional Hamiltonian
ones. Similarly to the problem of the influence of periodic perturbations on a limit cycle,
we consider the problem of the passage of an invariant torus through a resonance zone. The conditions
for synchronization of quasiperiodic oscillations are established. We illustrate our results
using the Duffing –Van der Pol equation as an example.

Maslov D. A., Merkuryev I. V.
Abstract
The dynamics of a vibrating ring microgyroscope operating in the forced oscillation mode
is investigated. The elastic and viscous anisotropy of the resonator and the nonlinearity of oscillations
are taken into consideration. Additional nonlinear terms are suggested for the mathematical
model of resonator dynamics. In addition to cubic nonlinearity, nonlinearity of the fifth
degree is considered. By using the Krylov – Bogolyubov averaging method, equations containing
parameters characterizing damping, elastic and viscous anisotropy, as well as coefficients of
oscillation nonlinearity are deduced. The parameter identification problem is reduced to solving
an overdetermined system of algebraic equations that are linear in the parameters to be
identified. The proposed identification method allows testing at large oscillation amplitudes
corresponding to a sufficiently high signaltonoise ratio. It is shown that taking nonlinearities
into account significantly increases the accuracy of parameter identification in the case of large
oscillation amplitudes.

Andreev A. S., Peregudova O. A.
Abstract
In this paper, the stability and stabilization problems for nonlinear Volterra integrodifferential
equations with unlimited delay are considered. The development of the direct Lyapunov
method in the study of the limiting properties of the solutions of these equations is carried
out by using Lyapunov functionals with a semidefinite time derivative. The topological dynamics
of these equations has been constructed revealing the limiting properties of their solutions.
The assumption of the existence of a Lyapunov functional with a semidefinite time derivative
gives a more complete solution to the positive limit set localization problem. On this basis new
theorems on sufficient conditions for the asymptotic stability and instability of the zero solution
of nonlinear Volterra integrodifferential equations are proved. These theorems are applied to the
problem of the equilibrium position stability of the hereditary mechanical systems as well as the
regulation problem of the controlled mechanical systems using a proportionalintegrodifferential
controller. As an example, the regulation problem of a mobile robot with three omnidirectional
wheels and a displaced mass center is solved using the nonlinear integral controllers without
velocity measurements.

Kiselev O. M.
Abstract
We obtain criteria for the stability of fast straightline motion of a wheeled robot using
proportional or proportional derivative feedback control. The motion of fast robots with discrete
feedback control is defined by the discrete dynamical system. The stability criteria are obtained
for the discrete system for proportional and proportionalderivative feedback control.
