Vol. 19, no. 3
Vol. 19, no. 3, 2023
Diyorov A. M., Rozikov U. A.
Abstract
In this short note we study a dynamical system generated by a twoparametric quadratic
operator mapping a 3dimensional simplex to itself. This is an evolution operator of the frequencies
of gametes in a twolocus system. We find the set of all (a continuum set of) fixed points and
show that each fixed point is nonhyperbolic. We completely describe the set of all limit points of
the dynamical system. Namely, for any initial point (taken from the 3dimensional simplex) we
find an invariant set containing the initial point and a unique fixed point of the operator, such
that the trajectory of the initial point converges to this fixed point.

Fakhretdinov M. I., Samsonov K. Y., Dmitriev S. V., Ekomasov E. G.
Abstract
The $\varphi^4$ theory is widely used in many areas of physics, from cosmology and elementary
particle physics to biophysics and condensed matter theory. Topological defects, or kinks, in
this theory describe stable, solitary wave excitations. In practice, these excitations, as they
propagate, necessarily interact with impurities or imperfections in the onsite potential. In this
work, we focus on the effect of the length and strength of a rectangular impurity on the kink
dynamics. It is found that the interaction of a kink with an extended impurity is qualitatively
similar to the interaction with a wellstudied point impurity described by the delta function,
but significant quantitative differences are observed. The interaction of kinks with an extended
impurity described by a rectangular function is studied numerically. All possible scenarios of
kink dynamics are determined and described, taking into account resonance effects. The inelastic
interaction of the kink with the repulsive impurity arises only at high initial kink velocities. The
dependencies of the critical and resonant velocities of the kink on the impurity parameters are
found. It is shown that the critical velocity of the repulsive impurity passage is proportional to
the square root of the barrier area, as in the case of the sineGordon equation with an impurity.
It is shown that the resonant interaction in the $\varphi^4$ model with an attracting extended impurity,
as well as for the case of a point impurity, in contrast to the case of the sineGordon equation, is
due to the fact that the kink interacts not only with the impurity mode, but also with the kink’s
internal mode. It is found that the dependence of the kink final velocity on the initial one has
a large number of resonant windows.

Frick P., Shestakov A.
Abstract
Developed turbulent flows in which the intervention of external forces is fundamentally
important at scales where the inertial range should exist are quite common. Then the cascade
processes are not conservative any more and, therefore, it is necessary to adequately describe
the external forces acting in the whole range of scales. If the work of these forces has a power
law scaling, then one can assume that the integral of motion changes and the preserving value
becomes a quadratic quantity, which includes the dependence on the scale. We develop this idea
within the framework of shell models of turbulence. We show that, in terms of nonconservative
cascades, one can describe various situations, including (as a particular case) the Obukhov – Bolgiano scaling proposed for turbulence in a stratified medium and for helical turbulence with
a helicity injection distributed along the spectrum.

Meloni F., Nakamura G., Grammaticos B., Martinez A. S., Badoual M.
Abstract
The farreaching consequences of ecological interactions in the dynamics of biological communities
remain an intriguing subject. For decades, competition has been a cornerstone in
ecological processes, but mounting evidence shows that cooperation does also contribute to the
structure of biological communities. Here, we propose a simple deterministic model for the study
of the effects of facilitation and competition in the dynamics of such systems. The simultaneous
inclusion of both effects produces rich dynamics and captures the context dependence observed
in the formation of ecological communities. Our findings reproduce relevant aspects in plant succession
and highlight the role of facilitation mechanisms in species coexistence and conservation
efforts.

Vaskova V. S., Rodnikov A. V.
Abstract
Motion of a particle modeling a spacecraft with a solar sail along a handrail joining two
heliocentric space stations is considered under the assumption that the sail is a perfect reflecting
plane that can be located at any angle with respect to the direction of solar rays, the particle
does not leave the plane of the orbit of the stations, the handrail is a tether that realizes an
ideal unilateral constraint whose boundary is some ellipse, and the particle motion is sufficiently
fast with respect to the orbital motion of the stations to neglect noninertiality of the orbital
frame of reference. The equations of particle motion are written in dimensionless form without
parameters, and the existence of an energy integral for the case of the sail orientation depending
only on the spacecraft location is established. This integral is used for complete integration of
the equations of motion for the particle relocations along the constraint boundary. The optimal
length of the tether for the fastest relocation of a particle between the most remote points of
the constraint boundary is computed for the case of the sail being orthogonal to the solar rays
throughout the motion. Such a relocation time is computed in dimensionless form and for some
real and hypothetical situations. A set of pairs of points in the constraint boundary between
which relocation along the constraint boundary with zero initial and final velocities and with
the invariably oriented sail is possible is constructed depending on the eccentricity of the ellipse.
The result is presented as several plots that illustrate the evolution of the pairs’ regions as the
eccentricity of the ellipse changes.

Pochinka O. V., Shubin D. D.
Abstract
In the present paper, nonsingular Morse – Smale flows on closed orientable 3manifolds are
considered under the assumption that among the periodic orbits of the flow there is only one
saddle and that it is twisted. An exhaustive description of the topology of such manifolds is
obtained. Namely, it is established that any manifold admitting such flows is either a lens space
or a connected sum of a lens space with a projective space, or Seifert manifolds with a base
homeomorphic to a sphere and three singular fibers. Since the latter are prime manifolds, the
result obtained refutes the claim that, among prime manifolds, the flows considered admit only
lens spaces.

Polekhin I. Y.
Abstract
A sufficient condition for the existence of forced oscillations in nonautonomous systems on
a plane is presented under the assumption that the magnitude of the nonautonomous perturbation
is small. An advantage of the results presented over analytic methods is that they can be
applied in degenerate systems as well.

Suddalai Kannan K., Abdul Kader S. M., Chinnathambi V., Sethu Meenakshi M. V., Rajasekar S.
Abstract
This study examines the phenomenon of vibrational resonance (VR) in a classical positiondependent mass (PDM) system characterized by three types of singlewell potentials. These potentials are influenced by an amplitudemodulated (AM) signal with $\Omega\gg\omega$. Our analysis is limited to the following parametric choices: (i) $\omega_0^2$, $\beta$, $m_0$, $\lambda>0$ (type1 singlewell), (ii) $\omega_0^2>0$, $\beta <0$, $2< m_0 <3$, $1< \lambda <2$ (type2 singlewell), (iii) $\omega_0^2>0$, $\beta <0$, $0< m_0 <2$, $0<\lambda<1$ (type3 singlewell). The system presents an intriguing scenario in which the PDM function significantly contributes to the occurrence of VR. In addition to the analytical derivation of the equation for slow motions of the system based on the highfrequency signal’s parameters using the method of direct separation of motion, numerical evidence is presented for VR and its basic dynamical behaviors are investigated. Based on the findings presented in this paper, the weak lowfrequency signal within the singlewell PDM system can be either attenuated or amplified by manipulating PDM parameters, such as mass amplitude ($m_0$) and mass spatial nonlinearity $\lambda$. The outcomes of the analytical investigations are validated and further supported through numerical simulations. 
Maslov D. A.
Abstract
This article is concerned with investigating the nonlinear dynamics of the cylindrical resonator
of a wave solidstate gyroscope. The nonlinearity of oscillations caused by the nonlinear
properties of electrostatic control sensors is considered. This nonlinearity is derived by taking
into account the finite ratio of resonator flexure to the small gap of electrostatic control sensors.
The equations of the electromechanical system that in interconnected form describe the nonlinear
mechanical oscillations of the gyroscope resonator and electrical oscillations in the control circuit
are derived. The resulting differential equations belong to the class of Tikhonov systems, since
the equation of electrical processes in the control circuit is singularly perturbed. By taking into
account the low electrical resistance of the oscillation control circuit, which determines a small
parameter at the derivative in the singularly perturbed equation of electrical processes, the nonlinear
oscillations of the wave solidstate gyroscope resonator are studied. The small parameter
method is used to obtain a mathematical model of the resonator dynamics, which jointly takes
into account the nonlinearity of the resonator oscillations and the electrical resistance of the
oscillation control circuit. A special method is proposed to reduce the nonlinear equations of the
resonator dynamics to the standard form of the system of differential equations for averaging
and the equations of the dynamics of the wave solidstate gyroscope resonator are averaged. It is
shown that, in the case of nonlinear oscillations, consideration of the electrical resistance of the
oscillation control circuit does not affect the angular velocity of the gyroscope drift, but causes
slight dissipation of the oscillations, which also leads to an insignificant correction of the resonant
frequency.
