Vol. 18, no. 2
Vol. 18, no. 2, 2022
Krasil'nikov P. S., Maiorov A. Y.
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.

Chernyshov A. V., Chernyshova S. A.
Abstract
The phenomenon of “bifurcation memory”, which can be detected during the steering of
river vessels, is considered by researchers with the questions of equilibrium point’s bifurcations.
It was noticed that in some dynamic systems during saddlenode bifurcation, areas are formed
on the phase plane (socalled “phase spot”), passing through which the phase velocity of the
representative point decreases. A decrease in phase velocity (e. g. during the steering of river
vessel) can cause navigation accidents during shallow waters navigation. In order to improve
sailing safety, it is necessary to investigate topological features of phase spots under any possible
environmental conditions and steering angle values. In this paper, we propose a new method
that makes it easy to get information about the localization of regions with decreased phase
velocity. We gained the results related to the topology of areas of different types of motion
(accelerated or decelerated) of representative point. In addition, the paper presents the process
of evolution of these regions, according to the change of steering angles, as well as before and
after the bifurcation. The new method, offered in the article, is more accurate for determining
the boundaries of the areas compared to the methods of Feigin and Chirkova. This will allow
us to make more correct predictions of changes in the dynamics of the object. The practical
implication of the suggested method is that by using it, we can get information about the
location of areas with different types of motion of the representative point on the phase plane
for different values of steering angle and environmental conditions. This information can be used
in the control algorithm of the driving object, for example, in order to predict the decreasing of
phase velocities.

Surulere S. A., Shatalov M. Y., Ehigie J. O., Adeniji A. A., Fedotov I. A.
Abstract
The oscillatory motion in nonlinear nanolattices having different interatomic potential energy
functions is investigated. Potential energies such as the classical Morse, Biswas – Hamann
and modified Lennard – Jones potentials are considered as interaction potentials between atoms
in onedimensional nanolattices. Noteworthy phenomena are obtained with a nonlinear chain,
for each of the potential functions considered. The generalized governing system of equations for
the interaction potentials are formulated using the wellknown Euler – Lagrange equation with
Rayleigh’s modification. Linearized damping terms are introduced into the nonlinear chain. The
nanochain has statistical attachments of 40 atoms, which are perturbed to analyze the resulting
nonlinearities in the nanolattices. The range of initial points for the initial value problem
(presented as secondorder ordinary differential equations) largely varies, depending on the interaction
potential. The nanolattices are broken at some initial point(s), with one atom falling off
the slender chain or more than one atom falling off. The broken nanochain is characterized by an
amplitude of vibration growing to infinity. In general, it is observed that the nonlinear effects in
the interaction potentials cause growing amplitudes of vibration, accompanied by disruptions of
the nanolattice or by the translation of chaotic motion into regular motion (after the introduction
of linear damping). This study provides a computationally efficient approach for understanding
atomic interactions in long nanostructural components from a theoretical perspective.

GonzálezGaxiola O., LeónRamírez A., ChacónAcosta G.
Abstract
Recently, motivated by the interest in the problems of nonlinear dynamics of cylindrical
shells, A. I. Zemlyanukhin et al. (Nonlinear Dyn, 98, 185–194, 2019) established the socalled
Schamel – Kawahara equation (SKE). The SKE generalizes the wellknown nonlinear Schamel
equation that arises in plasma physics problems, by adding the highorder dispersive terms
from the Kawahara equation. This article presents families of new solutions to the Schamel –
Kawahara model using the Kudryashov method. By performing the symbolic computation,
we show that this method is a valuable and efficient mathematical tool for solving application
problems modeled by nonlinear partial differential equations (NPDE).

Ekomasov E. G., Nazarov V. N., Samsonov K. Y.
Abstract
Possibility of changing the dynamic parameters of localized breather and soliton waves for
the sineGordon equation in the model with extended impurity, variable external force and dissipation
was investigated using the autoresonance method. The model of ferromagnetic structure
consisting of two wide identical layers separated by a thin layer with modified values of magnetic
anisotropy parameter was taken as a basis. Frequency of external field is a linear function of time.
The sineGordon equation (SGE) was solved numerically using the finite differences method with
explicit scheme of integration. For certain values of the extended impurity parameters a magnetic
inhomogeneity in the form of magnetic breather is formed when domain wall passes through it
with constant velocity. The numerical simulation showed that using special variable force and
small amplitude it is possible to resonantly increase the amplitude of breather. For each case
of the impurity parameters values, there is a threshold value of the magnetic field amplitude
leading to resonance. Geometric parameters of thin layer also have influence on the resonance
effect — for decreasing layer width the breather amplitude grows more slowly. For large layer
width the translation mode of breather oscillations is also excited. For certain parameters of
extended impurity, a soliton can form. For a special type of variable field with frequency linearly
dependent on time, soliton is switched to antisoliton and vice versa.

Hosseini K., Mirzazadeh M., Dehingia K., Das A., Salahshour S.
Abstract
In the present paper, the authors are interested in studying a famous nonlinear PDE referred
to as the $(2 + 1)$dimensional chiral Schrцdinger (2DCS) equation with applications in
mathematical physics. In this respect, the real and imaginary portions of the 2DCS equation
are firstly derived through a traveling wave transformation. Different wave structures of the
2DCS equation, classified as bright and dark solitons, are then retrieved using the modified
Kudryashov (MK) method and the symbolic computation package. In the end, the dynamics of
soliton solutions is investigated formally by representing a series of 3Dplots.

Shamin A. Y.
Abstract
This paper addresses the problem of the motion of the Chaplygin sleigh, a rigid body moving
with three points in contact with a horizontal plane. One of them is equipped with a knife edge
along which there is no slipping. Special attention is given to the case where dry friction is
present at one of the points of support without the knife edge. The equations of motion of the
body are written, the normal reactions are calculated, and the behavior of the phase curves in
the neighborhood of an equilibrium point, depending on the geometric and mass characteristics
of the body, is investigated by the method of introducing a small parameter.

Dzhalilov A., Mayer D., Aliyev A.
Abstract
Let $T\in C^{2+\varepsilon}(S^{1}\setminus\{x_b^{}\})$, $\varepsilon>0$, be an orientation preserving circle homeomorphism with rotation number $\rho_T^{}=[k_1^{},\,k_2^{},\,\ldots,\,k_m^{},\,1,\,1,\,\ldots]$, $m\ge1$, and a single break point $x_b^{}$. Stochastic perturbations $\overline{z}_{n+1}^{} = T(\overline{z}_n^{}) + \sigma \xi_{n+1}^{}$, $\overline{z}_0^{}:=z\in S^1$ of critical circle maps have been studied some time ago by DiazEspinoza and de la Llave, who showed for the resulting sum of random variables a central limit theorem and its rate of convergence. Their approach used the renormalization group technique. We will use here Sinai's et al. thermodynamic formalism approach, generalised to circle maps with a break point by Dzhalilov et al., to extend the above results to circle homemorphisms with a break point. This and the sequence of dynamical partitions allows us, following earlier work of Vul at al., to establish a symbolic dynamics for any point~${z\in S^1}$ and to define a~transfer operator whose leading eigenvalue can be used to bound the Lyapunov function. To prove the central limit theorem and its convergence rate we decompose the stochastic sequence via a Taylor expansion in the variables $\xi_i$ into the linear term $L_n^{}(z_0^{})= \xi_n^{}+\sum\limits_{k=1}^{n1}\xi_k^{}\prod\limits_{j=k}^{n1} T'(z_j^{})$, ${z_0^{}\in S^1}$ and a higher order term, which is possible in a neighbourhood $A_k^n$ of the points $z_k^{}$, ${k\le n1}$, not containing the break points of $T^{n}$. For this we construct for a certain sequence~$\{n_m^{}\}$ a series of neighbourhoods $A_k^{n_m^{}}$ of the points $z_k^{}$ which do not contain any break point of the map $T^{q_{n_m^{}}^{}}$, $q_{n_m^{}}^{}$ the first return times of $T$. The proof of our results follows from the proof of the central limit theorem for the linearized process.

Frick P., Okatev R., Sokoloff D.
Abstract
The solar 11year activity cycle is a famous manifestation of magnetic activity of celestial
bodies. The physical nature of the solar cycle is believed to be largescale magnetic field excitation
in the form of a wave of a quasistationary magnetic field propagating from middle solar latitudes
to the solar equator. The power spectrum of solar magnetic activity recorded in sunspot data
and underlying solar dynamo action contains quite a stable oscillation known as the 11year
cycle as well as the continuous component and some additional weak peaks. We consider a loworder
model for the solar dynamo. We show that in some range of governing parameters this
model can reproduce spectra with pronounced dominating frequency and wide spectral peaks in
the lowfrequency region. The spectra obtained are qualitatively similar to the observed solar
activity spectrum.

Lee K., Rojas A.
Abstract
In this paper we study the almost shadowable measures for homeomorphisms on compact
metric spaces. First, we give examples of measures that are not shadowable. Next, we show
that almost shadowable measures are weakly shadowable, namely, that there are Borelians with
a measure close to 1 such that every pseudoorbit through it can be shadowed. Afterwards, the
set of weakly shadowable measures is shown to be an $F_{\sigma \delta}$ subset of the space of Borel probability
measures. Also, we show that the weakly shadowable measures can be weakly* approximated
by shadowable ones. Furthermore, the closure of the set of shadowable points has full measure
with respect to any weakly shadowable measure. We show that the notions of shadowableness,
almost shadowableness and weak shadowableness coincide for finitely supported measures, or,
for every measure when the set of shadowable points is closed. We investigate the stability of
weakly shadowable expansive measures for homeomorphisms on compact metric spaces.

Ghosh U., Das T., Sarkar S.
Abstract
A brief outline of the derivation of the timefractional nonlinear Schrödinger equation (NLSE) is furnished. The homotopy analysis method (HAM) is applied to study timefractional NLSE with three separate trapping potential models that we believe have not been investigated yet. The first potential is a double cosine potential $[V(x)=V_1^{}\cos x+V_2^{}\cos 2x]$, the second one is the Morse potential $[V(x)=V_1^{}e^{2\beta x}+V_2^{}e^{\beta x}]$, and a hyperbolic potential $[V(x)=V_0^{}\tanh(x)sech(x)]$ is taken as the third model. The fractional derivatives and integrals are described in the Caputo and Riemann Liouville sense, respectively. The solutions are given in the form of convergent series with easily computable components. A physical analysis with graphical representations explicitly reveals that HAM is effective and convenient for solving nonlinear differential equations of fractional order.
