Volume 25, Number 4
Volume 25, Number 4, 2020
de Menezes Neto J. L., Cabral H. E.
Parametric Stability of a Pendulum with Variable Length in an Elliptic Orbit
Abstract
We study the dynamics of a simple pendulum attached to the center of mass of a
satellite in an elliptic orbit. We consider the case where the pendulum lies in the orbital plane
of the satellite. We find two linearly stable equilibrium positions for the Hamiltonian system
describing the problem and study their parametric stability by constructing the boundary curves
of the stability/instability regions.

Bulatovic R. M.
On the Stability of Potential Systems under the Action of Nonconservative Positional Forces
Abstract
The stability of linear mechanical systems with finite numbers of degrees of freedom
subjected to potential and nonconservative positional forces is considered. The positive semidefiniteness
of the potential energy is assumed. Three new stability criteria which are in a
simple way related to the properties of the system matrices are derived. These criteria improve
previously obtained results of the same type. Several simple examples are given to illustrate the
correctness and applicability of the obtained results.

GonzálezGaxiola O., Biswas A., Asma M., Alzahrani A. K.
Optical Dromions and Domain Walls with the Kundu – Mukherjee – Naskar Equation by the Laplace – Adomian Decomposition Scheme
Abstract
This paper numerically addresses optical dromions and domain walls that are
monitored by Kundu – Mukherjee – Naskar equation. The Kundu – Mukherjee – Naskar equation
is considered because this model describes the propagation of soliton dynamics in optical fiber
communication system. The scheme employed in this work is Laplace – Adomian decomposition
type. The accuracy of the scheme is $O(10^{8})$ and the physical structure of the obtained solutions
are shown by graphic illustration in order to give a better understanding on the dynamics of
both optical dromions and domain walls.

Andrade R. M.
Bernoulli Property for Some Hyperbolic Billiards
Abstract
We prove that hyperbolic billiards constructed by Bussolari and Lenci are Bernoulli
systems. These billiards cannot be studied by existing approaches to analysis of billiards that
have some focusing boundary components, which require the diameter of the billiard table to
be of the same order as the largest curvature radius along the focusing component. Our proof
employs a local ergodic theorem which states that, under certain conditions, there is a full
measure set of the billiard phase space such that each point of the set has a neighborhood
contained (mod 0) in a Bernoulli component of the billiard map.

Hosseini K., Samavat M., Mirzazadeh M., Ma W., Hammouch Z.
A New $(3+1)$dimensional Hirota Bilinear Equation: Its Bäcklund Transformation and Rationaltype Solutions
Abstract
The behavior of specific dispersive waves in a new $(3+1)$dimensional Hirota bilinear (3DHB) equation is studied. A Bäcklund transformation and a Hirota bilinear form of the model are first extracted from the truncated Painlevé expansion. Through a series of mathematical analyses, it is then revealed that the new 3DHB equation possesses a series of rationaltype solutions. The interaction of lumptype and 1soliton solutions is studied and some interesting and useful results are presented.

Borisov A. V., Mikishanina E. A.
Two Nonholonomic Chaotic Systems. Part II. On the Rolling of a Nonholonomic Bundle of Two Bodies
Abstract
The problem of rolling a nonholonomic bundle of two bodies is considered:
a spherical shell with a rigid body rotating along the axis of symmetry, on which rotors spinning
relative to this body are fastened. This problem can be regarded as a distant generalization of the
Chaplygin ball problem. The reduced system is studied by analyzing Poincaré maps constructed
in Andoyer – Deprit variables. A classification of Poincaré maps of the reduced system is carried
out, the behavior of the contact point is studied, and the cases of chaotic oscillations of the
system are examined in detail. To study the nature of the system’s chaotic behavior, a map of
dynamical regimes is constructed. The Feigenbaum type of attractor is shown.

Polekhin I. Y.
The Method of Averaging for the Kapitza – Whitney Pendulum
Abstract
A generalization of the classical Kapitza pendulum is considered: an inverted planar
mathematical pendulum with a vertically vibrating pivot point in a timeperiodic horizontal
force field. We study the existence of forced oscillations in the system. It is shown that there
always exists a periodic solution along which the rod of the pendulum never becomes horizontal,
i.e., the pendulum never falls, provided the period of vibration and the period of horizontal
force are commensurable. We also present a sufficient condition for the existence of at least two
different periodic solutions without falling. We show numerically that there exist stable periodic
solutions without falling.
