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.

The stability of linear mechanical systems with finite numbers of degrees of freedom subjected to potential and non-conservative 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.

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.

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.

The behavior of specific dispersive waves in a new $(3+1)$-dimensional Hirota bilinear (3D-HB) 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 3D-HB equation possesses a series of rational-type solutions. The interaction of lump-type and 1-soliton solutions is studied and some interesting and useful results are presented.

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.

A generalization of the classical Kapitza pendulum is considered: an inverted planar mathematical pendulum with a vertically vibrating pivot point in a time-periodic 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.