# Applied Problems of Nonlinear Oscillation and Wave Theory

## Dynamic damping of vibrations of a solid body mounted on viscoelastic supports

The study of the problem of damping vibrations of a solid body mounted on viscoelastic supports is an urgent task. The paper considers the problem of reducing the level of vibrations on the paws of electric machines using dynamic vibration dampers. For this purpose, the paw of electric machines is represented in the form of a subamortized solid body with six degrees of freedom mounted on viscoelastic supports.

## Stability thresholds of attractors of the Hopfield network

Purpose of the work is the detailed study of the attractors of the Hopfield network and their basins of attraction depending on the parameters of the system, the size of the network and the number of stored images. To characterize the basins of attraction we used the method of the so-called stability threshold, i.e., the minimum distance from an attractor to the boundary of its basin of attraction. For useful attractors, this value corresponds to the minimum distortion of the stored image, after which the system is unable to recognize it.

## Approach to nonlinearity parameter in liquids calculation based on the scaling theory of thermodynamic fluctuations

The nonlinearity parameter *B/A* is a characteristic of liquids and soft matter, which gains growing attention due to its sensibility to the composition of materials. This makes it a prospective indicator for nondestructive testing applications based on the ultrasound sounding suitable for a variety of applications from physic chemistry to biomedical studies.

## Hybrid SIRS model of infection spread

Purpose of this work is to build a model of the infection spread in the form of a system of differential equations that takes into account the inertial nature of the transfer of infection between individuals. Methods. The paper presents a theoretical and numerical study of the structure of the phase space of the system of ordinary differential equations of the mean field model. Results. A modified SIRS model of epidemic spread is constructed in the form of a system of ordinary differential equations of the third order.

## Criteria for internal fixed points existence of discrete dynamic Lotka–Volterra systems with homogeneous tournaments

Purpose of the work is to study the dynamics of the asymptotic behavior of trajectories of discrete Lotka–Volterra dynamical systems with homogeneous tournaments operating in an arbitrary (*m* − 1)-dimensional simplex. It is known that a dynamic system is an object or a process for which the concept of a state is uniquely defined as a set of certain quantities at a given time, and a law describing the evolution of initial state over time is given.

## Resonant and nonlinear phenomena during the propagation of magnetostatic waves in multiferroid, semiconductor and metallized structures based on ferromagnetic films and magnonic crystals

Purpose of this work is to compile an overview of a new and fruitful scientific direction in magnonics, which grew out of the works of Ph.D., Professor Yuri Pavlovich Sharaevsky, and related to the study of resonant and nonlinear phenomena during the propagation of magnetostatic waves in ferromagnetic films, ferromagnetic films with periodic inhomogeneities (magnonic crystals), coupled (layered and lateral) ferromagnetic structures, as well as ferromagnetic structures with layers of a different physical nature (semiconductor, ferroelectric, piezoelectric, normal metal l

## Experimental methods for the study of spin waves

Purpose of this paper is to give an overview of various experimental methods for investigation of spin waves characteristics. Methods. The paper presents a description of a number of experimental techniques, such as the probing method, the phase shift method, the method of measure of equiphase dependences, the method of intersecting wave beams, and the use of Fourier analysis of the complex transfer coefficient of spin waves to determine their spatial spectrum.

## On the conditions for safe connection to hub-cluster power grids

Purpose of this work is studying of the dynamics of a power grid model that results from the expansion of a highly centralized grid, i.e. a hub-cluster, by adding a small subgrid. The main attention is paid to the study of possible power grid operation regimes and their characteristics. Methods. Numerical simulation of power grid operation, the dynamics of which is described by the Kuramoto equations with inertia, is used. Results. Various power grid operation regimes and the boundaries of their existence in the parameter space are given.

## Variational approach to the construction of discrete mathematical model of the pendulum motion with vibrating suspension with friction

The main purpose of this work is, first, a construction of the indirect Hamilton’s variational principle for the problem of motion of a pendulum with a vibration suspension with friction, oscillating along a straight line making a small angle with the vertical line. Second, the construction on its basis of the difference scheme. Third, to carry out its investigation by methods of numerical analysis. Methods.

## Reconstruction of integrated equations of periodically driven phase-locked loop system from scalar time series

Purpose of this work is to develop a reconstruction technique for the equations of a phase-locked loop system under periodic external driving from a scalar time series of one variable. Methods. Instead of the original model, we reconstructed a time-integrated model. So, since it is not necessary to evaluate the second derivative of the observable numerically, the method sensitivity to observation noise has significantly decreased.