# Applied Problems of Nonlinear Oscillation and Wave Theory

## Nonstationary discrete theory of excitation of periodic structures and its application for simulation of traveling-wave tubes

Aim. This article presents a review of the nonstationary (time-domain) discrete theory of excitation of periodic electromagnetic structures and discusses applications of the theory for simulation of traveling-wave tube (TWT) microwave power amplifiers with slow-wave structures (SWS) of different kind. Methods. The discrete theory is based on a representation of a periodic SWS as a chain of coupled cells.

## Reconstructing the neuron-like oscillator equations modeled by a phase-locked system with delay from scalar time series

The purpose of this work is to develop the reconstruction technique for the neuron-like oscillator equations descibed by a phase-locked system model with delay from scalar time series. Methods. We reconstruct the state vector given a scalar series of only one variable corresponding to the transmembrane potential. The second variable is obtained by numerical differentiation with smoothing by a polynomial. The third variable is obtained by numerical integration using the Simpson method.

## Synchronization of infections spread processes in populations interacting: Modeling by lattices of cellular automata

Purpose. Study of synchronization of oscillations in ensembles of probabilistic cellular automata that simulate the spread of infections in biological populations. Method. Numerical simulation of the square lattice of cellular automata by means of the Monte Carlo method, investigation of synchronization of oscillations by time-series analisys and by the coherence function. Results. The effect has been found of synchronization of irregular oscillations, similar to the phenomenon of synchronization of chaos in dynamical systems.

## Analysis of steady-state stability for intracavity optical parametric oscillator: Method of small-parameter expansion

The aim of the study is to analytically determine the linear stability of a steady-state operation point for an optical parametric oscillator (OPO) intracavity pumped by a semiconductor disk laser (SDL). Methods. In order to build the analytic approximation to the characteristic equation roots, the method of small-parameter expansion is used. The results of analytic and numerical methods are compared with each other. Results.

## Normalized boundary value problems in the model of optoelectronic oscillator delayed

Purpose of this work is reduction of differential-difference-model of optic-electronic oscillator to more simple normalized boundary value problems. We study the dynamics of an optoelectronic oscillator with delayed feedback in the vicinity of the zero equilibrium state. The differential-difference-model contains a small parameter with the derivative. It is shown that in a certain neighborhood of the bifurcation point, the number of roots of the characteristic equation that have a real part close to zero increases unlimitedly with decreasing small parameter.

## Nonlinear problem of temperature distribution inside the Earth

The purpose of this research is to obtain a nonlinear heat conduction equation based on the Stefan–Boltzmann law and energy balance to study the temperature distribution inside the Earth, taking into account usual conductivity and radiant heat conductivity. Resulting equation with a fourth-degree nonlinearity allows to consider the heat transfer between a layer of matter and the environment. Methods to obtain the equation are based on the energy conservation law, taking into account the Kirchhoff’s law and introducing the variation of the blackness coefficient.

## Threshold stability of the synchronous mode in a power grid with hub cluster topology

The main purpose of this paper is to investigate the dynamics of the power grid model with hub cluster topology based on the Kuramoto equations with inertia. It is essential to study the stability of synchronous grid operation mode and to find conditions of its global stability. The conditions that ensure establishment of the synchronous mode instead of coexisting asynchronous ones are considered. Methods. In this paper we use numerical modelling of different grid operation modes.

## Link between the self-organization of dynamic cutting system and tool wear

Purpose of this work is improvement the efficiency of the metal cutting process through the agreement of external control from the CNC system with internal dynamics of the system, its evolution manifested through development of tool wear and influenced on parameters and dynamic properties of interacting subsystems of the tool and workpiece. Methods. Mathematical model of the forces is provided to reveal the dynamic connection formed by the cutting process between subsystems of the tool and workpiece.

## The influence of the output power of the generators on the frequency characteristics of the grid in a ring topology

Great interest in the field of dynamic systems and nonlinear processes is caused by research in the field of energy networks. A power grid is a complex network of coupled oscillators that demonstrates collective behavior by synchronizing

network elements at the base frequency of a power grid.

## Effect of nonlinearity on coupling estimations between oscillators using partial directed coherence approach

The purpose of this work is to determine the ability of the partial directed coherence method to identify directed interactions between nonlinear systems correctly in presence of nonlinear couplings between systems, as well as in the

case of measured signals generated by objects of high dimension. The other purpose is to determine the dependence of the coupling estimation results on the parameters: series length, sampling rate, model dimension and coupling architecture.