# Applied Problems of Nonlinear Oscillation and Wave Theory

## Spatial and temporal dynamics of the emergence of epidemics in the hybrid SIRS+V model of cellular automata

Purpose of this work is to construct a model of infection spread in the form of a lattice of probabilistic cellular automata, which takes into account the inertial nature of infection transmission between individuals. Identification of the relationship between the spatial and temporal dynamics of the model depending on the probability of migration of individuals.

Methods. The numerical simulation of stochastic dynamics of the lattice of cellular automata by the Monte Carlo method.

## Transfer of passive particles in the velocity field of vortex tripole moving on a plane

Purpose of this article is to study the transport of passive particles in the velocity field of a vortex tripole with a change in the parameter that determines the speed of the configuration movement. A structure consisting of a central vortex and satellite vortices rotating around it with the opposite vorticity is understood as a tripole. We employ a system of three point vortices, the most simple mathematical representation of a vortex tripole, which may be expressed as a system of nonlinear ordinary differential equations with a parameter.

## Application of joint singularity spectrum to analyze cooperative dynamics of complex systems

Purpose of this work is to generalize the wavelet-transform modulus maxima method to the case of cooperative dynamics of interacting systems and to introduce the joint singularity spectrum into consideration.

The research method is the wavelet-based multifractal formalism, the generalized version of which is used to quantitatively describe the effect of chaotic synchronization in the dynamics of model systems. Models of coupled Rossler systems and paired nephrons are considered.

## Coupled economic oscillations — synchronization dynamical model

Purpose of this work is the research of the dynamical processes and in particular the phenomenon of the synchronization in an ensemble of coupled chaotic economic oscillators.

Methods. The research methods are the qualitative and numerical methods of the theory of nonlinear dynamical systems and the theory of the bifurcations.

## Mechanisms leading to bursting oscillations in the system of predator–prey communities coupled by migrations

The purpose is to study the periodic regimes of the dynamics for two non-identical predator–prey communities coupled by migrations, associated with the partial synchronization of fluctuations in the abundance of communities. The combination of fluctuations in neighboring sites leads to the regimes that include both fast bursts (bursting oscillations) and slow oscillations (tonic spiking). These types of activity are characterized by a different ratio of synchronous and non-synchronous dynamics of communities in certain periods of time.

## 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.