Vol 30, No 6 (2022)

Congratulations to Aleksei Aleksandrovich Koronovskii on his anniversary.

Purpose of this work is a numerical study of the Rabinovich–Fabrikant system and its generalized model, which describe the occurrence of chaos during the parametric interaction of three modes in a nonequilibrium medium with cubic nonlinearity, in the case when the parameters that have the meaning of dissipation coefficients take negative values. These models demonstrate a rich dynamics that differs in many respects from what was observed for them, but in the case of positive values of the parameters. Methods. The study is based on the numerical solution of the differential equations, and their numerical bifurcation analysis using the MatCont program. Results. For investigated models we present a charts of dynamic regimes in the control parameters plane, Lyapunov exponents depending on the parameters, attractors and their basins. On the parameters plane, which have the meaning of dissipation coefficients, bifurcation lines and points are numerically found. They are plotted for equilibrium point and period one limit cycle. For both models we compared dynamics observed in the case when the parameters that have the meaning of dissipation coefficients take negative values, with the one observed in the case when these parameters take positive values. And it is shown that in the first case parameter space has a simpler structure. Conclusion. The Rabinovich– Fabrikant system and its generalized model were studied in detail in the case when the parameters which have the meaning of dissipation coefficients take negative values. It is shown that there are a number of differences in comparison with the case of positive values of these parameters. For example, a new type of chaotic attractor appears, multistability that is not related to the symmetry of the system disappears, etc. The obtained results are new, since the Rabinovich–Fabrikant system and its generalized model were studied in detail for the first time in the region of negative values of parameters which have the meaning of dissipation coefficients.

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. It differs from standard models by considering the inertial nature of the infection transmission process between individuals of the population, which is realized by introducing a «carrier agent» into the model. The model does not take into account the influence of the disease on the population size, while population density is regarded as a parameter influencing the course of the epidemic. The dynamics of the model shows a good qualitative correspondence with a variety of phenomena observed in the evolution of diseases. Discussion. The suggested complication of the standard SIRS model by adding to it an equation for the dynamics of the pathogen of infection presents prospects for its specification via more precise adjustment to specific diseases, as well as taking into account the heterogeneity in the distribution of individuals and the pathogen in space. Further modification of the model can go through complicating the function which defines the probability of infection, generation and inactivation of the pathogen, the influence of climatic factors, as well as by means of transition to spatially distributed systems, for example, networks of probabilistic cellular automata.

Purpose of this work is to use analytical and numerical methods to consider the problem of the structure and dynamics of coupled localized nonlinear waves in the sine-Gordon model with impurities (or spatial inhomogeneity of the periodic potential). Methods. Using the analytical method of collective coordinates for the case of the arbitrary number the same point impurities on the same distance each other, differential equation system was got for localized waves amplitudes as the functions on time. We used the finite difference method with explicit scheme for the numerical solution of the modified sine-Gordon equation. We used a discrete Fourier transform to perform a frequency analysis of the oscillations of localized waves calculate numerically. Results. We found of the differential equation system for three harmonic oscillators with the elastic connection for describe related oscillations of nonlinear waves localized on the three same impurity. The solutions obtained from this system of equations for the frequencies of related oscillation well approximate the results of direct numerical modeling of a nonlinear system. Conclusion. In the article shows that the related oscillation of nonlinear waves localized on three identical impurities located at the same distance from each other represent the sum of three harmonic oscillations: in-phase, in-phase-antiphase and antiphase type. The analysis of the influence of system parameters and initial conditions on the frequency and type of associated oscillations is carried out.

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