Shiyan O. Phenomenon of buffer in the parabolic equations of the Van-der-Pol type

The dissertation is devoted to research phenomena of buffer for a system of parabolic equations of the Van-der-Pol type. The dynamics of traveling waves for a system of parabolic equations of the Van-der-Pol type with small diffusion on a circle with radius r is studied. The theorem of existence, interaction, asymptotic form, and stability of these waves are proved. It is proved that the number of stable traveling waves increases with the radius r. The auto-oscillating system of connected diffusionally Van-der-Pole oscillators is studied. This system describes the front movement of the combustion on the segment with isolated edges. The theorem of existence, interaction, asymptotic form, and stability of periodic spatially inhomogeneous solutions that bifurcate from the losing stability of spatially homogeneous periodic solution - in-phase mode, are proved. We investigate problems of the form and the stability of this periodic solution in the deeply supercritical domain.