Construction of exact solutions and analysis of stability of complex systems by reduction to ordinary differential equations with power nonlinearities

Complex systems described by nonlinear partial differential equations of parabolic type or large-scale systems of ordinary differential equations with switching right-side are considered. The reduction method is applied to the corresponding problem for the system of ordinary differential equations without switching. A parametric family of time-periodic and anisotropic on spatial variables exact solutions of the reaction-diffusion system is constructed. The stability conditions of a large-scale system with switching are obtained, which consist in checking the stability of the reduced system without switching. The conditions for the existence of the first integrals for the reduced system of ordinary differential equations expressed by a combination of power and logarithmic functions are found. For the cases of two-dimensional and three-dimensional reduced systems, these conditions are written in the form of polynomial equations relating the system parameters.