О вырожденных особых точках динамических систем, имеющих отношение к нормализованному потоку Риччи на обобщенных пространствах Уоллаха

Abstract

In this paper, we study degenerated singular points (equilibrium points) of a dynamical system obtained by reduction of the normalized Ricci flow on generalized Wallach spaces. It is known that every generalized Wallach space is characterized by a triple of positive numbers satisfying well-defined inequalities. Therefore, the corresponding system of differential equations also depends on three real parameters. In the works of N.A. Abiev, A. Arvanitoyeorgos, Yu.G. Nikonorov, and P. Siasos a new approach was developed for studying of singular points. This approach is based on the idea of building a surface of parameters providing the normalized Ricci flow degenerate singular points. At natural (geometric) values of parameters, we established that for the normalized Ricci flow the nilpotent case never occurs, and the linearly zero case can occur only at a unique special combination of parameters. As a consequence, any other degenerate singular point of the system may be only semi-hyperbolic. In this paper, we remove the previous restrictions and study an abstract dynamical system abstracted from geometric essence. It is proved that some results of the mentioned works also hold for arbitrary values of the real parameters.