Degenerate equilibria at infinity in the generalized brusselator

In this paper, we consider a polynomial differential system of p + q degree, which was given from a general multi-molecular reaction in biochemistry as a theoretical problem of concentration kinetics. We analyze qualitative properties of its equilibria at infinity, determining characteristic directions and the numbers of orbits which go towards or away from those equilibria in characteristic directions. In the analysis, the high degree of polynomials and the high degeneracy of equilibria at infinity make so much trouble that both the known blowing-up method and the normal sector method are not effective in some cases. Our difficulties are overcome by discussing a kind of angular regions, which extends the classic normal sectors to more general.