Local Analysis of a Singularity of an Invertible ODE System. Simple Cases

Near its statiinary point we study solutions of an invertible system of ordinary differential equations with a square nonlinearity and with parameters υ ∈ IR and σ = ±1. The system appeared from the water-wade problem after its reduction on the center manifold and a selection of the basic first approximation and a power transformation of coordinates. We study the system by means of its normal form. In the set of values of parameters υ and σ we have selected and studied cases, where the normalizing transformation converges, and cases, where it diverges but there is a smooth normalizing transformation.