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

A. D. Bruno, A. Soleev


Abstract: 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. In a neighbourhood of a stationary point we study the system by means of its normal form for cases σ=-1, υ∈ [-5/4,1), when all eigenvalues are pure imaginary. We develop a theory of the structure of the normal form for resonant cases. Using it, we have found local families of periodic solutions and conditionally periodic solutions.