1:?3 1 : ? 3 resonant centers on C2 C 2 with homogeneous cubic nonlinearities
We characterize the collection of systems of differential equations on C2 C 2 of the form x ? = x + p ( x , y ) , y ? = ? 3 y + q ( x , y ) , where p p and q q are homogeneous polynomials of degree three (either of which may be zero), that possess a first integral in a neighborhood of (0,0) ( 0 , 0 ) of the form x3y+? x 3 y + ? , where omitted terms are of order at least five. Such systems are called 1:?3 1 : ? 3 resonant centers.
resonant center first integral polynomial systems of differential equations center variety darboux integration