Generalized normal form and the formal equivalence of systems with zero approximation (χ23 , - χ13)

We consider formal systems of differential equations of the form ẏ1 = y23 + ∑p = 4∞ Y 1(p) (y1, y2), ẏ2 = - y13 + ∑p = 4∞ Y2(p) (y1, y2), where Y i (p) are homogeneous polynomials of order p. Such systems are obtained from initial systems of the same form by using formal invertible changes of variables x i = y i + h i(y 1,y 2 (i = 1 , 2). For any p ≥ 4, we explicitly write n p = {5 , if p = 4r + 1;4 , if p ≠ 4r + 1}linear resonant equations. The initial system is formally equivalent to the above system if the coefficients of the polynomials Y i (p) satisfy the specified resonant equations. © 2005 Springer Science+Business Media, Inc.