A Generalized Normal Form and Formal Equivalence of Systems of Differential Equations with Zero Characteristic Numbers

摘要

Consider an autonomous system of differential equations whose right-hand side does not contain free terms, the matrix of the linear part is reduced to a Jordan form, and the nonlinearities are power series, either formal or convergent at zero: x_i = λ_ix_i + σ_ix_(i-1) + X_i(x) (i = 1,…,n), where σ_1 = 0, σ_2,…,σ_n = {0 or σ > 0}, and X_i = ∑_(q_1 + … + q_n = 2)~∞X_i~((q_1))x_1~(q_1)…x_n~(q_n). Throughout the following, we assume that the components q_1,…,q_n of the vector q are nonnegative integers, and we set |q| = q_1 + … + q_n and x~q = x_1~(q_1)…x_n~(q_n).