An explicit recursive formula for computing the normal forms associated with semisimple cases

•An explicit, recursive formula for computing normal forms of differential systems is developed.•This formula is for general n-dimensional systems associated with semisimple cases.•This formula contains the computation of center manifold.•This formula and the Maple program developed on the basis of this formula have been proved computationally efficient.

This paper presents an explicit, computationally efficient, recursive formula for computing the normal forms, center manifolds and nonlinear transformations for general n-dimensional systems, associated with semisimple singularities. Based on the formula, we develop a Maple program, which is very convenient for an end-user who only needs to prepare an input file and then execute the program to “automatically” generate the result. Several examples are presented to demonstrate the computational efficiency of the algorithm.