Symbolic computation of normal forms for semi-simple cases

This paper presents a method and computer programs for computing the normal forms of ordinary differential equations whose Jacobian matrix evaluated at an equilibrium involves semi-simple eigenvalues. The method can be used to deal with systems which are not necessarily described on a center manifold. An iterative procedure is developed for finding the closed-form expressions of the normal forms and associated nonlinear transformations. Computer programs using a symbolic computer language Maple are developed to facilitate the application of the method. The programs can be conveniently executed on a main frame, a workstation or a PC machine without any interaction. A number of examples are presented to demonstrate the applicability of the method and the computation efficiency of the Maple programs. (C) 1999 Elsevier Science B.V. All rights reserved. AMS classification: 34C20; 58F36; 34A34.