Computation of normal forms of differential equations associated with non-semisimple zero eigenvalues

This paper presents a method to compute the normal forms of differential equations whose Jacobian evaluated at an equilibrium includes a double zero or a triple zero eigenvalue. The method combines normal form theory with center manifold theory to deal with a general n-dimensional system. Explicit formulas are derived and symbolic computer programs have been developed using a symbolic computation language Maple. This enables one to easily compute normal forms and nonlinear transformations up to any order for a given specific problem. The programs can be conveniently executed on a main frame, a workstation or a PC machine without any interaction. Mathematical and practical examples are presented to show the applicability of the method.