The simplest normal forms associated with a triple zero eigenvalue of indices one and two

This paper considers the computation of the simplest normal forms of differ ential equations, with the particular attention oil the systems whose Jacob ian evaluated at an equilibrium has a triple zero eigenvalue of index one o r two. The computation is based oil near identity nonlinear transformations . Explicit formulas are derived for computing the simplest normal forms and the associated nonlinear transformations, which facilitates a direct imple mentation oil computer systems using Maple.