An algebraic approach for obtaining nilpotent normal forms in dimension 4

In this paper, an algebraic approach is presented for determining the norma l forms of four-dimensional systems with a nilpotent linear part. Certain t ransformations introduced in this case result in a simplified procedure for the calculation of normal forms. Thus, one does not need to solve a series of partial differential equations as usually required by the normal-form t heory; indeed, algebraic calculations are sufficient. The approach can be a pplied to higher-order systems with a nilpotent linear part as well. To ill ustrate the new approach, five examples are presented. Normal forms and the associated coefficients of two physical systems, an electric network and a mechanical system, are fully analyzed.