Formal Decomposition Method and Parametric Normal Forms

We introduce a formal decomposition method for efficiently computing the parametric normal form of nonlinear dynamical systems with multiple parameters. Recently introduced notions of formal basis style and costyle are applied through formal decomposition method to obtain the simplest parametric normal form for degenerate nonlinear parametric center. The necessary formulas are derived and implemented using Maple to compute the simplest parametric normal form of degenerate and nondegenerate nonlinear centers. Our program computes the order of any planar parametric systems associated with this singularity.