Computation of the simplest normal form for the singularity of two non-semisimple double zero eigenvalues

In this paper a method is presented for computing the simplest normal form for a vector field associated with the singularity of two non-semisimple double zero eigenvalues. The method developed here uses the lower order nonlinear terms in the normal form for the simplifications of higher order terms. An explicit formulae are derived, which can be used to compute the coefficients of the simplest normal form and the associated nonlinear transformation. The theoretical model for the nonlinear oscillation of a simply supported rectangular thin plate is given to demonstrate the computational efficiency of the method.