Normal forms of vector fields with perturbation parameters and their application

This paper is concerned with the normal forms of vector fields with perturbation parameters. Usually, such a normal form is obtained via two steps: first find the normal form of a “reduced” system of the original vector field without perturbation parameters, and then add an unfolding to the normal form. This way, however, it does not yield the relation (transformation) between the original system and the normal form. A study is given in this paper to consider the role of near-identity transformations in the computation of normal forms. It is shown that using only near-identity transformations cannot generate the normal form with unfolding as expected. Such normal forms, which contain many nonlinear terms involving perturbation parameters, are not very useful in bifurcation analysis. Therefore, additional transformations are needed, resulting in a further reduction of normal forms – the simplest normal form. Examples are presented to illustrate the theoretical results.