A matching pursuit technique for computing the simplest normal forms of vector fields

This paper presents a matching pursuit technique for computing the simplest normal forms of vector fields. First a simple, explicit recursive formula is derived for general differential equations, which reduces computation to the minimum. Then a matching pursuit technique is introduced and applied to the Takens–Bogdanov dynamical singularity. It is shown that unlike other methods for computing normal forms, the technique using matching pursuit does not need any algebraic constraints which are required for the existence of the simplest normal form. The efficient method and matching pursuit technique, which have been implemented using Maple, can be “automatically” executed on various computer systems. A number of examples are presented to demonstrate the advantages of the technique.