The simplest normal form for the singularity of a pure imaginary pair and a zero eigenvalue

摘要

This paper is concerned with further reduction of the conventional normal forms of differential equations. The attention is focused on the case when the Jacobian of a system evaluated at an equilibrium has a pair of purely imaginary eigenvalues and a simple zero. Explicit, recursive formulas are derived, which can be used to compute the coefficients of the simplest normal form and the associated nonlinear transformation up to any order. It is shown that unlike conventional normal forms, the simplest normal form is unique and invariant for a fixed "form", and different "forms" of simplest normal forms are equivalent in the sense of simplicity. The explicit formulas have been implemented using the computer algebra system Maple, and an example is given to demonstrate the efficiency of the computer software.