Computation of normal forms via a perturbation technique

Considering qualitative behavior of a non-linear dynamical system often leads to first simplifying the differential equations or finding their normal forms. A perturbation technique for computing normal forms is presented. This technique, associated with the method of multiple scales, can be used to systematically find a unique form for a given set of differential equations. The technique is discussed in detail through the analysis of Hopf bifurcation. It is shown that for Hopf bifurcation, the method only requires solving two dimensional matrix systems for any higher order normal forms of a general n-dimensional system. With the aid of the symbolic language Maple, this approach is straightforward, and is computationally efficient and fast. Furthermore, a simple Verification scheme is given for verifying the normal forms and associated non-linear transformations obtained using any methodology. Examples are presented to demonstrate the applicability of the perturbation technique and the computation efficiency of the Maple program. (C) 1998 Academic Press Limited.