COMPUTATION OF NORMAL FORMS FOR COUPLED N33,…,3 SYSTEMS

Abstract

The concept of normal form was used to study the dynamics of non-linear systems. The problem of describing the normal forms for a system of di erential equations at equilibrium with nilpotent linear part is solvable once the ring of invariants associated to the system is known. This study was concerned with the description of the normal form for di erential system with nilpotent linear part made up of n 3 3 Jordan blocks. The normal form of the systems with nilpotent linear part has the structure of a module of equivariants and is best described by giving its Stanley decomposition. An algorithm based on the notion of transvectants from classical invariant theory was used to determine the Stanley decomposition for the ring of invariants for the coupled systems when the Stanley decompositions of the Jordan blocks of the linear part are known at each stage. The Stanley decomposition for the ring of invariants was then veri ed by developing a table function denoted by T(3)n, where (3)n is the dimension of the linear part. The normal form have been obtained by boosting the Stanley decomposition for the ring of invariants to Stanley Decomposition of the module of equivariants. To put the normal form into practical use, asymptotic unfolding for a single block was included as an exposition to show the inclusion of arbitrarily parameters. The asymptotic unfolding was observed to exhibit all behavior which can be detected in perturbation of the original system up to a given degree, such as existence and stability