Abstract
The concept of normal form is used to study the dynamics of non-linear systems. In this work we describe the normal form for vector fields on 3 × 3 with linear nilpotent part made up of coupled R3n Jordan blocks. We use an algorithm based on the notion of transvectants from classical invariant theory known as boosting to equivariants in determining the normal form when the Stanley decomposition for the ring of invariants is known.
Highlights
The idea of simplification near an equilibrium goes back at least to Poincare (1880), who was among the first to bring forth the theory in a more definite form
We use an algorithm based on the notion of transvectants from classical invariant theory known as boosting to equivariants in determining the normal form when the Stanley decomposition for the ring of invariants is known
Our concern in this paper is to describe the normal form of the systemm x Ax v x, that is the set of all v such that Ax v x is in normal form where A is the linear part N33,3 from the Stanley decomposition of the ring of invariants
Summary
The idea of simplification near an equilibrium goes back at least to Poincare (1880), who was among the first to bring forth the theory in a more definite form. Cushman et al [1], using a method called covariant of special equivariant solved the problem of finding Stanley decomposition of N22, ,2. Their method begins by creating a scalar problem that is larger than the vector problem and their procedures are derived from classical invariant theory it was necessary to repeat calculations of classical invariants theory at the levels of equi-. The normal form for such a system contains only terms that belong to both the semisimple part of A and the normal form of the nilpotent, which is a coupled Takens- This example illustrates the physical significance of the study of normal forms for systems with nilpotent linear part.
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