|Title||Zeros of Equivariant Vector Fields: Algorithms for an Invariant Approach|
|Author(s)|| Patrick A. Worfolk|
|Type||Article in Journal|
|Abstract||This paper exploits the symmetry structure of the solution set. The orbit of any zero under the symmetry group of the vector field gives a set of solutions. Each orbit of points is collapsed to a single point by considering polynomials invariant under the action of the symmetry group. The goal is to derive a set of equations in the invariants whose zeros coincide with the zeros of the vector field. The number of solutions to these new equations will correspond to the number of orbits of zeros of the original equations. This approach has been used by Jaric et al. (1984) with crystallographic point groups. The algorithms needed naturally lend themselves to symbolic computer algebra techniques. This paper emphasizes computable results. The linearization of a vector field at an equilibrium point determines local stability and bifurcation. Frequently the most complicated dynamical behavior of a vector field|
occurs when parameter values are close to those where the linearization is degenerate. Therefore we are also interested in computing relations on the parameters which result in degenerate linearizations at the equilibrium points. We show how these relations can be written in terms of the invariants. In Section 2, we present our notation and summarize the mathematical theory of polynomial invariants. This discussion includes little more than the material we will need for our analysis and we do not give any proofs of the results which may be found easily in the literature. We have relied heavily here on the work of Stanley (1979) and
Sturmfels (1993). In Section 3, we give a brief summary of the results from Grobner basis theory on which some of our techniques rely. Grobner bases are the computational tool developed by Buchberger (1985).
|Journal||Journal of Symbolic Computation|