The aim of this work is the presentation of a special sort of algorithms named by computer algebra in applications to equivariant dynamical systems. These algorithms are based on the notion of Gröbner basis, which is a set of polynomials with special properties having the same set of solutions than the given polynomials. Obviously, they are very useful for bifurcation theory.
In the first chapter the recent progress on Gröbner basis is presented. The main points here are the author’s implementation of the Hilbert series driven Buchberger algorithm, the exploitation of the sparsity of polynomials in the structural Gröbner basis detection, and the dynamic version of the Buchberger algorithm. In the conclusion the author gives the standard application of Gröbner bases to the solution of systems of polynomial equations.
The results from algorithmic invariant theory are presented in the second chapter. They are needed in order to construct a generic equivariant vector field and guarantee the “unique representation”. First the computations of invariants and equivariants using Hilbert series are described. Derksen’s algorithm using the algebraic group structure is given, and also the algorithms exploiting the Cohen-Macaulay structure. Then algorithms follow which give generators such that a unique representation is guaranteed. The computation of a Hironaka decomposition by algorithmic Noether normalization and the computation of a Stanley decomposition of the module of equivariants are presented.
In the third chapter the author demonstrates the usage of computer algebra in symmetric bifurcation theory. As a simple example secondary Hopf bifurcation with \(D_3\)-symmetry is considered. Here the results of the second chapter about algorithmic determination of invariants and equivariants are applied in order to find a generic equivariant vector field.
The concluding chapter is devoted to an orbit space reduction method in equivariant dynamical systems theory with application of the suggested algorithms. The methods for computation of Gröbner bases are applied here in multiples ways: for the computation of the generic equivariant vector field, for the computations of the second chapter, rewriting an invariant in terms of a Hilbert basis, checking radicals and many other aspects.
The work gives a survey on the author’s results linking very different mathematical branches and points out the future developments.