Symmetriereduktion polynomieller Vektorfelder

  • Symmetry reduction of polynomial vector fields

Schröders, Raphael; Walcher, Sebastian (Thesis advisor); Scheurle, Jürgen (Thesis advisor)

Aachen : RWTH Aachen University (2021)
Dissertation / PhD Thesis

Dissertation, RWTH Aachen University, 2021

Abstract

In this doctoral thesis we modify the familiar orbit space method for symmetry reduction of polynomial ordinary differential equations on $\mathbb{R}$$^{n}$ or $\mathbb{C}$$^{n}$, with the goal of developing a more feasible reduction method. The symmetries of such differential equations form a linear algebraic group. When this group is non-trivial and the algebra of polynomial invariants of the symmetry group can be finitely generated, a generator set of the invariant algebra yields a reduction map. This approach is known as orbit space reduction. This reduction method can often be problematic in practice, since even for relatively simple groups minimal generator systems of the invariant algebra may be very large, therefore the image of the reduction map is complicated and can only be embedded in a high-dimensional vector space. Thus this method is often unfeasible. On the other hand it is also possible to reduce via rational invariants. Here the minimal number of generators of the field of rational invariants is limited by n-s+1, where n is the initial dimension and s the generic orbit dimension of the symmetry group. But with rational invariants the problem of uncontrolled denominators appears. For instance, interesting invariant sets, which are induced by the symmetry, may be lost, and the existence of a reduction at a certain point cannot be guaranteed. In view of the specific problems of both reduction methods we develop a middle ground approach, with the goal of resolving the issues of very large generator sets as well as uncontrolled denominators, thus combining the advantages of the two approaches. For this purpose we apply suitably chosen localizations. Using invariant sets which are induced by the symmetry group, we show that at a given point, at which the reduction is to be carried out, there exists a localization such that the appearing denominators do not vanish at this point, and the number of required generators for the invariants is limited to at most n-s+2. Such a generator set thus yields a local reduction map to an algebraic variety, which has a codimension of at most 2 in its ambient space. In applications one is often interested in the symmetric differential equations of a given symmetry group. Therefore, in order to apply the reduction method a generator system for the symmetric polynomial vector fields, which form a module over the invariant algebra, is also required. We show that by using suitable localizations the number of generators for this module can be limited to n, provided that we impose minor restrictions on the symmetry group. For the invariants as well as for the symmetric vector fields one can use one and the same localization at a given point. The existence proofs in the general case will then be specialized to toral groups. Considering the initial problem of very large generator systems of the invariant algebra, the best possible scenario occurs for this type of group, for the number of generators can even be limited to n-s by our reduction method. Furthermore we show how to compute the required generators (for the invariants as well as symmetric vector fields), without a priori knowledge of a generator system of the invariant algebra. We illustrate the results with some concrete examples.

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