Applications of SAGBI-bases in dynamics. (English) Zbl 1028.65137
From the abstract: The classical reduction techniques of bifurcation theory, Lyapunov-Schmidt reduction and centre manifold reduction, are investigated where symmetry is present. The symmetry is given by the action of a finite or continuous group. The symmetry is exploited systematically by using the algebraic structure of the module of equivariant polynomial tuples. We generalize the concept of SAGBI-bases to module-SAGBI basis and explain how to use this concept within the two reduction techniques. Examples illustrate the theoretical results. In particular the reduction onto centre manifold is performed for the Taylor-Couette problem with \(SO(2)\times O(2)\)-symmetry.
In this interdisciplinary paper we suggest the application of computational algebra to the theory of dynamical systems. This is on the line of previous work on orbit space reduction [cf. K. Gatermann, Computer algebra methods for equivariant dynamical systems. Lect. Notes Math. 1728 (2000; Zbl 0944.65131)] and singularity theory for Hamiltonian systems.
In this interdisciplinary paper we suggest the application of computational algebra to the theory of dynamical systems. This is on the line of previous work on orbit space reduction [cf. K. Gatermann, Computer algebra methods for equivariant dynamical systems. Lect. Notes Math. 1728 (2000; Zbl 0944.65131)] and singularity theory for Hamiltonian systems.
Reviewer: Jan Sieber (Bristol)
MSC:
| 65P30 | Numerical bifurcation problems |
| 68W30 | Symbolic computation and algebraic computation |
| 37M20 | Computational methods for bifurcation problems in dynamical systems |
| 37J20 | Bifurcation problems for finite-dimensional Hamiltonian and Lagrangian systems |
| 37C80 | Symmetries, equivariant dynamical systems (MSC2010) |
| 37G40 | Dynamical aspects of symmetries, equivariant bifurcation theory |
| 13P10 | Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) |
| 34C20 | Transformation and reduction of ordinary differential equations and systems, normal forms |
| 34C23 | Bifurcation theory for ordinary differential equations |
| 35B32 | Bifurcations in context of PDEs |
| 76E06 | Convection in hydrodynamic stability |
Keywords:
SAGBI basis; bifurcation theory; Lyapunov-Schmidt reduction; centre manifold reduction; symmetry; Taylor-Couette problem; computational algebra; dynamical systems; singular theory; Hamiltonian systemsCitations:
Zbl 0944.65131Software:
Macaulay2References:
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