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Invariant sets forced by symmetry. (English) Zbl 1417.37113

Summary: Given a linear (algebraic) group \(G\) acting on real or complex \(n\)-space, we determine all the common invariant sets of \(G\)-symmetric vector fields. It turns out that the investigation of certain algebraic varieties is sufficient to characterize these invariant sets forced by symmetry. Toral, compact and reductive groups are discussed in some detail, and examples, including a Couette-Taylor system, are presented.

MSC:

37C80 Symmetries, equivariant dynamical systems (MSC2010)
37C15 Topological and differentiable equivalence, conjugacy, moduli, classification of dynamical systems
34C14 Symmetries, invariants of ordinary differential equations
34C20 Transformation and reduction of ordinary differential equations and systems, normal forms
58J70 Invariance and symmetry properties for PDEs on manifolds
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