Invariants of the Rotation Group

Valery Romanovski () and Douglas Shafer ()
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Valery Romanovski: University of Maribor, Center for Applied Mathematics & Theorectical Physics
Douglas Shafer: University of North Carolina, Mathematics Dept.

Chapter Chapter 5 in The Center and Cyclicity Problems, 2009, pp 1-35 from Springer

Abstract: In Section 3.5 we stated the conjecture that the center variety of family (3.3), or equivalently of family (3.69), always contains the variety V(Isym) as a component. This variety V(Isym) always contains the set R that corresponds to the time-reversible systems within family (3.3) or (3.69), which, when they arise through the complexiӿcation of a real family (3.2), generalize systems that have a line of symmetry passing through the origin. In Section 3.5 we had left incomplete a full characterization of R. To derive it we are led to a development of some aspects of the theory of invariants of complex systems of differential equations. Using this theory, we will complete the characterization of R and show that V(Isym) is actually its Zariski closure, the smallest variety that contains it. In the ӿnal section we will also apply the theory of invariants to derive a sharp bound on the number of axes of symmetry of a real planar system of differential equations.

Keywords: Characteristic Vector; Phase Portrait; Rotation Group; Polynomial System; Conjugate Variable (search for similar items in EconPapers)
Date: 2009
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DOI: 10.1007/978-0-8176-4727-8_5

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