| We investigate three separate but related problems. The motivating problem is the elliptical instability in inviscid fluid dynamics. The presence of a family of reversing symmetries in the fluid dynamical formulation motivates a second problem---the study of reversible dynamical systems via normal-form theory. The construction of normal-forms requires that the solutions to a certain PDE be polynomials, and this leads to the third problem---the interplay between algebra and analysis in finding these solutions via classical invariant theory.; There are two parameters in the elliptical instability problem, and these occur as well in the finite dimensional dynamical systems that are used as models of the fluid dynamical system. A complete normal-form analysis therefore cannot be restricted to the most common case (of codimension one), but must also include those of codimension two.; The normal-form analysis is general in that it applies to a wider class of reversible dynamical systems than that representing the fluid dynamical problem. However, where it is necessary to choose among several reversing symmetries, we choose the ones that arise from the fluid dynamical problem. We also incorporate in our analysis the relevant symmetries that arise from the fluid dynamical problem. Emphasis is given to a special parameter range ("weak perturbation of rotational symmetry") since much of the literature of the subject refers to this range. |
