Groebner basis methods for stationary solutions of a low-dimensional model for a shear flow

Abstract

We use Groebner basis methods to extract all stationary solutions for the 9-mode shear flow model that is described in Moehlis et al, New J. Phys. 6, 54 (2004). Using rational approximations to irrational wave numbers and algebraic manipulation techniques we reduce the problem of determining all stationary states to finding roots of a polynomial of order 30. The coefficients differ by 30 powers of 10 so that algorithms for extended precision are needed to extract the roots reliably. We find that there are eight stationary solutions consisting of two distinct states that each appear in four symmetry-related phases. We discuss extensions of these results for other flows.