Precession driven flows are found in any rotating container filled with
liquid, when the rotation axis itself rotates about a secondary axis that is
fixed in an inertial frame of reference. Because of its relevance for planetary
fluid layers, many works consider spheroidal containers, where the uniform
vorticity component of the bulk flow is reliably given by the well-known
equations obtained by Busse in 1968. So far however, no analytical result on
the solutions is available. Moreover, the cases where multiple flows can
coexist have not been investigated in details since their discovery by Noir et
al. (2003). In this work, we aim at deriving analytical results on the
solutions, aiming in particular at, first estimating the ranges of parameters
where multiple solutions exist, and second studying quantitatively their
stability. Using the models recently proposed by Noir \& C{\'e}bron (2013),
which are more generic in the inviscid limit than the equations of Busse, we
analytically describe these solutions, their conditions of existence, and their
stability in a systematic manner. We then successfully compare these analytical
results with the theory of Busse (1968). Dynamical model equations are finally
proposed to investigate the stability of the solutions, which allows to
describe the bifurcation of the unstable flow solution. We also report for the
first time the possibility that time-dependent multiple flows can coexist in
precessing triaxial ellipsoids. Numerical integrations of the algebraic and
differential equations have been efficiently performed with the dedicated
script FLIPPER (supplementary material)
