  Linear stability analysis of the whole spectrum of static hedgehog solutions
of the Skyrme model on the three-sphere of radius L is carried out. It turns
out that only solutions that in the limit of infinite L tend to skyrmions
(localized at the poles) are linearly stable. The other solutions are unstable
and, for a given solution, the number of instabilities, for L sufficiently
large, is equal to the index of a harmonic map to which this solution tends
pointwise in the limit of infinite L. Solutions which tends pointwise to
harmonic maps and which in addition have a definite parity, undergo a
transition by +1 in the number of instabilities as L grows. Due to the
instability, new solutions, with spontaneously broken reflection symmetry,
appeare by bifurcations. In the case of the 1-skyrmion this critical phenomenon
can be fully described analytically. As a result, in some neighbourhood of
critical radius at which 1-skyrmion bifurcates from the identity solution, one
gets unique series expansions for the profile of the 1-skyrmion and for its
energy, though the series coefficients, due to nonlinearity, are not known in
general form. To the author's best knowledge the series were not given in
literature so far. A similar mechanism of spontaneous breaking of parity is
also observed when other solutions appear by bifurcations from symmetric
solutions.
