  Bifurcation phenomena are ubiquitous in elasticity, but their study is often
limited to linear perturbation or numerical analysis since second or higher
variations are often beyond an analytic treatment. Here, we review two key
mathematical ideas, namely, the splitting lemma and the determinacy of a
function, and show how they can be fruitfully used to derive a reduced
function, named Landau expansion in the paper, that allows us to give a simple
but rigorous description of the bifurcation scenario, including the stability
of the equilibrium solutions. We apply these ideas to a paradigmatic example
with potential applications to various softly constrained physical systems and
biological tissues: a stretchable elastic ring under pressure. We prove the
existence of a tricritical point and find bistability effects and hysteresis
when the stretching modulus is sufficiently small. These results seem to be in
qualitative agreement with some recent experiments on heart cells.
