  An autonomous system of ordinary differential equations in the plane with a
centre-saddle bifurcation is considered. The influence of time damped
perturbations with power-law asymptotics is investigated. The particular
solutions tending at infinity to the fixed points of the limiting system are
considered. The stability of these solutions is analyzed when the bifurcation
parameter of the unperturbed system takes critical and non-critical values.
Conditions that ensure the persistence of the bifurcation in the perturbed
system are described. When the bifurcation is broken, a pair of solutions
tending to a degenerate fixed point of the limiting system appears in the
critical case. It is shown that, depending on the structure and the parameters
of the perturbations, one of these solutions can be stable, metastable or
unstable, while the other solution is always unstable.
