  Pleshkan proved in 1969 that, up to a linear transformation and a constant
rescaling of time, there are four isochrones in the family of cubic centers
with homogeneous nonlinearities $\mathscr C_3.$ In this paper we prove that if
we perturb any of these isochrones inside $\mathscr C_3,$ then at most two
critical periods bifurcate from its period annulus. Moreover we show that, for
each $k=0,1,2,$ there are perturbations giving rise to exactly $k$ critical
periods. As a byproduct, we obtain a partial result for the analogous problem
in the family of quadratic centers $\mathscr C_2.$ Loud proved in 1964 that, up
to a linear transformation and a constant rescaling of time, there are four
isochrones in $\mathscr C_2.$ We prove that if we perturb three of them inside
$\mathscr C_2,$ then at most one critical period bifurcates from its period
annulus. In addition, for each $k=0,1,$ we show that there are perturbations
giving rise to exactly $k$ critical periods. The quadratic isochronous center
that we do not consider displays some peculiarities that are discussed at the
end of the paper.
