  The global bifurcation diagrams for two different one-parametric
perturbations ($+\lambda x$ and $+\lambda x^2$) of a dissipative scalar
nonautonomous ordinary differential equation $x'=f(t,x)$ are described assuming
that 0 is a constant solution, that $f$ is recurrent in $t$, and that its first
derivative with respect to $x$ is a strictly concave function. The use of the
skewproduct formalism allows us to identify bifurcations with changes in the
number of minimal sets and in the shape of the global attractor. In the case of
perturbation $+\lambda x$, a so-called global generalized pitchfork bifurcation
may arise, with the particularity of lack of an analogue in autonomous
dynamics. This new bifurcation pattern is extensively investigated in this
work.
