  In this paper we obtain a detailed description of the global and cocycle
attractors for the skew-product semiflows induced by the mild solutions of a
family of scalar linear-dissipative parabolic problems over a minimal and
uniquely ergodic flow. We consider the case of null upper Lyapunov exponent for
the linear part of the problem. Then, two different types of attractors can
appear, depending on whether the linear equations have a bounded or an
unbounded associated real cocycle. In the first case (e.g.~in periodic
equations), the structure of the attractor is simple, whereas in the second
case (which occurs in aperiodic equations), the attractor is a pinched set with
a complicated structure. We describe situations when the attractor is chaotic
in measure in the sense of Li-Yorke. Besides, we obtain a non-autonomous
discontinuous pitchfork bifurcation scenario for concave equations, applicable
for instance to a linear-dissipative version of the Chafee-Infante equation.
