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In this paper we obtain a detailed description of the global and cocycleattractors for the skew-product semiflows induced by the mild solutions of afamily of scalar linear-dissipative parabolic problems over a minimal anduniquely ergodic flow. We consider the case of null upper Lyapunov exponent forthe linear part of the problem. Then, two different types of attractors canappear, depending on whether the linear equations have a bounded or anunbounded associated real cocycle. In the first case (e.g.~in periodicequations), the structure of the attractor is simple, whereas in the secondcase (which occurs in aperiodic equations), the attractor is a pinched set witha complicated structure. We describe situations when the attractor is chaoticin measure in the sense of Li-Yorke. Besides, we obtain a non-autonomousdiscontinuous pitchfork bifurcation scenario for concave equations, applicablefor instance to a linear-dissipative version of the Chafee-Infante equation.

Global an cocycle atractor for non-utonomous reaction-difussion equations . The case of null upper Lyapunov exponente