  The uncorrelated (``sequential'') and correlated (``nonsequential'') double
ionization of the H2 molecule in strong laser pulses is investigated using the
tools of nonlinear dynamics. We focus on the phase-space dynamics of this
system, specifically by finding the dynamical structures that regulate these
ionization processes. The emerging picture complements the recollision scenario
by clarifying the distinct roles played by the recolliding and core electrons.
Our analysis leads to verifiable predictions of the intensities where
qualitative changes in ionization occur. We also show how these findings depend
on the internuclear distance.
