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.
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