Using quantum mechanics calculations, we theoretically study the dissociation and ionization dynamics of the hydrogen-molecule ion in strong laser fields. Having prepared the nuclear wave packet of H2+ in a specific vibrational state, a pump laser is used to produce a vibrational excitation, leading to quasi-periodical vibration without ionization. Then, a time-delayed few-cycle laser is applied to trigger the dissociation or ionization of H2+. Both the time delay and the intensity of the probe laser alter the competition between dissociation and ionization. We also explore the dependence of kinetic-energy release spectra of fragments on the time delay, showing that the channels of above-threshold dissociation and below-threshold dissociation are opened and closed periodically. Also, dissociation from different channels is influenced by nuclear motion. The dissociation mechanism has been described in detail using the Floquet picture. This work provides a useful method for steering the electronic and nuclear dynamics of diatomic molecules in intense laser fields.
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