We study wave packet dynamics of a Bose condensate in a periodically shaken trap. Dichotomy, that is, dynamic splitting of the condensate, and dynamic stabilization are analyzed in analogy with similar phenomena in the domain of atoms in strong laser fields. 03.75.Fi,32.80.RM,05.30.Jp Recently it has become possible to prepare BoseEinstein condensates of alkali gases [1] with up to 10 7 magnetically trapped atoms. The condensed state is a macroscopically populated quantum state well localized in the magnetic trap. It is, therefore, an ideal tool to study wave packet dynamics under experimentally feasible conditions. There are many interesting quantum phenomena resulting from the electronic wave packet dynamics; in this Letter we refer, in particular, to the phenomena of wave packet dichotomy and stabilization exhibited by an electron bound by an atomic potential in presence of a strong laser field [2]. We argue that the same phenomena occur in the dynamics of the condensate wave function. The analogy is based on the fact that in the frame of reference moving with the free electron oscillating in the field, the Kramers-Henneberger frame, the effect of the laser is equivalent to a periodic shaking of the atomic potential along the laser polarization axis. A condensate in a periodically shaken trap could, therefore, a priori show a similar behavior.As the intense laser field drives the electron, the process of ionization of the atom occurs. By increasing the laser intensity, one normally increases the ionization rate. However, for very intense fields of high frequency, this rate eventually starts to decrease with intensity -this is called atomic stabilization [2]. In this process the electronic wave packet remains bounded, i.e. well localized in space (without spreading), although highly distorted due to the combined effects of the laser field and the atomic potential . This effective atom-laser potential exhibits a double well structure which splits the electronic wave packet into two spatially separate parts; this effect is called dichotomy. To achieve stabilization it is necessary to turn on the laser adiabatically in order to ensure that the atomic ground state will evolve to the ground state of this effective atom-laser potential. This type of stabilization [3] has never been observed experimentally [4], since it requires very intense high frequency fields, which currently can only be generated in a form of a very short pulse; as an electron in an atom is highly unstable, it would thus be most likely ionized during the turn-on of such a pulse.Let us analyze the analogy between the electron and the condensate in more detail. The electron bound by an atomic potential U ( r) and interacting with a laser field of amplitude E e z (polarized along the z-direction) is, for our purposes, best described in the Kramers-Henneberger frame of reference, in which the interaction with the laser field results in an effective time dependent "atomic" potential:where α L = (eE)/(m e ω 2 L ) is the electron excursion amplitude...