  In order to study certain questions concerning the distribution of the
overlap in Sherrington--Kirkpatrick type models, such as the chaos and
ultrametricity problems, it seems natural to study the free energy of multiple
systems with constrained overlaps. One can write analogues of Guerra's replica
symmetry breaking bound for such systems but it is not at all obvious how to
choose informative functional order parameters in these bounds. We were able to
make some progress for spherical pure $p$-spin SK models where many
computations can be made explicitly. For pure 2-spin model we prove
ultrametricity and chaos in an external field. For the pure $p$-spin model for
even $p>4$ without an external field we describe two possible values of the
overlap of two systems at different temperatures. We also prove a somewhat
unexpected result which shows that in the 2-spin model the support of the joint
overlap distribution is not always witnessed at the level of the free energy
and, for example, ultrametricity holds only in a weak sense.
