  This paper develops the analysis needed to set up a Morse-Bott version of
embedded contact homology (ECH) of a contact three-manifold in certain cases.
In particular we establish a correspondence between "cascades" of holomorphic
curves in the symplectization of a Morse-Bott contact form, and holomorphic
curves in the symplectization of a nondegenerate perturbation of the contact
form. The cascades we consider must be transversely cut out and rigid. We
accomplish this by studying the adiabatic degeneration of $J$-holomorphic
curves into cascades and establishing a gluing theorem. We note our gluing
theorem satisfying appropriate transversality hypotheses should work in higher
dimensions as well. The details of ECH applications will appear elsewhere.
