  Dynamical billiards, or the behavior of a particle traveling in a planar
region $D$ undergoing elastic collisions with the boundary, has been
extensively studied and is used to model many physical phenomena such as a
Boltzmann gas. Of particular interest are the dispersing billiards, where $D$
consists of a union of finitely many open convex regions. These billiard flows
are known to be ergodic and to possess the $K$-property. However, Turaev and
Rom-Kedar (1998) proved that for dispersing systems permitting singular
periodic orbits, there exists a family of smooth Hamiltonian flows with regions
of stability near such orbits, converging to the billiard flow. They conjecture
that systems possessing such singular periodic orbits are dense in the space of
all dispersing billiard systems and remark that if this conjecture is true then
every dispersing billiard system is arbitrarily close to a non-ergodic smooth
Hamiltonian flow with regions of stability. In this paper, we consider billiard
tables consisting of the complement to a union of open unit disks with disjoint
closures. We present a partial solution to this conjecture by showing that if
the system possesses a near-singular periodic orbit satisfying certain
conditions, then it can be perturbed to a system that permits a singular
periodic orbit. We comment on the assumptions of our theorem that must be
removed to prove the conjecture of Turaev and Rom-Kedar for these systems.
