  In a one-parameter study of a noninvertible family of maps of the plane
arising in the context of a numerical integration scheme, Lorenz studied a
sequence of transitions from an attracting fixed point to "computational
chaos." As part of the transition sequence, he proposed the following as a
possible scenario for the breakup of an invariant circle: the invariant circle
develops regions of increasingly sharper curvature until at a critical
parameter value it develops cusps; beyond this parameter value, the invariant
circle fails to persist, and the system exhibits chaotic behavior on an
invariant set with loops [Lorenz, 1989]. We investigate this problem in more
detail and show that the invariant circle is actually destroyed in a global
bifurcation before it has a chance to develop cusps. Instead, the global
unstable manifolds of saddle-type periodic points are the objects which develop
cusps and subsequently "loops" or "antennae." The one-parameter study is better
understood when embedded in the full two-parameter space and viewed in the
context of the two-parameter Arnold horn structure. Certain elements of the
interplay of noninvertibility with this structure, the associated invariant
circles, periodic points and global bifurcations are examined.
