A route to computational chaos revisited: noninvertibility and the breakup of an invariant circle

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.