A recent article in Chaos [Barreto et al., Chaos, 13(1):151-164, 2003] describes a drive-response system coupling a parametrized family of noninvertible modified baker maps with a nonlinear response. The article describes the effect of noninvertibility on the synchronization set (i.e. the attractor). Namely, at the bifurcation parameter, the synchronization set has a cusp. For every parameter after the bifurcation, there is a multivalued synchronization set with self-intersections, or loops. See the first animation. This example is a special case in which the noninvertibility occurs at a critical point of the synchronization set. In contrast, the synchronization set generically continues to be a smooth nonintersecting curve even after the onset of noninvertibility. See Animations 2 and 3. This is described more fully in an upcoming article by Josic and Sander.

\(u\) versus \(y\) in the exceptional case of Barreto et al. As the parameter varies, a smooth manifold forms a cusp, which turns into a loop.

\(u\) versus \(y\) in the typical case; the noninvertibility does not occur at a critical point. Therefore the synchronization set becomes multivalued as a graph but retains its smooth manifold structure.

A closeup view of the main loop in the previous animation.