  We numerically study quasiperiodic normally hyperbolic attracting invariant
circles that appear for certain parameter values in a family of
three-dimensional Henon-like maps. These parameter values make up contour
segments in the parameter space where the invariant circles have constant
irrational rotation number. At the edges of these segments we find a breakdown
of the hyperbolicity of the invariant circle. We observe the collision and loss
of smoothness of two of the invariant Lyapunov bundles while the Lyapunov
exponents all remain distinct. This is very similar to the breakdown of
quasiperiodic normally hyperbolic invariant circles studied in previous works
that have mostly focused on skew product type systems along with a few other
special types of systems.
  The numerical tools we use for finding the invariant circles and calculating
rotation numbers, Lyapunov exponents and bundles are based on the recently
developed Weighted Birkhoff method. To apply all of these tools we need for the
invariant circles to be attracting (or repelling) and for the system to be
invertible. This is a severe restriction compared to alternative methods, but
it is very numerically efficient and allows us to study even highly irregular
circles.
