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We numerically study quasiperiodic normally hyperbolic attracting invariantcircles that appear for certain parameter values in a family ofthree-dimensional Henon-like maps. These parameter values make up contoursegments in the parameter space where the invariant circles have constantirrational rotation number. At the edges of these segments we find a breakdownof the hyperbolicity of the invariant circle. We observe the collision and lossof smoothness of two of the invariant Lyapunov bundles while the Lyapunovexponents all remain distinct. This is very similar to the breakdown ofquasiperiodic normally hyperbolic invariant circles studied in previous worksthat have mostly focused on skew product type systems along with a few otherspecial types of systems.  The numerical tools we use for finding the invariant circles and calculatingrotation numbers, Lyapunov exponents and bundles are based on the recentlydeveloped Weighted Birkhoff method. To apply all of these tools we need for theinvariant circles to be attracting (or repelling) and for the system to beinvertible. This is a severe restriction compared to alternative methods, butit is very numerically efficient and allows us to study even highly irregularcircles.

Breakdown of hyperbolicity for quasiperiodic attracting invariant circles in a family of three-dimensional Henon-like maps