Different scenarios for hyperbolicity breakdown in quasiperiodic area preserving twist maps

We present a computer-assisted numerical study of different bifurcations of saddle invariant tori in quasiperiodic area preserving twist maps. We detect three bifurcation scenarios. In the first scenario, the smooth bifurcation, the regularity of the invariant torus is preserved, and the stable and unstable invariant bundles collide smoothly. In the other scenarios, the spiky and folding breakdowns, the invariant torus looses smoothness, and the invariant bundles collide non-smoothly. In the former, the C(1) seminorm of the torus does not blow up while in the latter it does. Numerics suggest that after the breakdowns non-uniformly hyperbolic invariant objects persist. These are qualitatively different depending on the type of breakdown. Finally, using anti-integrable limit theory, we provide a proof of existence of non-uniformly hyperbolic invariant objects for systems very far from the integrable regime.