Collision of invariant bundles of quasi-periodic attractors in the dissipative standard map.

We perform a numerical study of the breakdown of hyperbolicity of quasi-periodic attractors in the dissipative standard map. In this study, we compute the quasi-periodic attractors together with their stable and tangent bundles. We observe that the loss of normal hyperbolicity comes from the collision of the stable and tangent bundles of the quasi-periodic attractor. We provide numerical evidence that, close to the breakdown, the angle between the invariant bundles has a linear behavior with respect to the perturbing parameter. This linear behavior agrees with the universal asymptotics of the general framework of breakdown of hyperbolic quasi-periodic tori in skew product systems.