The creation of strange non-chaotic attractors in non-smooth saddle-node bifurcations

We propose a general mechanism by which strange non-chaotic attractors (SNA) can be created during the collision of invariant tori in quasiperiodically forced systems, and then describe rigorously how this mechanism is implemented in certain parameter families of quasiperiodically forced interval maps. In these families a stable and an unstable invariant circle undergo a saddle-node bifurcation, but instead of a neutral invariant circle there exists a strange non-chaotic attractor-repellor pair at the bifurcation point. This is accompanied by the existence of a ‘sink-source-orbit’, meaning an orbit with positive Lyapunov exponent both forwards and backwards in time, in the intersection of the attractor and the repellor. Unlike previous proofs for the existence of SNA, which are all restricted to very specific classes and depend on very particular properties of the considered systems, the approach developed here gives a clear geometric intuition about what happens and should allow to treat a number of different situations in a similar way. As an example, we add the description of strange non-chaotic attractors with a certain inherent symmetry, as they occur in non-smooth pitchfork bifurcations.