Robust attractor of non-twist systems

We present a new kind of one-dimensional attractor, which has not yet been predicted in the non-linear dynamics theory. We consider a non-linear map, which presents typical non-twist manifestations, as isochronous resonances and shearless torus. It is known that this torus corresponds to a very sturdy barrier in the phase space of some area-preserving systems. We show that when dissipation is present in the system, the shearless curve carries its robustness to the dissipative scenario. It becomes a powerful attractor, which we call shearless attractor, which is persistent under the variation of the parameters and it exchanges its stability from chaotic to quasi-periodic, or vice-versa, depending on the set of parameters.