is the counterpart of SNA.

By the use of segment map a similarity between attractors with riddled basin and SNAs become apparent, and the relevance between various phenomena including “fractalization of torus” and “emergence of ring-like shaped repeller” and (easily observable) bifurcations in the segment map (i.e., bifurcations accompanied with qualitative changes of attractors) are also indicated.

At parameter values where the “fractalization” occur, it seems that the transformation from a smooth torus to SNA and the emergence of a new repeller occur simultaneously. We will report detailed analysis on this phenomenon in the forthcoming paper.

Our method make the most of the fact that the dynamics on the fiber is one dimensional map. Thus it would not be straight forward to apply our method to systems with higher dimensional fiber dynamics. On the other hand, the dynamics of the forcing could be easily generalized, for example, to chaotic ones. As the attraction to SNA in quasi-periodically forced systems could be regarded as a special class of the weak generalized synchronization phenomena [@Singh-etal-08; @Keller-Otani-13; @Keller-etal-13], application of our method to systems with chaotic driving would help to obtain some new perspective for generalized synchronization phenomena between non-identical chaotic elements.

We 