systems: the existence of SNA is indicated for the case where the existence of continuous attractor is excluded by topological constraint[@Hunt-Ott-01; @Kim-ETAL-03] and the emergence of SNA at non-uniform saddle-node bifurcation is proven [@Jaeger-09]. As for higher dimensional and/or irreversible systems, also where large part of the numerical studies have been done, there still many phenomena which have not been understood clearly.

When a quasi-periodic external force is applied to such systems that possibly exhibit chaotic behavior even without external forcing, it seems natural to expect that unstable invariant sets with complicated structure would play essential role in the dynamics[@Badard-08; @Glendinning-ETAL-06]. In quasi periodically forced circle map system with relatively strong modulation (thus it is no longer reversible), existence of such complicated structure is indicated using densely distributed winding number of the orbits[@Glendinning-ETAL-09]. Also in numerical researches, repelling invariant sets with complicated structure are observed, for example, as a basin boundaries between two distinct attractors, and the boundary crisis has been identified as one of the routes to SNA. Thus such repelling invariant sets are considered to be directly relevant with the emergence of SNA at least for such cases.

In some other cases, the existence of repelling sets is 