also strongly suggested from observation of the parameter dependence of the size or the smoothness of the attractor (i.e., crisis-like drastic enlargement of an attractor or “fractalization” of a torus), while there seem to be no visible basin boundary in its neighborhood. In such cases it is not trivial task to identify and to locate such presumed repellers. One possible approach to locate such invariant sets is to consider a perturbation to the invariant sets in the system without external forcing. However, this method is applicable only for limited class of invariant sets (i.e., a torus or composition of multiple tori). Another useful approach is to approximate the quasi-periodic forcing with periodic one[@Pikovsky-Feudel-95; @Chastell-etal-95], and thus approximate 1+1-dimensional skew-product system with 1-parameter family of 1-dimensional autonomous systems. This method, known as rational approximation (RA) method, have been used as a major tool to observe such invariant sets in numerical researches. It has been used for the classification of the bifurcation to create SNA in QPLM [@Heagy-Hammel-94; @Nishikawa-Kaneko-96; @Witt-etal-97; @Prasad-etal-97; @Feudel-etal-98; @Kim-etal-03], and corresponding types of bifurcation are observed also in QP-forced higher dimensional systems [@Osigna-Feudel-00; @Kim-Lim-05; @Lim-Kim-05]. Though the observed features of the “repeller” obtained in RA are consistent with 