(b), and $[0.84,0.85]$ for (c) respectively.\[FIG:fractalized\]](figure-6b.eps "fig:"){width="46.00000%"} ![Repeller and SNA emerged via fractalization route, $(a,\epsilon)=(0.75, 0.46673)$. The pull-back attractor (SNA) and the repeller, as well as boundary of absorbing areas, are plotted for 4000 (a), 2000 (b and c) target fibers, with $\theta$ values in $[0,1]$ for (a), $[0.8,0.9]$ for (b), and $[0.84,0.85]$ for (c) respectively.\[FIG:fractalized\]](figure-6c.eps "fig:"){width="46.00000%"}

summary
=======

We proposed a method to observe a kind of “repeller” in quasi-periodically forced logistic map system and a method to distinguish “smooth” attractors and SNAs, using auxiliary dynamical system describing evolution of a “line segment” on a fiber. This method enables to obtain images of repellers and bifurcation diagrams efficiently.

Some numerical results obtained with these methods are exhibited, namely,

-   phase diagram with some revision in the borderline between smooth torus attractor and SNA,

-   detailed images of a repellers with non-trivial ring-like structure,

-   absorbing area surrounded by curves with many peaks observed nearly at the emergence of a non-trivial repeller, which exhibits a symptom of non-uniform saddle-node bifurcation in the segment map system,

-   nested structure of multiple absorbing areas and associated multiple repellers observed near TDT critical point,

-   image of repeller with apparently noisy structure which 