W^+ \circ \tilde M$ where $\tilde W^+ \ | \ \tilde \Omega
\rightarrow \tilde \Omega$, $\tilde
W^+(\theta,z,w)=(\theta,z,\min\{w,\delta^+\})$, where $\delta^+$ denotes the amplitude of the perturbation. ( $\delta$ is set as $5.0\times 10^{-7}$ in the examples shown in this report.)

Numerical results
=================

In this section, we present some of the numerical results, namely, phase diagrams in $(a,\epsilon)$-plane and visualized images of attractors and repellers for some selected parameter values.

![Phase diagram in $a$-$\epsilon$ plane](figure-1a-s.eps "fig:"){width="45.00000%"} ![Phase diagram in $a$-$\epsilon$ plane](figure-1b-s.eps "fig:"){width="45.00000%"} ![Phase diagram in $a$-$\epsilon$ plane](figure-1c-s.eps "fig:"){width="45.00000%"}

Marks in the box (b) indicate parameter values corresponding to the following figures. Black star for fig. \[FIG:typical-example\], black $\times$ for fig. \[FIG:repeller-onset\], white $\times$ for fig. \[FIG:fractalized\]. Those plots are obtained with mesh size $(\Delta a,\Delta\epsilon)$ taken as $(0.002,0.002)$ for (a), $(0.001,0.001)$ for (b), $(0.0005,0.0002)$ for (c) and $(0.000002,0.000001)$ for the inset of (c). The observed type of the attractor or the repeller is indicated with color as illustrated in the legend box in box (b). C, T, 2T and S respectively stand for chaotic, torus, period 2 tori and strange non-chaotic attractor, and “+r” indicates the existence of a “non-trivial” repeller which is not a part of the basin boundary with 