set does not belong to any of inner attractor’s shadow.

The obtained set of points would be the section of the repeller whose basin of repulsion is given by the shadow of the outer attractor. In practice, we try to obtain the image of multiple sections of the repellers using the information of pull back attractors on one fiber. The detail of the procedure is described in the appendix.

It should be noted that a pull back attractor on a fiber does not simply correspond to a section of the attractor in autonomous 2-dimensional system.[@Kloeden-Rasmussen-11] If $A\subset \Omega$ is an asymptotically stable attractor of $M$ in the view the 2-dim autonomous system, boundary of $A$ would consist of graph(s) of continuous and piece-wise smooth function of $\theta$. In such cases, the intersection of $A$ with a certain fiber $\{\theta=c\}$ should coincide with a pull back attracting set on the fiber (which may consists of multiple pull back attractors). On the other hand, for the cases with SNA, although pull-back attractor(s) for almost all fibers can be obtained just like the above case, the shape of their union over the fibers is not continuous. Thus SNA does not simply correspond to 