Considering the much-studied [4] logistic map, which is not piecewise-linear but consists of an inverted parabola of height a, it is possible to embed traps in a region which is only visited during the initial transient. On the chaotic attractor there is no chance of trapping. Whether or not such a system traps will then require the initial state to lie in a certain region, which may be fractal. Then, if such a system is knocked off its final attracting chaotic solution by some external stimulus, the transient may be re-entered and the chance of trapping revived. Thus, in a real world example where attracting solutions are the exception rather than the norm, and where most of the chaotic behaviour consists of repeatedly-excited transients, traps outside the chaotically attracting regions may be important.
It is also possible to envisage a regression of nested maps; a small logistic map may be constructed within the main logistic map with the new zero on the fixed point where the iteration line intersects the main map. One can then have trapping which is itself chaotic; and the added noise will perturb the motion from the smaller chaotic region to the larger. Thus we can have fixed point traps, limit cycle traps and chaotic traps; these can lie within or outside a chaotic attractor.