Hi Erico,

The purpose of Definition 6.2 is to introduce the concept of chaotic set in order to define the chaotic attractor. The concept of forward limit set is described in section 6.1. At the end of Definition 6.2, it is written: "A chaotic attractor is a chaotic

set that is also an attractor.". If the chaotic orbit would not be contained in its forward limit set, then the forward limit set would not be the chaotic attractor. Note also that a chaotic attractor is an invariant set.

I am sure that if you read carefully the previous pages to definition 6.2, you would understand it without problem, at least intuitively.

I hope this might be of some help.

Miguel

Thank you, Miguel.

I was thinking in terms of an initial condition x0 outside of its omega limit set, w(x0), whose orbit under f is chaotic. Thus, if x0 is in the basin of attraction of a chaotic attractor (or in the stable manifold of a chaotic saddle) then fn(x0) converges asymptotically to a chaotic invariant set, that is w(x0). I would think that such w(x0) is a chaotic set, but not according to Alligood, since x0 was outside w(x0). What am I missing?

Erico

Sorry Erico for the delay...

As I visuaiize it, your x0 is simply a point belonging to the basin of attraction, but it does not belong  to the chaotic attractor which is itself the chaotic set which at the same time is an invariant set.

Miguel

 That's right, Miguel. x0 does not belong to the chaotic attractor, but in my view, the omega limit set, w(x0), coincides with the chaotic attractor if x0 is in its basin. I don't see why x0 has to belong to its own chaotic w(x0) to define a chaotic set. What is wrong with a definition like "A chaotic set is the forward limit set of a chaotic orbit" ?