Does this dynamical system show an “absorbing area” or a “chaotic area”?

Question

I am following the technical report by C.Mira: "Noninvertible maps: notion of chaotic area vs that of strange attractor" in order to characterize the behavior some dynamical systems of my own.

In the mentioned article, there is a definition of absorbing area as a region such that the successive images of all points of a neighborhood, from a finite number of iterations, enter into it and cannot get away after entering. In the other hand a chaotic area is an "invariant absorbing area, containing a fractal set made up of repelling cycles with increasing period... giving rise to orbits with the property of sensitivity to initial conditions".

The context and the question: I have a family of discrete-time, nonlinear, dynamical system (the same one described on this preceding question):

$$S_{D,t,f}=\left\{(x,y): (x_{n+1},y_{n+1}) = \left( \Im\left(\frac{1}{(x_{n} + y_n i+1)^{t}}\right)\sin{\frac{D}{f}}, \Re\left(\frac{1}{(x_{n} + y_n i+1)^{t}}\right)\cos{\frac{D}{f}} \right)\right\}$$

Usually most of the systems of the family arrive to a periodic orbit in relatively few iterations (hundreds or few thousands). But some of them not. This is one of those that do not arrive to an attractor, $S_{D=4, t=2, f=1}$. It is part of the region inside the red frame (the picture shows the whole family and $S_{D=4, t=2, f=1}$ is one of the systems only, click to enlarge):

Specifically $S_{D=4, t=2, f=1}$ standalone is visualized as follows for $n=64 \cdot 10^3$,$n= 10^5$, and $n=2 \cdot 10^5$ iterations (click to enlarge):

Looking at the image, it seems to have three symmetries: it is easy to see that the system generates some clusters of points in different regions of the space and they are similar, there is rotation and specular images inside the structures generated by the system. The more iterations I add, the more points get accumulated in those areas, so I tend to think that this is an absorbing area and it is not a chaotic area.

My main reason is that I cannot see "orbits" in the classical meaning of a visual "orbit". I just can see clusters of points. Unfortunately I do not reach any periodic set of points. It seems that the collection of points grows without finding a sink, always around those clusters of points. The visual result is that as $n$ growths, it gets brighter due to the accumulation of points.

Question: can I say that this is a chaotic area, or it is just a nonchaotic absorbing area? or is there another concept that express this kind of cluster-like accumulation better than those ones? Thank you.

Answer 1

I think the definition of chaotic set is somewhat imprecise : "containing a fractal set made up of repelling cycles with increasing period" should probably be "containing a fractal set in which repelling cycles are dense". And it is usual to add in the definition that there is a dense orbit in the attractor (it can be proved that if there is one dense orbit, then there is a dense $G_\delta$ of dense orbits, so from a topological point of view generic starting points in the attractor give rise to dense orbits; the repelling cycles, though dense, are only countable).

It looks like your regions are indeed chaotic attractors or at least chaotic absorbing sets.