It has been shown [1], that in some cases, in chaos set **A** the
sub-set **B** can exist, in which the chaos behavior can be stabilized
by external parametrical perturbations.

My main goal was to create a computer model, which represent the conditions
of stabilization, try to find the **B** set and analyze it.

I have explored the **f(x)=ax(1-x)** map. As the first step I've
found the **A** set (the set of chaotic behavior) and as a second step,
by using different parameters combination (that can be consider as parameter
perturbations of the system) I've found the **B** set (the set, where
some stable cycles exists).

In the first model, at the first time step I used **a1**, on
the second - **a2**, on the third - **a3**, on the fourth - again
**a1**, etc.

In the other model I used more complicated sequence, but also only three different values of the map parameter.

All parameters have been taken from the interval, where the probability
of chaotic behavior is more high. For the **f(x)=ax(1-x)** map this
is [3,8 4,0] interval. So, all values of parameter **a** have
been taken from this interval.

As result, I've got several solutions, which can be represent as a points
of different colors in the space of **a1**, **a2** and **a3**
parameters. The color of point corresponds to the period of a stable cycle.
Or they can be represent as a 2D layers with **a1** and **a2** as
axis with fixed **a3**.

Ok, I think it's enough.

1. Here is the (PDF and PS) article which is related to this work (in Russian)

2. Abstract only. In English (PDF and PS)

2. Here is low-res AVI file (2MB), which
was created from 2D layers. (axis- **a1** & **a2**, different
layers - **a3**). Size is ~2MB

3. Here is 3D model of **B** set, as result
of more complicated perturbation. This is FLC file, inside ZIP archive.
You can install and use Autodesk Animation Player for Windows to see and
playback FLC file (AAWIN.ZIP archive 203KB).

[1] A.Yu.Loskutov, S.D.Rybalko. Parametric perturbations and suppression of chaos in n-dimensional maps. Preprint ICTP IC/94/347, Trieste, Italy, 1994.