Master degree research (diploma)
"On the properties of quadratic maps under parametric perturbations".

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.