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Quadratic Coupled Map

The basic one dimensional quadratic map is given by:


Where a is a constant and also acts as the only bifurcation parameter for the system. The period one fixed points are found setting tex2html_wrap_inline438 :


We can see that for fixed points to exist, tex2html_wrap_inline440 because x is a real map and thus a fixed point cannot be complex. Now we can go a couple steps further and find the higher period points. The period two points are given by:


We see once again that there is a constraint on a for the existence of period two points, namely tex2html_wrap_inline446 . This observation leads one to wonder, what values of a lead to higher period orbits? The plot above shows stable orbits of higher period developing for increasing a. At high enough a, we see the situation which is referred to as chaos: any two initial points, no matter how close, will eventually diverge.

This brings us to the main focus of this paper: given two independent systems which vary slightly in initial conditions, we seek a way to modify the systems so that they do not diverge under iteration. One proposed solution is as follows, namely the coupled system:


We have effectively changed the pair of one dimensional maps into a two dimensional map. The system is uncoupled when the parameter tex2html_wrap_inline454 , that is to say that the two systems act independently. The goal is to keep these two systems as loosely coupled as possible, but still have them synchronize. By this we mean that tex2html_wrap_inline456 . We call this synchronizing the two chaotic systems. The first method of attack on this problem is analytical. For a given a, we want to find values of tex2html_wrap_inline400 so that for close initial values of x,y, synchronization must occur. To simplify this task, we introduce the new variable tex2html_wrap_inline464 , and we are interested in when tex2html_wrap_inline466 . Through a little algebra, the map becomes:


Since tex2html_wrap_inline468 depends on x, we want to get some idea of when x itself remains bounded. In other words, we want to solve for b such that tex2html_wrap_inline476 (that is, tex2html_wrap_inline478 ). When x is at the maximum value of b, we get tex2html_wrap_inline484 , and the solution tex2html_wrap_inline486 . On the other hand, |a| must not exceed b or else when tex2html_wrap_inline492 , then tex2html_wrap_inline494 . This leads us to the requirement tex2html_wrap_inline496 . Unless this is satisfied, all solutions which do not start out at a fixed point eventually diverge. The information just obtained is useful in understanding the 1D quadratic map, but now we wish to apply it to the coupled system. We ideally would like tex2html_wrap_inline498 . Thus we impose the condition tex2html_wrap_inline500 , which can be satisfied using the triangle inequality and our bound on tex2html_wrap_inline360 as follows:


By assuming that our initial tex2html_wrap_inline504 is arbitrarily small (remember we only need to show synchronization for small initial differences), we can neglect tex2html_wrap_inline506 and proceed:


Substituting an tex2html_wrap_inline400 in this range into the above, we find the range of tex2html_wrap_inline504 for which synchronization is guaranteed:


It is reassuring to know that given a quadratically coupled map, there is a coupling parameter which will force synchronization for small enough initial variations. However, plugging in various a in the range [1.5,2.0], where interesting behavior occurs, we find that the analytically obtained values of tex2html_wrap_inline400 are between roughly 0.28 and 0.25. Remembering that lower tex2html_wrap_inline400 means stronger coupling, we see that these are very strongly coupled systems. The question arises, can we improve on this result? After all, two very restrictive relations were used, namely the maximum value of tex2html_wrap_inline360 and the triangle inequality.

The answer to this question may be sought through numerical methods. One such method is DSTOOL, a software package which allows us to see fixed points, periodic orbits, and stable and unstable manifolds, which are the points that fall into and ``originate from'' (in the sense that they fall in under the inverse map) the fixed point. DSTOOL was used mostly for these tasks, as well as to look at the iterations of specific initial points.

The second technique was a bit more involved. The most direct test for synchronization is to iterate the map several times, and see if for a given point tex2html_wrap_inline504 approaches 0. By breaking the plane into a fine grid, and iterating at each point in the grid, we can obtain a fairly good picture of the synchronization of the system. This was done using a FORTRAN program, and the results were fairly interesting. As the diagrams below show, there is good synchronization for coupling much weaker than the analytic work would indicate. Further, there is a distinct critical threshold of tex2html_wrap_inline400 at which the basin of attraction loses its connectivity. The synchronization near this value of tex2html_wrap_inline400 appears to be highly sensitive, as the following diagrams show for the value a = 1.5, which is in the chaotic regime.

Figure 3:   Basin of Attraction; a = 1.5, tex2html_wrap_inline326 , tex2html_wrap_inline328

Figure 4:   Basin of Attraction; a = 1.5, tex2html_wrap_inline332

Figure 5:   Basin of Attraction; a = 1.5, tex2html_wrap_inline336

next up previous
Next: Conclusions Up: Synchronization of Chaotic Maps Previous: Background