Different Attractors for Even/Odd Iterations

## Question:

Hi. My name is Joe. I think this is wonderful of you to put this info on the web for free. I am not able to go "back to school" as I would like, so this is the best access that I have to this kind of info. Thanx.

I do have a question in the "Phase control maps" chapter. (If this isn't the right place to ask questions or if this isn't allowed please let me know and I will go away). While playing with the "attractor demonstration" applet, I notice that with g set at 3.2, all the even iterations show the attractor split between two points (.5, .5) and (.8, .8). This makes sense to me and seems normal. (and it goes along with your pictures) However, all the odd iterations seem to settle out at one point, (.5, .5). This I do not understand. Why do the even iterations show both points of the attractor while the odd iterations only show one? This is not a problem with the applet, as I have another program that allows me to set up the same graph and it shows the same result. It also shows me that if the Xo value is below .68, y settles at .5. If Xo is above .68, it settles at .8.

Please help me to understand this. If this whole thing is a stupid question please let me know and I won't bother you any more. Thank you for your time. "j"

Hi Joe,
It's J. D. Jones again...Remember, the guy who researched the Power Mac - Netscape problem. I am now wearing my nonlinear dynamics hat. This is the place to get your questions answered, within the limits of the time available to me.

It is not clear to me what level of mathematics education you are working from. We will have to speak calculus and vector fields to be rigorous with your question. In my opinion life is too short to deal with that now so I will offer a sort of hand waving, common sense argument for why even and odd iterations behave differently and may depend on the starting point of the iteration track.

To begin with, go to the attractor demo http://mcanv.com/oaad.html, click on Plot to set up the function and set the gain to g=3.2. Then click on the action button enough times to settle out on the attractor. Make a note of the y value of the upper branch (about 0.8) and the lower branch (about 0.5). Then shift to the i=1 for the base iteration and note the y value of the intersections of this iteration curve with the y=x line. You should find that the outside intersections have the same y values that we got for upper and lower branches when iterating with the zeroth iteration (the function itself). The center intersection is at the value about 0.68.

Now click on the Action button a few times to set up an iteration track with that first (i=1) iteration as the base. You should see the iteration track captured by the loop between the middle and lower intersections. If the starting point for the iteration track had hit in the upper loop, the track would have been trapped in that upper loop and drawn to the upper intersection. Unfortunately in the online version of this program you do not have control of the iteration starting point.

It is as though the odd iterations can see the individual attracting fixed points and so be captured by one or the other depending on the starting point. The even iterations act as though they are unaware that individual fixed points exist. The only fixed point seen by them is the central one and it attracts them, keeping them from getting too far away from that point. Play around with higher base iterations(i=2,3,4,5...etc) to confirm that those curves intersect y=x the same way as do the zeroth and first.

By experimenting with different gains(g) and base iterations(i) you can get an intuitive feel for the effect of these parameters, all without resort to excessive mathematics.

This information is brought to you by M. Casco Associates, a company dedicated to helping humankind reach the stars through understanding how the universe works. My name is James D. Jones. If I can be of more help, please let me know.

JDJ