"To err is human"

And guess what?! I'm human. My error this time was such a small thing but it could have had huge consequences if I had not caught it when I had. Actually it would have been far better to have caught it much much earlier but I had no reason to see problems until this point.

I had successfully combined my manifold examination with the Julia Set calculations into one succinct image for each value of the parameter lambda. Nothing twigged until I sat there showing them to my supervisor and he commented that surely the attractors should actually lie within the prisoner set (the points that remained after the iteration). My manifolds also did not line up with the areas they would be expected to. This meant one of two things. Either I had some scaling issue (quite possible with all the jiggling around I had done to get them on the same image) or something was wrong with either side of the analysis (the MATLAB or DSTool code).

Since the DSTool code had been written by someone far more experienced in the area, it was to my code that I turned my attention to find the problem. Took me a while but (as the error rule says) when I found it it was easy to fix. I like to program quite efficiently and as I was testing the absolute value of Z at each iteration I had set
absZ = abs(Z)*abs(Z);
Now instead of square rooting this after I had tested it I had left it which meant that instead of just dividing by the square of abs(Z) at each iteration, it was dividing by the fourth power of abs(Z). Doh!
Thus all my Julia Set images were actually wrong. Re-running the (easily) corrected code showed just how wrong it had been. But it is fixed now and since then I have been working like stink to analyse the correct progression of Julia Sets.

One of the areas that I have now been progressing into was a direct result of the correction. Once the correct maps were generated, I added a further capability to zoom the section around the interesting circle region. This showed that the points within the Julia Set boundary were actually behaving very much like the stable set patterns seen by the PhD student I have been working with. So I have been expanding my analysis to include calculation of the stable set within the Julia Set boundary. This is likely to contribute to connectedness and other behaviours of the map I have already examined.

As far as I understand it, the stable set is the set of points that go to the attractor(s) via the stable manifold due to the existence of a saddle point. Some points, in the course of being iterated, will end up on the stable manifold or on the saddle itself before slowly reaching the attractor. The stable set is all the points back in time that forward in time end up at the stable manifold (thus more pre-image calculations are involved!).

I have calculated this for a few values and they show promising information - I would have inserted images here but the server is rejecting them...
They should help to determine what the hole actually does to the map at least in the limited sense that I am examining it.

Oh, also I would also have shown you the bifurcation progression through lambda but since the server is rejecting my images I won't be. I'll try next week. Sufficed to say that there is a lot going on. I'm not entirely sure I'll find it easy to stick to the 45 page limit. If anyone has any recommendations for that (things that they would find interesting so think should be included) please let me know. :-)

Inconvenient Delays

So here we are a month since I last posted. I would explain but that would feel like giving excuses and I'm trying to get out of the habit of feeling guilty for things which were out of my hands. I do however have something for you to read and perhaps enjoy.

My most recent deadline has been for a draft masters thesis for my project. This is available for you to read via this link.

What I have written covers about a third of the area of work I have been doing. It is very minimalist and basic with very little of the style or continuity that I duly put in my work. My introduction and background are perhaps not as full as I would like but maybe you can provide some thoughts on that since it is aimed at bringing people of different ranges of knowledge on the subject up to speed before I share my analysis. If there is any point of logic missing please do let me know.

The version I am sharing is the version I would have like to share with the examiners but my supervisor had other ideas so I actually submitted a stripped down version that is even worse so I will not be sharing that.

One point which you should all love is all the images generated that I have included in the report. In my time I have analysed the Julia set changes, the stability changes, the manifolds and most recently the full bifurcations in one half with the other half due to be completed this week. I have finally reached the crux of my examination of the map which is a strange one. The circle hole in one half has an important bearing on the location of the attractor which "disappears" and yet still points remain in the set - this is only possible with an attractor still present yet at some point there doesn't even exist a higher period attractor, just saddles and repellors. Thus this week my question to answer is whether an attractor still appears to exist for the points under the mapping despite the properties attributed to it or if the boundary of the hole itself becomes the attractor or neither of these things (which would mean either a waste of half a year's work or something totally cool and new).

I'll let you know next week hopefully and also provide you will the full bifurcation diagram.