Nonlinearity gets all the fun

Nonlinearity is inherent in most systems before you make an assumption and reduce the beautiful complexity down to a boring bog standard linear system. There is a section in Strogatz that shows probably the most interesting thing you can do with linear systems (Chapter 5 section 3 page 138 onwards). And I only read it once because linear systems were dull and pretty repetitive long before A level.

Nonlinearity is pretty easy to understand. Take for example square numbers. You can make 4 by either multiplying 2 and 2 or -2 and -2. This means that, under the square operator, 4 has two (well, four really, but two are identical to the other two) possible previous values. This makes squaring a number nonlinear because tracking back you do not know whether two 2s or two -2s made the 4 you see at the moment. This particular brand of nonlinearity is called noninvertibility. You still following me?

Good. Right, to take this further we call the previous values of each number pre-images and the operation, a mapping. Now the quadratic (squaring) mapping that I am examining is:

what happens to z when a=2 and I vary lambda. This mapping does funny things depending on what I set lambda to. If it is between 0 and 1 then it behaves just like the squaring operator - each value of z has two pre-images as I iterate the map but in a circle radius=1-lambda around C the values have no pre-images. Now if I make lambda greater than 1, some values have two pre-images as before but the z values in the circle now have four pre-images.

Still with me? We can take this further but placing this property on curves not just points. A curve is just a line on a graph (an infinite set of points). Instead of looking backwards as we did with the points, we instead go forwards and see how the curve changes. It's first image is a circle, say. This curve then becomes the pre-image of the set of points of the circle under the mapping. And so on, forwards in time. It helps to look at the image below:

You can see the curves collecting on (heading towards) one curve called a manifold (it's attracting). Now consider the relationship of the points and the curves. Because a curve is just a set of points. With noninvertibility inherent in the system, what happens it you take the manifold and reverse the mappings for all the points? Wild Chaos! Tracking the possible points and curves and how and why they interact is the next stage of my project - at least for certain ranges of values for C, a and lambda. This limitation is purely because my project has a time limit of a year.

I am currently working with a PhD student, Stefanie Hittmeyer (who incidentally produced the image above) who is working on the full problem in all its scope. She has been focused on the manifold part of it for the moment while I have been examining the fractal structure side. My work nicely compliments hers. This work is on the fore-front of chaos research. Now it may seem that all this has little value in the real world but the great thing about nonlinear and chaotic systems is that it can accurately mimic real world systems. This is a major advantage over linear systems because when taking a real world problem you have to reduce the picture you are looking at so you can describe it using the simplest tools. With this kind of research, no such reduction is necessary and you can understand the system in it's full complexity. So maybe it'll turn out that neurological signals in the brain will turn out to have similar properties to the mapping that I am examining and my research will contribute to a fuller understanding of how we came to have consciousness and help neurosurgeons get better at fixing the biochemical circuitry that makes us who we are. Or maybe, and more likely since this mapping is a reduction of the five dimensional Lorenz model, it will just help meteorologists get really, really good at predicting weather patterns.

Who knows? It's sort of like the laser - a solution created with no problem that has now become integral to our lifestyle. Just check out how many things it is used in! I hope you managed to follow all that to the end. If you have any specific questions please feel free to email me but I would much rather you follow that urge to take a deeper look yourself (even if that is just reading all the wiki pages). It's way more fun!