On a less serious note

So, I realised just a few minutes ago that I haven't written about chaotic attractors in all their splendour. An incredible oversight on my part. These objects are achingly beautiful and intricately complex in form. They arise in phase space (I believe I have covered this) in certain parameter regions of systems. Their fractal geometry means that dynamics is incredibly sensitive on them and while trajectories will follow the path of the attractor (duh! attracting :-) ), there is no predicting which exact part of the attractor it will land on and follow round or ultimately where it will end up. The attractor is fixed, the trajectories are not. A nice succinct explanation is available here.

The one that ties in closest to my work is the Lorenz attractor. The three dimensional system for describing turbulence gives birth to an amazing structure. This attractor is where the 'butterfly effect' expression originated. The attractor itself is geometrically between two and three dimensions and winds itself around two points.

The image (from wiki) shows exactly this. Where the trajectories 'cross' in this two-dimensional visualisation is actually where they layer over each other - manifolds cannot intersect (the unstable manifold exactly describes the attractor). The view is better in 3d. The fractal (Hausdorff) dimension of the attractor is about 2.06.

Other systems with strange attractors are the Rossler system and Henon map. This page gives a nice description and a few images.

As a side note: for those of you who read and enjoyed 'Harry Potter and the Methods of Rationality' or 'Luminosity' I would recommend this strangely named (but you'll understand all too soon) 'Baby Eating Aliens' also from the Less Wrong family and the (there is a theme here, no?) 'Harry Potter and the Wastelands of Time' if you like a bit of complex, epic writing with lots of action and magic!

Enjoy! Also, something to look forward to: I hope to have my poster available for you lot next time I post. :D

Doh! Almost forgot. One of the lecturers at my university (Hinke Osinga) actually crocheted the Lorenz stable manifold! If you want to know how, read this!

Incidentally, my project supervisor is the Bernd Krauskopf mentioned in the article. And yes, it is one of their Christmas decorations. ;)

Happy New Year!

This post I want to introduce my main project. The title is "Dynamics and Bifurcations of Two-Dimensional Noninvertible Maps". My plan for the project is available here to give you an idea of what I would like to achieve with it. Don't worry about all the scary words in it - the main idea is to find out what happens if I slightly permute the Mandelbrot set (a two dimensional map of Julia sets represented as a number set). Currently I'm focused on looking at how the Julia sets change. So far I have made lots of pretty pictures! as well as some observations on what happens.

Fractals are the most beautiful objects in chaos theory (wiki has a great introduction to them here). Simple definition is that fractals are shapes that are self-similar at all levels of magnitude - you cannot tell how zoomed in or out you are. The Mandelbrot is the most famous fractal and I love it because it is defined so simply but produces infinite complexity. My project considers the Mandelbrot as a single instance of a wider group of two-dimensional non-invertible maps. Noninvertible here means a many-to-one mapping forward in time i.e. if you back track there are many points that could have produced your original point.

Fractals were first observed in nature in the form of leaves on the trees. Ferns are made up of smaller ferns which are made up of even smaller ferns. The simplicity of defining a fractal has suggested that the information contained within the DNA of a plant to tell it what shape to grow into is in fact encoding fractal information. Another example would be one used recently in one of the Royal Institution Christmas Lectures - lungs. They translate a large volume of air onto a large surface area of blood vessels and they do it by being fractal. And recently there has been research that has found fractal patterns in semiconductor material at the quantum scale.

Fractals can be produced by nonlinear dynamics. This returns us back to the Lorenz equations and the butterfly attractor. The attractor found to describe the behaviour of the system is a fractal. Such a combination of parts of chaos theory is typical and continuing to find more links is very exciting. I have been finding bifurcation points and their stability in the fractal Julia sets as I permute the Mandelbrot set. Nonlinear dynamics can be produced by fractals. Thus my research is far reaching across the whole field of study. A field of study that is far reaching, interlinked and incredibly beautiful.