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.
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