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Homeomorphism on reconstructed attractor space

  1. Jul 7, 2012 #1
    Hi,
    I am a computer programmer actually involved in a project dealing with dynamical systems;
    I have math and physics background, though not advanced, so please have patience if I make coarse logical errors. I am going to explain my little problem in detail, in hope someone could help.
    I am doing phase space reconstruction from a very particular time series whose elements are very large positive integer numbers in the range [0; 2^1200+] . I have already obtained the time delay and the embedding dimension by means of mutual information and false nearest neighbors algorithms, respectively, by help of my own custom software properly made to deal with such big numbers.
    The problem now is in the graphical representation of the attractor: as you may know, there is no general way to plot large numbers on screen (due to display size and resolution), given that these large numbers are effective coordinates of the system. So I've thought of a simple stratagem in order to easily accomplish this. Intuitive description follows.
    Takens' theorem ensures that my attractor would be mapped into euclidean space R^k, so my (properly delayed) large numbers would be components in R of each point of the reconstructed attractor. The logarithm function is an isomorphism from R+ to R, and so it could be a homeomorphism (must be uniform?) that, when applied to the set of the reconstructed attractor points, would preserve all the topological properties of the attractor, giving me a set with extremely shorter numbers as well. Think of it as a way to 'compress' the coordinates, without altering the topology.
    However, I am not completely sure to be right, because I am not a mathematician and many doubts arises. Maybe a formal proof or disproof would be really simple to make, although not in my reach at the moment.
    Could anyone please review my reasoning? Any feedback would be really appreciated, and credits would be granted in my little humble work.
    Thank you.
     
  2. jcsd
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