Generating Fractal EquationsDate: 7/1/96 at 22:26:3 From: Anonymous Subject: Generating Fractal Equations Is there any way to generate a fractal equation from a set of numbers? For example, taking a million or so seemingly random numbers and "reversing" the fractal process to provide a formula that will generate the same series of numbers every time? Date: 7/7/96 at 0:43:6 From: Doctor Pete Subject: Re: Generating Fractal Equations In short, the answer is no. Before I explain in further detail, let me set down some preliminary notions; these you may or may not already be familiar with. I am assuming what you refer to as "fractal process" is the iteration of some mapping such as z |--> f(z). That is, we start with an initial value z0, and calculate the values f(z0), f(f(z0)), f(f(f(z0))), ... and plot these iterates. The set of iterates under the mapping defined above is called the *forward orbit* (or for simplicity, orbit) of z0 under the mapping z |--> f(z). The *inverse orbit* or *backward orbit* is the set of iterates of the inverse mapping, z |--> f^-1(z), where f^-1(f(z))=z. Note that such a mapping may be multivalued, such as in the case where f(z)=z^2+c; another term for the iterates of the inverse mapping of f is the *preimages* of z0 under f. So this is basically how we generate sets of points (and pretty pictures). Take some function f, and a point z0, and we plot the orbit of z0 under f. There are other ways of plotting points, such as numerical solution of differential equations, but in general the concept is the same: we use iteration to get the next point from the previous. Now for the explanation: One of the things about iteration is that it has the property of magnifying errors - that is, if two points z1 and z2 are initially very close together, their difference will increase with every iteration. This is also the exact same property that gives fractals their complex and unpredictable nature. So after, say, fifteen iterations, the roundoff error created by the computer will be too large to make the value of the point have any specific meaning; that is, if you calculated 50 iterations, and then take 50 inverse iterations, you will not get your original point back. This is why you cannot take a set of data and represent it by a mapping if the data is fractal. However, if you knew the order in which the points were plotted, you have orbit information which *might* let you *approximate* the data with some sort of mapping. But this relies on understanding the dynamics of the mapping beforehand, which is highly unlikely. -Doctor Pete, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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