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%                       Computational Physics
%                             SPRING 2008
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%                     Instructor: Kevin H. Knuth
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% HW 6
% The Quadratic Family and Bifurcations
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% In this homework you will investigate the bifurcations and fractal
% properties of the quadratic family: Q(x) = x*2 - c
% 
% 1. Consider parameter values from c = 0 to 4
%    Iterate the Quadratic map at many values of c within that range to be
%    able to display the bifurcation diagram representing the period
%    doubling phenomenon.  Plot the diagram and note the point of the first
%    bifurcation from period 1 to period 2, as well as the period 3 window.
%
% 2. By increasing the resolution of your diagram at the key points, find
%    the values of c at which the period doubling bifurcations occur (do
%    this carefully).  Show that Feigenbaum's constant is appoximately:
%    4.669...
%
% 3. Now consider the complex plane, and the Quadratic map for z:
%    Q(z) = z*z' - c, where z' is the complex conjugate of z.  Iterate the
%    function for an array of c-values.  Show that the set of c-values that
%    does not diverge is called the Mandlebrot Set.  It can be shown that
%    if |z| > 2, then the future iterates will diverge to Infinity.  You can
%    use this to speed up your plotting.
%
% 4. Write code to detect the periodicity of various regions in C, when
%    iterated with the quadratic map. Label these regions on a map of the
%    Mandelbrot set.
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% Problem by: Kevin H. Knuth
% Created on: 20 April 2008
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