Mandelbrot

# Mandelbrot - x t 1 iy t 1 = x 2 t-y 2 t c 1 i 2 x t y t...

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Mandelbrot Set Instructor: Wayne M. Getz Department of Environmental Science, Policy, and Management, University of California, Berkeley, CA 94720 September 7, 2010 Bifurcation in discrete equations We have seen the bifurcation diagram for the quadratic map: x t +1 = x t + rx t (1 - x t /K ) . Consider the quadratic map in the complex variable z = x + iy z t +1 = z 2 t + c, z 0 = 0 , c = c 1 + ic 2 . This can be written as a two-dimensional model as follows: If z t = x t + iy t then z 2 t = ( x t + iy t ) = ( x 2 t - y 2 t ) + i (2 x t y t + c 2 ) . Hence z t +1 = z 2 t + c
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Unformatted text preview: x t +1 + iy t +1 = ( x 2 t-y 2 t + c 1 ) + i 2 x t y t Equate real and imaginary parts: x t +1 = x 2 t-y 2 t + c 1 y t +1 = 2 x t y t + c 2 Mandelbrot Set : By varying c , ﬁnd those values for which the solution remains bounded (i.e. does not diverges to ∞ ). Julia Sets : Allow z to vary. Then for given c ﬁnd all values of z for which the solutions is unstable. 1...
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