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Complexity Leadership Dynamical Systems & Leadership Jim Hazy July 19, 2007
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Moving toward Chaos As Alice explores the phase space for values of μ, the birthrate, She sees a series of bifurcations; here it is period doubling. But she is relieved because the system remains periodic.
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Moving toward Chaos But when the birthrate μ > a, the situation changes. For a while, Alice knows the system will be within certain parameters, but it never repeats… it’s in a chaotic attractor. Eventually, for high enough value for μ, the system is completely unpredictable.
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Bifurcation & the Mandelbrot Set Bifurcations to Chaos (right to left) for the real component of the family of functions: Q c (z) = z 2 + c Mandelbrot Set is the bifurcation set on the complex plane; the Real component is the spine Bulbs in the set represent periods of attractors; 1 = fixed Values outside the set tend to infinity 12 3 3 3 4 8
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Mandelbrot
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Mandelbrot Close-up
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Julia Sets or Boundary Sets
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