1. Complexity Leadership Dynamical Systems & Leadership Jim Hazy July 19, 2007

2. 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.

3. 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.

4. 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

5. Mandelbrot

6. Mandelbrot Close-up

12. Julia Sets or Boundary Sets