Julia Sets
Some of the most famous fractals are Julia sets, and Mandelbrot sets (discussed on the next page). These are complex dynamical systems, and they deal with chaos and order at the same time, in both competition and coexistence. Julia studied the iteration of rational and polynomial functions at the beginning of the twentieth century. It was around this time that the biggest discoveries in Chaos were being made.
Julia showed that with a function f(x), various behavioural patterns can be observed when that function is iterated. For a starting value x, (for example x = k), consider the following sequence of values:
k, f(k), f(f(k)), ... , n(f)f(k)
Its possible that values could remain small or maybe tend to arbitrarily large values by repeatedly applying this function. In a similar way to the description using the hair on a wrist above, the Julia set of f is the division or boundary between these sets. The parts where the starting value k causes the iterated result to stay bounded are referred to as filled.
When imaginary numbers come into play with the real numbers, the resulting complex numbers are divided over the complex plane (again into two sets). Images formed this way can be very beautiful, such as this example:
Here is a Julia set for the function f(z) = z2 - 0.75. (See reference A8)
(Only the square in the complex plane including numbers z = x + iy where both x and y are between �1.5 is shown.)
Next: Mandelbrot Sets
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