Julia set

Julia sets, described by Gaston Julia, are fractal shapes defined on the complex number plane. Given an iterated map of the complex plane to itself, or a collection of such maps, the Julia set for this system can be defined (informally) as the set of points for which nearby points do not exhibit similar behaviour under repeated iterations of the map(s).

Quadratic maps

Given two complex numbers, c and z0, we define the following recursion:

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:zn+1 = zn2 + c

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This is sometimes referred to as a quadratic map, and is a type of dynamical system. Given a specific choice of c and z_0, the above recursion leads to a sequence of complex numbers z_1, z_2, z_3... called the orbit of z_0.

Related Topics:
Map - Dynamical system - Orbit

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Depending on the exact choice of c and z_0, a large range of orbit patterns are possible. For a given fixed c, most choices of z_0 yield orbits that tend towards infinity. (That is, the modulus | z_n | grows without limit as n increases.) For some values of c certain choices of z_0 yield orbits that eventually go into a periodic loop.

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Finally, some starting values yield orbits that appear to dance around the complex plane, apparently at random. (This is an example of chaos.) These starting values make up the Julia set of the map, denoted J_c. Some authors also define the filled-in Julia set, denoted K_c, which is the set of all z_0 with yield orbits which do not tend towards infinity. The "normal" Julia set J_c is the edge of the filled-in Julia set.

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If the modulus of zn becomes larger than 2 for some n then it is guaranteed that the orbit will tend to infinity. This test makes it straightforward to plot Julia sets for quadratic maps using a computer.

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