Mandelbrot set
Formal definition

The Mandelbrot set is the set of values of c in the complex plane for which the orbit of the critical point z = 0 {\textstyle z=0} under iteration of the quadratic map

remains bounded. Thus, a complex number c is a member of the Mandelbrot set if, when starting with z 0 = 0 {\displaystyle z_{0}=0} and applying the iteration repeatedly, the absolute value of z n {\displaystyle z_{n}} remains bounded for all n > 0 {\displaystyle n>0}.

For example, for c = 1, the sequence is 0, 1, 2, 5, 26,..., which tends to infinity, so 1 is not an element of the Mandelbrot set. On the other hand, for c = − 1 {\displaystyle c=-1}, the sequence is 0, −1, 0, −1, 0,..., which is bounded, so −1 does belong to the set.

The Mandelbrot set can also be defined as the connectedness locus of the family of quadratic polynomials f (z) = z 2 + c {\displaystyle f(z)=z^{2}+c}, the subset of the space of parameters c {\displaystyle c} for which the Julia set of the corresponding polynomial forms a connected set. In the same way, the boundary of the Mandelbrot set can be defined as the bifurcation locus of this quadratic family, the subset of parameters near which the dynamic behavior of the polynomial (when it is iterated repeatedly) changes drastically.
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