Robert P. Munafo, 2002 Apr 18.
"Bifurcation" can be used to refer to period doubling bifurcation, a property of Iteration and Orbital Dynamics, It has its origins in the iterative squaring of the iteration formula, and is responsible for the binary nature of the external angles.
However, I use the word more often to refer to the doubling of the order of rotational symmetry that occurs as you zoom in closer to any island. It shows up in all the filaments and is particularly notable in the embedded Julia sets.
In such bifurcation, the largest feature has 2-fold rotational symmetry. In the center of this is found a feature with 4-fold rotational symmetry. In the center of this is found a feature with 8-fold rotational symmetry. The doubling continues until to get to the island at the center. The rotational symmetry breaks down as you get closer to the island, because the island's shape (which, like the continent, has no rotational symmetry) begins to dominate. However, the doubling of number of features continues forever (see external angle).
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From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2012. Mu-ency index