This Demonstration shows symmetry-restoring bifurcations of a nonlinear complex map: , with , and and varying. The iteration starts at ; 10,000 points are visited and the first 10 iterations are discarded.
This Demonstration is an animation of the symmetric icons first seen in the beautiful book Symmetry in Chaos: A Search for Pattern in Mathematics, Art and Nature by M. Field and M. Golubitsky and developed in Complex Analysis with Mathematica, Chapter 7, "Symmetric chaos in the complex plane". Further information about the latter book is at its web page. This particular animation is tailored to showing the creation of symmetry through birfucation. You can also read about this in The Mathematica Journal (see the related links).