This plot shows the unstable (orange) and stable (blue) roots of the twice-iterated cubic logistic map as functions of . This is a polynomial of degree nine and has nine complex roots. For small values of λ there is one real and stable root. As you increase λ this root becomes unstable and two complex roots simultaneously coincide and become real and stable. They then move apart and branch again. This is the complex-variable explanation of bifurcation.
This example is a Mathematica 6 Demonstration version of the example given in Chapter 5 of Complex Analysis with Mathematica. The idea is to explain basic bifurcations by the use of complex variable theory. This topic is explored by finding out when the roots of high-degree polynomials are real, and stable or unstable under the cobweb map. Such an analysis shows precisely where and how the first few bifurcations occur. Further information about the book is at the author's website.