Complex Views of Logistic Maps

λ

This plot shows the unstable (orange) and stable (blue) roots of the twice-iterated cubic logistic map as functions of

λ

. 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.