English: Bifurcation of periodic points from period 1 to 2 for fc(z)=z*z +c. On the left of the following animation, we see the graph of ${\displaystyle f_{c}(x)=x^{2}+c}$ for ${\displaystyle 0\leq c\leq 1/2}$ together with the line ${\displaystyle y=x}$. This clearly shows the collision of two fixed points - one attractive and the other repulsive. They briefly merge into one neutral fixed point and then disappear all together. On the right, we move to the complex plane and plot those same fixed points in the Julia sets for those polynomials. From this perspective, they never really disappeared at all. I guess this is a rather special case of the fact that every complex polynomial has periodic points of every order while real polynomials do not.