The last bifurcation we illustrate with the quadratic map is a
transcritical bifurcation ,
in which an
unstable and stable periodic orbit collide and exchange stability. A
transcritical bifurcation occurs in the quadratic map when 
. As in a saddle-node bifurcation,
at a transcritical bifurcation.
However, a transcritical bifurcation also has an additional constraint not
found in a saddle-node bifurcation, namely,
For the quadratic map this
fixed point is just the
period one orbit at the origin, 
, found from equation (2.10).
A summary of these three types of bifurcations is presented in Figure 2.21. Other types of local bifurcations are possible; a more complete theory for both maps and flows is given in reference [7].
Figure 2.21: Summary of bifurcations.
