In a saddle-node bifurcation a pair of periodic orbits are created ``out of nothing.'' One of the periodic orbits is always unstable (the saddle), while the other periodic orbit is always stable (the node). The basic bifurcation diagram for a saddle-node bifurcation looks like that shown in Figure 2.15.
Figure 2.15: Saddle-node bifurcation diagram.
The saddle-node bifurcation is fundamental to the study of nonlinear systems since it is one of the most basic processes by which periodic orbits are created.
A saddle-node bifurcation is also referred to as a
tangent bifurcation because of the mechanism by which the orbits are born.
Consider the nth composite of some mapping function 
which is
near to a tangency with the line y = x.
Figure 2.16: Tangency mechanism for a saddle-node bifurcation.
Let 
be the value at which a saddle-node bifurcation occurs.
Notice in Figure 2.16 that 
is tangent to the line y =
x at . For 
, no period n orbits exist
in this neighborhood, but for 
two orbits are born.
The local stability of a point of a map is determined by
.
Since 
is tangent to y = x at a
bifurcation, it follows that at
,
Tangent bifurcations abound in the quadratic map. For instance, a pair of
period three orbits are created by a tangent bifurcation in the quadratic map
when 
. As illustrated in Figure
2.17
Figure 2.17: A pair of period three orbits created by a tangent bifurcation in the
quadratic map (shown with the unstable period one orbits).
for 
, there are eight points of intersection. Two
of the intersection points belong to the period one orbits, while the
remaining six make up a pair of period three orbits. Near to tangency, the
absolute value of the slope at three of these points is
greater than one--this is the unstable period three orbit.
The remaining three points form the
stable periodic orbit. The birth of this stable period three orbit is clearly
visible as the period three window in our numerically constructed bifurcation
diagram of the quadratic map, Figure 2.14. In
fact, all the odd-period orbits of the quadratic map are created by some sort
of tangent bifurcation.
