Poincaré used the term bifurcation to describe the ``splitting'' of asymptotic states of a dynamical system. Figure 2.14 shows a bifurcation diagram for the quadratic map (see Plate 2 for a color version of a quadratic map bifurcation diagram). As we examine Figure 2.14, we see that several different types of changes can occur. We would like to analyze and classify these bifurcations. At a bifurcation value, the qualitative nature of the solution changes. It can change to, or from, an equilibrium, periodic, or chaotic state. It can change from one type of periodic state to another, or from one type of chaotic state to another.
For instance, in the bouncing ball system we are initially in an
equilibrium state, with the ball moving in unison with the table.
As we turn up
the table amplitude we first find sticking solutions.
As we increase the amplitude further, we find a critical parameter value
at which the ball
switches from the sticking behavior to bouncing in a period one
orbit. Such a change from an equilibrium state to a periodic state is an
example of a saddle-node bifurcation. As we further turn up the table
amplitude, we find that there is a second critical
table amplitude at which the ball switches from a period one to a period two
orbit. The analogous period doubling bifurcation in the quadratic map
occurs at 
. Both the saddle-node and the period
doubling bifurcations
are examples of local bifurcations. At their birth (or death) all the orbits
participating are localized in phase space; that is, they
all start out close together. Global bifurcations can also
occur, although typically these are more difficult to analyze since they can give birth
to an infinite number of periodic orbits. In this section we analyze three
simple types of local bifurcations that commonly occur in nonlinear systems.
These are the saddle-node, period doubling, and transcritical bifurcations [6].