The quadratic family of equations includes all functions of the form f(x) = x2 + p, where p is any real constant parameter. So all of the following are examples of quadratic functions:
|f(x) = x2 + 2||2|
|f(x) = x2 - 4||-4|
|f(x) = .5 + x2||.5|
By using graphical analysis on various quadratic functions, you can see that as the parameter changes, the number of fixed points (where the function f(x) intersects the diagonal line y = x) changes as well. This changing in the number of fixed points is called a bifurcation. We will examine a particular type of bifurcation called a saddle-node.
As we decrease the value of the parameter, the saddle-node is the first bifurcation that we encounter. While p > .25, the function has no fixed points, as shown in the graphical analysis picture below.
|Using .1 as the seed and .5 as the parameter, we can see that there are no fixed points (before the bifurcation).|
When p = .25, the function has one fixed point, a saddle, as depicted below.
|Using .1 and .75 as seeds and .25 as the parameter, we can see that there is one neutral fixed point (during the bifurcation).|
While p < .25, the function has two fixed points, one attracting and one repelling, as shown below.
|Using 1.25 and 1.5 as seeds and -.5 as the parameter, we can see that there are two fixed points (after the bifurcation).|