The normal form is the simplest differential equation that captures the essential features of a system near a bifurcation point. To get the normal form, you need to perform a nonlinear change of variables. First, we perform a linear change of variables in terms of the eigenvectors. The eigenvectors of the Jacobian are and . Let
Multiplying through gives us
We want the equations in terms of x and y. Therefore we have
Now, we take the derivative and get
Simplifying, we get
Since x=A and , we can substitute in and get
These identities will allow for easier simplification. Our equations simplify to
where . We must make one more change of variables to get the constant out of . Let . The equations become
For convenience, drop the hat from y. Now, we are ready to calculate the normal form. To do this requires a technique called a nonlinear change of variables. In this calculation, we will attempt to remove the terms not proportional to y in the last equation. Let
This will allow us to choose a value for the parameter which cancels out in the y equation. In terms of y', the equation is
Taking the derivative, we get
Our next step is to substitute in for y. Then, we will pick so that the terms drop out. After plugging in for y, we find that the value of we want is
Now, since equals the above, y' goes to zero over time. So we set y' = 0. We get . Since
This simplifies to
This is the normal form of the bifurcation. It is of the form which is the normal form of a pitchfork bifurcation. This analysis was for the A and n variables. In fact, the full system undergoes a Hopf bifurcation towards periodic solutions of amplitude |A|.
Here, the normal form predicts a pitchfork bifurcation, the same as AUTO, the software package that drew the bifurcation diagram. Here are the values of the parabola when graphing J vs. x.
Figure 5: Pitchfork Bifurcation
This is the bifurcation diagram of the system. Notice the similarities with Figure .