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

Recall that

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

Simplifying,

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

This gives

Now, since equals the above, *y*' goes to zero over time. So we set *y*' = 0. We get . Since

we get

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 [4].