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Normal Form of the Bifurcation

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].

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