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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 tex2html_wrap_inline1436 and tex2html_wrap_inline1438 . 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 tex2html_wrap_inline1446 , we can substitute in and get


Recall that


These identities will allow for easier simplification. Our equations simplify to


where tex2html_wrap_inline1448 . We must make one more change of variables to get the constant tex2html_wrap_inline1450 out of tex2html_wrap_inline1452 . Let tex2html_wrap_inline1454 . 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 tex2html_wrap_inline1458 which cancels out tex2html_wrap_inline1460 in the y equation. In terms of y', the equation is


Taking the derivative, we get




Our next step is to substitute tex2html_wrap_inline1464 in for y. Then, we will pick tex2html_wrap_inline1458 so that the tex2html_wrap_inline1460 terms drop out. After plugging in for y, we find that the value of tex2html_wrap_inline1458 we want is


This gives


Now, since tex2html_wrap_inline1476 equals the above, y' goes to zero over time. So we set y' = 0. We get tex2html_wrap_inline1482 . Since


we get


This simplifies to


This is the normal form of the bifurcation. It is of the form tex2html_wrap_inline1484 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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Next: Laser Rate Equations With Up: Normal Form Previous: Normal Form