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With the four pictures discussed above firmly in hand, we now attempt to put all of this information together to discuss bifurcations. To bifurcate means to split apart: in one dimensional equations, it is the equilibrium points that undergo bifurcations. As an example, consider the simple autonomous equation

dy / dt = y2 - A

Clearly, this equation has two equilibrium points when A > 0, only one when A = 0, and none when A < 0. We thus say that this family undergoes a bifurcation as A passes through 0.

In our course, we expect students to understand what happens when a family of differential equations of the form

dy / dt = fA (y)

undergoes a bifurcation. This means they must first know where to look for these changes. As a consequence of the first derivative test, bifurcations only occur at equilibria at which f'A vanishes. Next they must determine how the equilibria change; this involves describing both the graphs of fA and the corresponding phase line at, before, and after the bifurcation.

We encourage our students to view this process dynamically. Often, this involves computer experiments and animations of bifurcations. In Figure 6 we have included an animation of the differential equation

dy / dt = fA (y) = y2 - A

that shows the changes in the phase line, the solution curves, and the graph of fA as A varies. We expect our students to have a mental image of these changes and to be able to write coherently about them.

Click on this icon to download the animation).

To understand why bifurcations only occur when the derivative vanishes, it is important for students to visualize how the graphs of fA vary as parameters change. In Figure 7, we have included an animation of all three cases: sinks, sources, and nodes, showing how one does not expect bifurcations in the sink or source case, but that bifurcations may indeed occur when a node is present.

Click on this icon to download the animation).

Of course, animations are difficult for students to do on homeworks or exams, so we encourage our students to draw the associated bifurcation diagram. This image is a plot of the phase lines for the differential equation versus the parameter. For example, the bifurcation diagram for

fA(y) = y2 - A

is shown in Figure 8. We include enough phase lines in this image so that students are able to view this process dynamically; they ``see'' the equilibrium point structure change as A increases through 0.

The type of bifurcation that fA undergoes is among the most common bifurcations. Using terminology borrowed from the higher dimensional analogue of this situation, we call this a saddle-node bifurcation.

There are many other types of bifurcations that may be analyzed by students using these qualitative methods. One is the pitchfork bifurcation illustrated by

dy / dt = fB (y) = y3 - By.

This equation has an equilibrium point at 0 for all values of the parameter B. Two new equilibrium points (at the positive and negative square roots of B) arise when B > 0. Hence a bifurcation occurs at B = 0. The bifurcation diagram shown in Figure 8 illustrates the reason for the name ``pitchfork.''

next An application: harvesting (Next Section)
up Bifurcations and Phase Lines (Cover Page)
previous Classification of equilibrium points. (Previous Section)

Robert L. Devaney
May 6, 1995