** Classification of equilibrium
points.** (Next Section)

Bifurcations and Phase Lines
(Cover Page)

Qualitative approach to
autonomous equations. (Previous Section)

One of the most difficult things for students to comprehend at this point in the course is the relation between the phase line and the graph of f as a function of y. Perhaps the reason for this is our predilection for drawing phase lines vertically (so that they line up nicely with the slope field), but drawing the y-axis horizontally when plotting the graph of f as a function of y. In Figure 3 we have sketched the phase line and graph of f for the differential equation

### dy / dt = y^{4} - y^{2}.

**Figure 3:** The graph of f(y) = y^{4} - y^{2} and the corresponding phase line

Students are expected to translate zeroes of f as equilibrium points, intervals where f>0 as y-values where solutions increase, and intervals where f<0 as y-values where solutions decrease. At first, even the better students try to argue that we need to consider places where ** the derivative** of f is positive in order to conclude something about increasing solutions. Understanding the subtle relation between the graph of f and the behavior of solutions is a difficult but rewarding experience for students.

As homework problems relating to these concepts, we provide students with a picture of the graph of f and ask for the phase line in return. Or one can ask the question in reverse: given the phase line, sketch (qualitatively), the graph of f as a function of y. On exams, we often include a matching question in which students have to determine which graphs and phase lines correspond. An example is provided in Figure 4.

**Figure 4:** Which phase line corresponds to the given graph?

** Classification of equilibrium points.** (Next Section)

Bifurcations and Phase Lines (Cover Page)

Qualitative approach to autonomous equations. (Previous Section)

*Robert L. Devaney *

May 6 1995