In our courses, the analytic solution of differential equations definitely takes a back seat to qualitative and numerical techniques. At the outset, when covering first order equations, we remind students how to solve separable equations. But that is essentially the only analytic technique we include until the end of the first order discussion when we finally solve first order linear equations and mention changes of variables.
Most of our effort deals with autonomous equations of the form
Although this type of equation is separable, we rarely take the time to carry out the integrations explicitly (when we do it is most often to show students how ``ugly'' and uninformative the resulting formulas are). Rather, we emphasize the interrelationships between four different pictures associated with this equation. These pictures are fairly easy to draw using only the formula for f(y), and which they do not give a completely accurate portrayal of the behavior of solutions. They do provide a good picture of the qualitative or long term behavior of solutions.
The four images are the slope (or direction) field, the ty-graphs of solutions, the phase line, and the graph of f as a function of y. Of course, the slope field and solution graphs are standard topics in most differential equations courses. Since we are dealing with autonomous equations, we can capture all of the information provided by the slope field in a simpler, one-dimensional picture. In this picture, called the phase line, we record the locations of all of the equilibrium points of f (the zeroes of f(y)) as well as the direction of solution curves y(t). Since the slope field generated by f(y) is independent of t, this is all the information we need to understand the fate of solutions. In Figure 1 we sketch the slopefield, solution graphs, and phase line for
Figure 1: The slope field, phase line, and solution graphs
In the phase line, circles indicate rest points, while arrows indicate the directions that solutions move as t increases. The existence and uniqueness theorem guarantees that solution curves do not cross the equilibrium points. Consequently, a phase line as depicted in Figure 2 implies that solutions between the equilibrium points p and q tend from q to p as t runs from plus to minus infinity.
Figure 2: Solutions tend from q to p as time increases
Two quick comments are in order. One has to argue that solution curves in Figure 2 actually do tend to p as t tends to infinity (and to q as t tends to minus infinity), but this is a relatively straightforward argument using the slope field. Also, these qualitative methods cannot tell us when a solution tends to infinity in finite time, but they do allow us to conclude that the solution is unbounded.
One important reason for introducing the phase line early is the fact that much of our emphasis later in the course is on systems of equations rather than higher order equations. This means that students need to be able to understand parametrized curves in the xy-plane and to interpret the behavior of the x(t) and y(t) components of the solution. The phase line gives the students early practice doing this which serves them well in this later activity.
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
Figure 3: The graph of f(y) = y4 - y2 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?
The phase line and graph of f also provide an easy method to classify equilibrium points for autonomous differential equations. There are only three basic types: sinks (nearby solutions converge to the equilibrium point), sources (nearby solutions diverge), and nodes (all other behavior). With an eye toward the classification of equilibria in systems, we discuss the ``first derivative test'' for autonomous equations: Suppose p is an equilibrium point for the equation
Figure 5: Sinks, sources, and nodes
See Figure 5. As above, students need to understand the relationship between the sign of f'(p), the rise or fall of the graph of f near p, and the behavior of solutions to appreciate this result.
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
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
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
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
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
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.''
Having used the logistic population model as earlier in the course as one of our fundamental models of a nonlinear differential equation, it is natural to augment this model to take into account the effect of harvesting on the population. Recall that the logistic population model is given by
where we have normalized both the growth constant and carrying capacity to be 1. This equation has a source at P = 0 and a sink at P = 1. Students easily interpret this to mean that any nonzero initial population eventually leads to a population that settles down to a (normalized) value P = 1.
Now we introduce harvesting. Suppose that we remove k ``units'' of population per time period. If we assume that these individuals are removed continuously, the differential equation becomes
In Figure 8 we see that this graph is tangent to the P-axis when K = 1/4. For K < 1/4, this equation has 2 equilibrium points but when K > 1/4 both equilibrium points disappear. More importantly, since the graph of fK is negative when K > 1/4, we conclude that the population necessarily dies out for these parameter values. More importantly, we see that, as long as we start with a sufficiently large population, the population never dies out when K < 1/4. As soon as harvesting exceeds K = 1/4, then disaster strikes -- the population necessarily becomes extinct. Although oversimplified, this example provides an excellent example of the importance of bifurcation phenomena in differential equations.
This lab is due Thursday, November 14 at 5 PM. Turn it in to your TF or to the TF in the lab.
In this lab you will discuss the phase portraits and bifurcations for the linear system of differential equations
dy/dt = x - 0.5 y
where A is a parameter and -0.4 < A < 0.4. If you click on the following icon, you will see a QuickTime animation of the phase planes for this system as A changes.
Click here to download the animation.
Incidentally, contrary to what is printed in the animation, you cannot click on the buttons to control the movie. Depending upon which Quick Time viewer you have, you can stop and start the motion by using the control panel at the bottom of the screen, or by double clicking in the window.
In this animation, we have selected eight different initial points (marked by asterisks) and plotted the corresponding solution curves through these points. Note that the behavior of these solutions changes as A varies. Your job is to explain the bifurcations that this system undergoes as the parameter changes.
Answer each of the following questions about this system.
1. List below all values of A for which there are bifurcations for this system. Sketch the phase portraits just before, at, and just after each bifurcation. Use one color pen to indicate solutions that tend toward an equilibrium point, another for those that tend away from an equilibrium point, and a third for solutions that do neither. Also, indicate what happens to the eigenvalues of the system as the parameter passes through the bifurcation.
2. Sketch the path in the trace-determinant plane that this family of matrices is following as A increases from -0.4 to 0.4. Indicate on this path where the two bifucations have occurred.
In this demo you will see the motion of three solutions of the logistic differential equation
On the left is a plot of the graph of P vs t. On the right is the phase line for the system. You will see the motion of 3 solutions (red, blue, and green) simultaneously on the graph and the phase line.
In this demo you will see the motion of a particular solution of the differential equation for the nonlinear pendulum
dy/dt = -0.2 y - sin(x)
On the left is the phase plane. On the right are the graphs of x(t) and y(t). The idea is to explain the motion that you are seeing. At various points the motion in the animation stops. At these points, you should quickly stop the video using the video controls at the bottom of the screen. The questions I ask at this point are: Where is the actual pendulum when animation stops? What is the direction of motion? Students should act out the motion using their arms as simulators of the pendulum. It is nice not to swing your arms so wildly so as to endanger the lives of your neighbors.