In this chapter we look at models in which something is changing continuously over time. These models are described mathematically by differential equations of the form
When you read the words "differential equations" and look at the equation above, your first reaction is likely to be -- "What has this got to do with middle school or high school science and mathematics?" Traditionally differential equations have been put off until the sophomore or junior year of college. Even more recently, differential equations have been taught as part of college or AP calculus. But, in fact, some models that are best described using the language of differential equations are accessible to high school and even middle school students.
Traditional college differential equations courses concentrate on algebraic methods for finding exact solutions of differential equations. We will do very little of this, although we will point out a few places where modeling provides important examples that can illustrate particular ideas or techniques in a modern AP calculus course. Instead we will be using the language of differential equations and graphical and numerical ways of looking at differential equations. These are extremely powerful techniques and yet they are well within the range of high school and even middle school students.
We begin with the simplest possible differential equations -- differential equations of the form
or, more compactly,
-- that is, the function on the right hand side of the general equation
is the linear function
Most of our work in this chapter will be graphical and we will use many different kinds of graphs. We will begin by applying some graphical tools to the differential equation
Click here to open a new window with a Java applet. Arrange these two windows so that they overlap and you can move easily back-and-forth between them by clicking on the exposed portion of the inactive window to make it active.
This applet has three graphs and a blue bar. Two of the graphs look familiar. The green graph is a graph that shows the function
The horizontal axis is used for the variable p and the vertical axis for the values of the function f(p). When the applet is first opened the constant m is 1/2. You can change the value of this constant by clicking along the right edge of the green graph. Try it. If you click at the top of the right edge, the constant m becomes +1. If you click at the bottom of the right edge, the constant m becomes -1. Clicking at intermediate points produces intermediate values for m.
Below the green graph is a one-dimensional "graph." You should think of this graph as being a map of a river. Different points along the river represent different values of the variable p. This line is a copy of the p-axis (the horizontal axis) in the green graph directly above. The points line up exactly. For example, the value p = 0 is marked by a purple dot on the green graph and on the river graph.
The arrows below the river indicate the direction in which the current in the river is flowing. Make the value of m approximately equal to +1/2 by clicking halfway between the p-axis and the top edge of the green graph along the right edge of the green graph. Remember the green graph shows the right hand side of the differential equation
Notice that when p is positive, so is f(p) and, thus, p' = f(p) is positive. This means that p will increase. Thus at these points the current is flowing to the right -- in the direction of increasing p. When p is negative, so is f(p) = m p and, thus, at these points p is decreasing and the current is going to the left. Notice the arrows below the river point to the right at positive values of p and to the left at negative values of p. These arrows indicate the direction of the current.
Now make the value of m approximately -1/2 by clicking along the right edge of the green graph roughly half-way between the p-axis and the bottom edge of the green graph. Notice that the arrows below the river change direction. This happens because now that m is negative, the right hand side of the differential equation is negative when p is positive. Thus for positive value of p the current now flows to the left. Similarly, for negative values of p the current now flows to the right.
Experiment with different values of m. Notice that when m is negative the arrows point toward the origin (the purple dot) and when m is positive they point away from the origin.
The purple dot marks a point (zero) where the current is stagnant. This is an equilibrium point for our differential equation. If the initial condition is at this point -- that is, if the initial condition is zero -- then p(t) will always remain at zero. Thinking about the river, if a cork is placed in the river at this point it will sit there forever.
When the arrows (the current) are pointing toward zero then this equilibrium point is attracting -- a cork dropped into the river will be carried toward the equilibrium point. When the arrows point away from zero then this equilibrium point is repelling and a cork in the river will be carried away from zero unless the cork is placed exactly at zero.
The blue graph will be a graph that shows the function p(t). The horizontal axis of this graph is used for time, the variable t, and the vertical axis is used for the variable p. This can be confusing at first. The variable p is marked on the horizontal axis in the green graph and in the horizontal river graph but it is marked on the vertical axis in the blue graph.
Notice there are two arrows, one pointing downward indicating a point on the river and one pointing to the right indicating a point on the vertical axis (the p-axis) in the blue graph. These arrows indicate the same value of p on the two different graphs. You can move either arrow by clicking just above the river graph or by clicking just to the left of the p-axis in the blue graph. When you move one arrow the other moves at the same time, so that the two arrows always indicate the same value of p on the two different graphs. Experiment a bit moving these arrows to see how the two different axes are used to represent the same points.
The two arrows you have been moving indicate the initial condition for a continuous dynamical system
To see the predictions made by this continuous dynamical system, click on the blue bar at the lower right of the applet. The solution will be represented two different ways. In the river graph in the lower left you will see a cork placed at the initial condition and then watch as it is carried along by the currents. At the same time you will see a graph of its location, p(t), drawn on the blue graph. Try it. Experiment using the applet to see what happens with various different initial conditions and various different values of the constant m.
Summarize your observations. What effects does the value of the constant m have on the behavior of this dynamical system? What effects does the value of the constant b have on the behavior of this dynamical system? What effects does the initial condition have?
Your observations should lead to the following theorem.
Classification Theorem for Linear Continuous Dynamcial Systems
The continuous dyamical system
has exactly one equilibrium point
and this equilibrium point is attracting if m is negative and repelling if m is positive. If m is negative then for any initial condition the solution p(t) approaches the equilibrium point.
Proof
This theorem is easy to prove in a calculus class. The solution of the initial value problem is
It is easy to check that this is the solution. Then the conclusion follows by finding the appropriate limit.
Now we want to look at more complicated continuous dynamical systems of the form
where the function f(p) on the right hand side of the differential equation is nonlinear. Click here to open another window with another Java applet. Arrange the two windows so that they overlap and you can go back-and-forth easily between the two windows by clicking on the exposed part of the inactive window to make it active.
This Java applet is similar to the first Java applet in this module but it has a new feature. Notice the green graph (the graph of the function f(p)) is nonlinear. This dynamical system has two equilibrium points -- that is, two places where f(p) = 0. Nonlinear dynamical systems can have many equilibrium points. Experiment with this applet, trying different initial conditions and watching what happens.
After you've experimented with the function f(p) in this applet, you can change the function. If you click on any of the vertical lines in the green graph the value of the function at the value of p corresponding to the vertical line will be changed to the point at which you clicked. You can change the values of the function at any or all of these values of p and the applet will fill in the intermediate values with straight lines. Notice as you change the function the dots marking equilibrium points and the arrows indicating the direction of the current change. The applet draws arrows that fit and may omit arrows that don't fit well.
Change the function f(p) as described above and experiment with your new function. Try different initial conditions and different functions.
Based on your experimentation and the linear classification theorem above, formulate a theorem about when an equilibrium point of a nonlinear dynamical system is attracting and when it is repelling. What does it mean for an equilibrium point of a nonlinear dynamical system to be attracting?
