Systems of Differential Equations (Part 7. Phase Plane)
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Phase Plane
Before proceeding with actually solving systems of
differential equations there’s one topic that we need to take a look at. This is a topic that’s not always taught in a
differential equations class but in case you’re in a course where it is taught
we should cover it so that you are prepared for it.
Let’s start with a general homogeneous system,

(1)

Notice that


is a solution to the system of differential equations. What we’d like to ask is, do the other
solutions to the system approach this solution as t increases or do they move away from this solution? We did something similar to this when we
classified equilibrium solutions in a previous section.
In fact, what we’re doing here is simply an extension of this idea to
systems of differential equations.
The solution is called an equilibrium solution for the system. As with the single differential equations
case, equilibrium solutions are those solutions for which


We are going to assume that A is a nonsingular matrix and hence will have only one solution,


and so we will have only one equilibrium solution.
Back in the single differential equation case recall that we
started by choosing values of y and
plugging these into the function f(y)
to determine values of . We then used these values to sketch tangents
to the solution at that particular value of y. From this we could sketch in some solutions
and use this information to classify the equilibrium solutions.
We are going to do something similar here, but it will be
slightly different as well. First we are
going to restrict ourselves down to the 2 x 2 case. So, we’ll be looking at systems of the form,


Solutions to this system will be of the form,


and our single equilibrium solution will be,


In the single differential equation case we were able to
sketch the solution, y(t) in the yt plane and see actual solutions. However, this would somewhat difficult in
this case since our solutions are actually vectors. What we’re going to do here is think of the
solutions to the system as points in the x_{1}x_{2}
plane and plot these points. Our
equilibrium solution will correspond to the origin of x_{1}x_{2} plane and the x_{1}x_{2} plane is called the phase plane.
To sketch a solution in the phase plane we can pick values
of t and plug these into the
solution. This gives us a point in the x_{1}x_{2} or phase
plane that we can plot. Doing this for
many values of t will then give us a
sketch of what the solution will be doing in the phase plane. A sketch of a particular solution in the
phase plane is called the trajectory
of the solution. Once we have the
trajectory of a solution sketched we can then ask whether or not the solution
will approach the equilibrium solution as t
increases.
We would like to be able to sketch trajectories without
actually having solutions in hand. There
are a couple of ways to do this. We’ll
look at one of those here and we’ll look at the other in the next couple of
sections.
One way to get a sketch of trajectories is to do something
similar to what we did the first time we looked at equilibrium solutions. We can choose values of (note that these will be points in the phase
plane) and compute . This will give a vector that represents at that particular solution. As with the single differential equation case
this vector will be tangent to the trajectory at that point. We can sketch a bunch of the tangent vectors
and then sketch in the trajectories.
This is a fairly work intensive way of doing these and isn’t
the way to do them in general. However,
it is a way to get trajectories without doing any solution work. All we need is the system of differential
equations. Let’s take a quick look at an
example.
Example 1 Sketch
some trajectories for the system,
Solution
So, what we need to do is pick some points in the phase
plane, plug them into the right side of the system. We’ll do this for a couple of points.
So, what does this tell us? Well at the point (1, 1) in the phase
plane there will be a vector pointing in the direction
Doing this for a large number of points in the phase plane
will give the following sketch of vectors.
Now all we need to do is sketch in some trajectories. To do this all we need to do is remember
that the vectors in the sketch above are tangent to the trajectories. Also the direction of the vectors give the
direction of the trajectory as t
increases so we can show the time dependence of the solution by adding in
arrows to the trajectories.
Doing this gives the following sketch.
This sketch is called the phase portrait. Usually
phase portraits only include the trajectories of the solutions and not any
vectors. All of our phase portraits
form this point on will only include the trajectories.
In this case it looks like most of the solutions will
start away from the equilibrium solution then as t starts to increase they move in towards the equilibrium solution
and then eventually start moving away from the equilibrium solution
again.
There seem to be four solutions that have slightly
different behaviors. It looks like two
of the solutions will start at (or near at least) the equilibrium solution
and them move straight away from it while two other solution start away from
the equilibrium solution and then move straight in towards the equilibrium
solution.
In these kinds of cases we call the equilibrium point a saddle point and we call the
equilibrium point in this case unstable
since all but two of the solutions are moving away from it as t increases.

As we noted earlier this is not generally the way that we
will sketch trajectories. All we really
need to get the trajectories are the eigenvalues and eigenvectors of the matrix
A.
We will see how to do this over the next couple of sections as we solve
the systems.
Here are a few more phase portraits so you can see some more
possible examples. We’ll actually be
generating several of these throughout the course of the next couple of
sections.
Not all possible phase portraits have been shown here. These
are here to show you some of the possibilities.
Make sure to notice that several kinds can be either asymptotically
stable or unstable depending upon the direction of the arrows.
Notice the difference between stable and asymptotically
stable. In an asymptotically stable
node or spiral all the trajectories will move in towards the equilibrium point
as t increases, whereas a center
(which is always stable) trajectory will just move around the equilibrium point
but never actually move in towards it.
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