Important Note:
If you see funny characters in your display, and if you are using Xwindows
(for instance, under linux), please see the following file
for information on configuring your browser:
Xfonts
(unfortunately, printing will still give funny fonts  please use the pdf or
postscript versions of this document for printing!)
Some Remarks on Phase Planes
Some Remarks on Phase Planes
A technique which is often very useful in order to analyze the phase plane
behavior of a twodimensional autonomous system
is as follows. We would like to understand the graphs of solutions
(
x(
t),
y(
t)) look like. One possibility is to try to see these graphs as the
level sets of some function
h(
x,
y).
For example, take
(that is,
f(
x,
y) =
x and
g(
x,
y) =
y).
If we could solve for
t as a function of
x, by inverting the function
x(
t), and substitute the expression that we obtain into
y(
t), we would end
up with an expression
y(
x) for the
ycoordinate in terms of the
x
coordinate, eliminating
t. This cannot be done in general, but it suggests
that we may want to look at
dy/
dx. Formally (or, more precisely, using the
chain rule), we have that
which is a differential equation for
y as a variable dependent on
x.
This equation is separable:
so we obtain, taking antiderivatives,
where
c is an undetermined constant.
In conclusion, the solutions (x(t),y(t)) all lie in the circles
x^{2}+y^{2} = c of different radii and centered at zero.
Observe that we have
not solved the differential equation, since we did
determine the forms of
x and
y as functions of
t (which, as a matter of
fact, are trigonometric functions).
What we have done is just to find curves (the abovementioned circles)
which contain all solutions.
Even though this is less interesting (perhaps) than the actual solutions, it
is still very interesting. We know what the general phase plane picture looks
like.
Another example is this:
Here,
dy/
dx =
x^{5}/
y^{5} so we get again a separable equation, and
we see that the solutions all stay in the curves
More interesting is the general case of predatorprey equations:
where
a,
b,
c,
d are all positive constants.
Then
so

ó õ


æ ç
è


a
y

b 
ö ÷
ø

dy = 
ó õ


æ ç
è

 
c
x

+d 
ö ÷
ø

dx 

and from here we conclude that the solutions all stay in the sets
a ln(y)  by + c ln(x)  d x = c 

for various values of the constant
c.
It is not obvious what these sets look like, but if you graph the level sets
of the function
h(x,y) = a ln(y)  by + c ln(x)  d 

you'll see that the level sets look like the orbits of the
predatorprey system shown, for the special values
a = 2,
b = 1.2,
c = 1, and
d = 0.9 in page 144 of the book.
(Of course, the scales will be different, for different values of the
constants, but the picture will look the same, in general terms.)
This argument is used to prove that predatorprey systems always lead to
periodic orbits, no matter what the coefficients of the equation are.
Homework:
In each of the following problems, a system
is given. Solve the equation
and use the information to sketch what the orbits of the original equation
should look like.
 [(dx)/( dt)] = y(1+x^{2}+y^{2}), [(dy)/( dt)] = x(1+x^{2}+y^{2}).

[(dx)/( dt)] = 4y(1+x^{2}+y^{2}), [(dy)/( dt)] = x(1+x^{2}+y^{2}).

[(dx)/( dt)] = y^{3}e^{x+y}, [(dy)/( dt)] = x^{3}e^{x+y}.

[(dx)/( dt)] = y^{2}, [(dy)/( dt)] = (2x+1)y^{2}.

[(dx)/( dt)] = e^{xy}cos(x), [(dy)/( dt)] = e^{xy}.