MA354 Lecture: Graphical Solutions of 2D Autonomous ODEs
Last class: For 1D autonomous ODEs
Sketching solution by drawing slope field (HW 1d, 1e)
Sketching solution by analyzing phase line (HW 3, 9 11e)
Another example using phase lines to sketch a solution.
Logistic Growth
dP
 r (M  P) P
dt
1. find rest points
2. find signs of P’ over
intervals
3. repeat steps for P’’
y’
y’’
(-)
--------------
zeros of P’: P=M and P=0
circle the rest points on the # line
draw little arrows!
  stable
 unstable
 semi-stable
P’ = rMP-rP2
P’’ = rM(P’)-[2rP(P’)]=P’(rM-2rP)=rP’*(M-2P)
zeros are zeros of P’: 0 and <
and also: P=M/2
(0)
(+)
------------(+) (M/2) (-)
(M)
(-)
------------(+)
Note that y’’ has a local maximum at M/2; this shows that the maximum rate of growth
occurs when the population is M/2. This has been used to find M in real populations.
When the change in the rate of growth is zero, P  M/2.
So far we’ve looked at 1d ODEs:
dy
 ( y  1)( y  2)
dx
…y(t)
yielded a phase line  y
dP
 r (M  P) P
dt
…P(t)
yielded a phase line  P
Now looking at 2D ODES (12.1):
2-dimensional ODES are often written in the form:
dy1
yields a phase plane (y1, y2)
 g ( y1 , y 2 )
dx
dy 2
 f ( y1 , y 2 )
dx
More physically intuitive is to express the variables spatially as a function of time:
dy
 g ( x, y )
dt
dx
 f ( x, y )
dt
These are again autonomous ODEs because the independent variable (t) doesn’t appear in
g or f.
The entire phase space is now 3D: x,y and t.
How do we represent this?
Let’s go back to the 2D case:
dy
 ( y  1)( y  2)
recall
dx
dy
 ( y  1)( y  2)
this is equivalent to
dt
Whether y is a function of x or t, we think
of the independent variable systematically
varying from - to +.
Thus in the 3D case, as t varies from - to +, the curves trace positions in an (x,y) plane,
requiring a 3D representation, unless we are willing to lose the time dimension. On a 2D
x-y plane, we can draw solutions on the (x-y) plane where the solution will be projected
for all times. (i.e., using parametric curves). Over time (as t varies), the solution moves
through the (x,y) plane. The solution to the ODE can be thought of as a pair of parametric
equations (x(t),y(t)) whose derivatives satisfy the ODE.
This could be potentially confusing! Consider the trajectory of myself through the xyplane over the course of a week:
Non-autonomous example:
In my case, a lot of information is lost due to the fact
that my trajectories really are time-dependent and I
travel in different directions along the same path.
However, I am not described by an autonomous diff
eq! Paths in autonomous ODEs have a special
properties that lend themselves to parametric
descriptions: namely, the departure from any point
(x,y) does not depend on t, as it is a function of x
and y only.
THUS: the trajectories of autonomous ODEs can be conveniently described
parametrically, on a 2D xy phase plane. Where you go from a point (x,y) is always the
same, it doesn’t depend on t.
Graphical Solutions for 2D ODEs
A solution curve (x(t),y(t)) in the xy-plane (phase plane) is called a trajectory, path or
orbit of the system..
Equilibrium points (rest points) occur where g(x,y)=0 and f(x,y)=0 simultanously. Then
y’=0 and x’=0 so neither variable is changing. Trajectories that “go through” rest points
are just a point.
Behavior of Trajectories
1) There is at most one trajectory through any point in the phase plane. (Only one
vector assigned to a point.) This implies trajectories cannot intersect.
2) No trajectory can meet itself unless it is a closed (periodic) curve.
3) A trajectory that starts at a point other than a rest point cannot reach a rest point in
a finite amount of time. (It can only approach the rest point asymptotically.)
Thus there are 3 possibilities for the long-term behavior of any trajectory:
(1) The trajectory is periodic.
(2) The trajectory asymptotically approaches a rest point.
(3) The trajectory is unbounded – at least one of x(t) and y(t) (possibly both) become
arbitrarily large.
Worksheet
PlotVectorField[{-x+y,-x-y},{x,-3,3},{y,-2,2}]
first quadrant:
dy
dt
 0 and
dx   0ifx  y

dt  0if ( y  x)
Notes: dynamical systems can be linearized at equilibrium points, and eigenvalues of the
linerized system can indicate whether the equilibrium point is stable or not.