MATH 304, section 2 -- Ordinary Differential Equations

Some Examples of ``phase portraits'' of  2 x 2  systems

September 27, 2004

First we consider the system   X' = matrix([[2, 3], [3, 2]])   X

By the Superposition Principle, we know that

all   X(t) = c[1]*exp(5*t)*matrix([[1], [1]])+c[2]*exp(-t)*matrix([[1], [-1]])   are

solutions.  Here are a few of these plotted together in

the   x - y   plane.

>    plot([[exp(5*t), exp(5*t),t=0..1],
[-2*exp(5*t),-2*exp(5*t),t=0..1], [.5*exp(5*t)-8*exp(-t),.5*exp(5*t)+8*exp(-t),t=0..1],
[exp(5*t)+7*exp(-t),exp(5*t)-7*exp(-t),t=0..1],
[2*exp(5*t)+3.5*exp(-t),2*exp(5*t)-3.5*exp(-t),t=0..1],
[-exp(5*t)-4*exp(-t),-exp(5*t)+4*exp(-t),t=0..1],
[-exp(5*t)+3*exp(-t),-exp(5*t)-3*exp(-t),t=0..1],
[10*exp(-t),-10*exp(-t),t=0..2], [-10*exp(-t),10*exp(-t),t=0..2]
],x=-20..20,y=-20..20,color=[red,red,black,black,black,black,black,blue,blue]);

[Maple Plot]

Now consider

X' = matrix([[5, -1], [2, 2]])   X .   The general solution here is

X(t) = c[1]*exp(4*t)*matrix([[1], [1]])+c[2]*exp(3*t)*matrix([[1/2], [1]])  .  Here are a few of these:

>    plot([[exp(4*t), exp(4*t),t=0..3],
[-2*exp(4*t),-2*exp(4*t),t=0..3], [exp(4*t)+exp(3*t)/2,exp(4*t)+exp(3*t),t=0..2],
[.2*exp(4*t)-exp(3*t)/2,.2*exp(4*t)-exp(3*t),t=0..2],
[-.2*exp(4*t)+exp(3*t)/2,-.2*exp(4*t)+exp(3*t),t=0..2],
[exp(3*t)/2,exp(3*t),t=0..10], [-exp(3*t)/2,-exp(3*t),t=0..10]
],x=-200..200,y=-200..200,color=[red,red,black,black,black,blue,blue]);

[Maple Plot]

>