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' =
X
By the Superposition Principle, we know that
all
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]); |
Now consider
X' =
X .
The general solution here is
. 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]); |
> |