Systems of firstorder differential equations
149
Mathematicians have demonstrated a number of properties for the trajectories
of autonomous systems. Here we shall simply list them.
(1)
There is no more than one trajectory through any point in the phase plane
(2)
A trajectory that starts at a point that is not a fixed point will only reach
a fixed point in an infinite time period
(3)
No trajectory can cross itself unless it is a closed curve. If it is a closed
curve then the solution is a periodic one.
4.3
Vectors of forces in the phase plane
We established in chapter 2, when considering single autonomous differential
equations, that we could establish the direction of
x
when
t
varies from the sign of
˙
x
. In the case of a system of two differential equations we can establish the direction
of
x
from the sign of ˙
x
and the direction of
y
from the sign of ˙
y
. Such movements
in
x
and
y
give us insight into the dynamics of the system around the equilibrium.
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 Fall '11
 Dr.Gwartney
 Economics, Differential Equations, Equations, Elementary algebra, Softwarepackages

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