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Victor Liou
Dr. McNamara
Math Lab 3.4
This project was to produce a picture of abcspace for the system shown below:
=
+
dxdt ax by
=
dydt cx
In this graph we would indicate areas where the system behaves differently such as spiral sink, center,
source in relation to the values of a, b, and c.
By making this “picture” we will have a better
understanding of this system.
The first part of the project asked to analyze the case where a=0.
We determine through
( 
det A
)
λI
that
=
λ2 bc
.
We know that when concerned with complex eigenvalues with equation
+
α βi
, If
=
α 0
, then the system behaves as a center around zero.
Therefore, If
<
bc 0
, then this system will be a
center.
This occurs in the 2
nd
and 4
th
quadrants when either b or c<0 but not both.
If
>
bc 0
, then
=±
λ
bc
and the system will be a saddle with straight line solutions
=
AY λY
.
This occurs in the 1
st
and 3
rd
quadrants when both b and c are positive and negative.
Finally if
=
bc 0
along the b and c axis, then the
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 Spring '08
 MCNAMARA
 Math

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