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7/29/2019 Math 275 Spring 2012 3-7
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3.7 Trace - Determinant Plane
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2
2
Again we consider the system: , .
With characteristic equation 0,
4giving eigenvalues .
2
a bdX AX Ac ddt
T D
T T D
= =
+ =
=
G
G
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22
The , , determines the nature
of the eigenvalues for the matrix
onsider the equation
.
This equation describes a parabola in the Trace - Determinant
plane,
4
C
4 0
4
.
discriminant T D
TT D D
A
= =
upon which, the eigenvalues of matrix are real and equal.
.,
2 2
A
T T =
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T
D
1R
2R3R
4R
5R 6R
Plane regions to analyzeT D
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5
2
2
2
We consider each of the regions in the - plane:
Region
In and the real eigenvalues are
What about the second eigenval
1:
1 4 0 0,
4
2
4
20 0.
T D
R
R T D T
T T D
T T DT
> >
=
+ > >
2
ue ?4
2
T T D
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6
2 2 2
2
2
Region 1:
If 0 4 4 since 0.
4So 0 ... both eigenvalues are positive so2
the origin is a .
4If 0 , so the matrix has one
2positive eigenvalue and one zero
0T
R
D T D T T D T T
T T D
source
T DD
> < < >
>
= =
eigenvalue. The
positive axis is an .T entire line of sources
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T
D
Sources
2R3R
4R
5R 6R
Lineof Sources
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2
2 2
Region 2:
In 2 4 0, and the eigenvalues arecomplex:
4 42 2 2
Since 0 0
The origin is a .
Re( )2
R
R T D
T i D T T D T i
T
spiral source
T
>=
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T
D
Sources
3R
4R
5R 6R
Lineof Sources
Spiral
Sources
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2
2 2
Region 3:
In 3 4 0, and again the eigenvalues arecomplex:
4 4
2 2 2
Since 0 0
The origin is a .
Re( )2
R
R T D
T i D T T D Ti
T
spiral sink
T
>
+
=
=
ues are zero.The origin is "algebraically unstable".
If 0 both eigenvalues are negative.
The origin is a .
T
sink
<
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T
D
Sources
Saddles
Lineof Sources
Spiral
Sources
Spiral
Sinks
Sinks
Lineof Sinks
Saddles
Centers
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