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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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2

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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ENDEND