Complex Eigenvalues and Phase Portraits

Fundamental Set of SolutionsFor Linear System of ODEs

y1y2

=a22

a12a11a21

� �y01y02

�

With Eigenvalues and Eigenvectors�1 v1and and v2�2

The General Solution Takes The Form

y1y2

�= C1 C2+e�1t v1 e�2tv2

Not All Matrices Have Real Eigenvalues/Eigenvectors

y1y2

=

� �y01y02

��11

�4�1

Has Eigenvalues and Eigenvectors

�1 v1and=�1 + 2i =2i1

�and= =

�,

1�2i

�2 v2�1� 2i

Has The General Solution

C1e(�1+2i)t C2e

(�1�2i)t+2i1

� �

1�2i

y =

Not All Matrices Have Real Eigenvalues/Eigenvectors

y1y2

=

� �y01y02

��11

�4�1

Has The General Solution

C1e(�1+2i)t C2e

(�1�2i)t+2i1

� �

1�2i

y =

Lots of Complex NumbersWe want a Real General Solution

Not All Matrices Have Real Eigenvalues/Eigenvectors

y1y2

=

� �y01y02

��11

�4�1

Has The General Solution

C1e(�1+2i)t C2e

(�1�2i)t+2i1

� �

1�2i

y =

Lots of Complex NumbersWe want a Real General Solution

Recall Euler’s Formula

Not All Matrices Have Real Eigenvalues/Eigenvectors

y1y2

=

� �y01y02

��11

�4�1

Has The General Solution

C1e(�1+2i)t C2e

(�1�2i)t+2i1

� �

1�2i

y =

y =�2 sin (2t)cos (2t)

�e�1tC1 C2+( )

�e�1tC1 C2( )+ i �

2 cos (2t)sin (2t)

Or

Just a Constant

Not All Matrices Have Real Eigenvalues/Eigenvectors

y1y2

=

� �y01y02

��11

�4�1

Has The General Solution

C1e(�1+2i)t C2e

(�1�2i)t+2i1

� �

1�2i

y =

y =�2 sin (2t)cos (2t)

�e�1t

�e�1tC1 C2( )+ i �

2 cos (2t)sin (2t)

Or

Just a Constant

B1

Not All Matrices Have Real Eigenvalues/Eigenvectors

y1y2

=

� �y01y02

��11

�4�1

Has The General Solution

C1e(�1+2i)t C2e

(�1�2i)t+2i1

� �

1�2i

y =

y =�2 sin (2t)cos (2t)

�e�1t

�e�1tC1 C2( )+ i �

2 cos (2t)sin (2t)

Or

Just a Constant

B1

Not All Matrices Have Real Eigenvalues/Eigenvectors

y1y2

=

� �y01y02

��11

�4�1

Has The General Solution

C1e(�1+2i)t C2e

(�1�2i)t+2i1

� �

1�2i

y =

y =�2 sin (2t)cos (2t)

�e�1t

�e�1t+

2 cos (2t)sin (2t)

Or

Just a Constant

B1 B2

Not All Matrices Have Real Eigenvalues/Eigenvectors

y1y2

=

� �y01y02

��11

�4�1

Has The General Solution

C1e(�1+2i)t C2e

(�1�2i)t+2i1

� �

1�2i

y =

y =�2 sin (2t)cos (2t)

�e�1t

�e�1t+

2 cos (2t)sin (2t)

Or

B1 B2

Some Things To Noticey1y2

=

� �y01y02

��11

�4�1

Has Eigenvalues and Eigenvectors

�1 v1and=�1 + 2i =2i1

�and= =

�,

1�2i

�2 v2�1� 2i

Are Complex Conjugates

Some Things To Noticey1y2

=

� �y01y02

��11

�4�1

Has Eigenvalues and Eigenvectors

�1 v1and=�1 + 2i =2i1

�and= =

�,

1�2i

�2 v2�1� 2i

Are Complex Conjugates

This Will Always Hold

If Has Complex Eigenvalues and Eigenvectors

�1 v1 then =�2

y1y2

=a22

a12a11a21

� �y01y02

�

and �1 and v1=v2

So In Practice, Only Need To Find

�1 v1and

This Will Always Hold

�1 v1 then =�2

y1y2

=a22

a12a11a21

� �y01y02

�

and �1 and v1=v2

So In Practice, Only Need To Find

�1 v1and

If This Has Complex Eigenvalues and Eigenvectors

General Solutiony1y2

=a22

a12a11a21

� �y01y02

�

�1 = a+ bi

v1=

+

�y =

The General Solution Is

C1e(a+bi)t

c1c2

d1d2

ii

++ C2e

(a�bi)t

�c1c2

d1d2

ii

��

�c1c2

d1d2

ii

++

If This Has Eigenvalue and Eigenvector

General Solutiony1y2

=a22

a12a11a21

� �y01y02

�

�1 = a+ bi v1=

�c1c2

d1d2

ii

++

The General Solution Is

y = eatB1

�c1cos (bt) sin (bt)� d1cos (bt) sin (bt)�c2 d2

+

eat �c1 cos (bt)sin (bt) d1

cos (bt)sin (bt)c2 d2

++

B2

If This Has Eigenvalue and Eigenvector

Questions?

Phase PortraitsRemember Direction Fields:

t

y

y0 = (1� t) y

2.1

0.5 1.0 1.5

3.1

4.1

We Can Draw Direction Fields For 2D Systems

y1y2

=

� �y01y02

��11

�4�1

y1

y2

We Can Draw Direction Fields For 2D Systems

y1y2

=

� �y01y02

��11

�4�1

y1

y2

� ��11

�4�1

11

We Can Draw Direction Fields For 2D Systems

y1y2

=

� �y01y02

��11

�4�1

y1

y2

=

� � ��11

�4�1

110

�5

We Can Draw Direction Fields For 2D Systems

y1y2

=

� �y01y02

��11

�4�1

y1

y2

=

� � ��11

�4�1

110

�5

We Can Draw Direction Fields For 2D Systems

y1y2

=

� �y01y02

��11

�4�1

y1

y2

� ��11

�4�1

�1�1

We Can Draw Direction Fields For 2D Systems

y1y2

=

� �y01y02

��11

�4�1

y1

y2

=

� � ��11

�4�1

�1�10

5

We Can Draw Direction Fields For 2D Systems

y1y2

=

� �y01y02

��11

�4�1

y1

y2

=

� � ��11

�4�1

�1�10

5

We Can Draw Direction Fields For 2D Systems

y1y2

=

� �y01y02

��11

�4�1

y1

y2

We Can Draw Direction Fields For 2D Systems

y1y2

=

� �y01y02

��11

�4�1

y1

y2

We Can Draw Direction Fields For 2D Systems

y1y2

=

� �y01y02

��11

�4�1

y1

y2

We Can Draw Direction Fields For 2D Systems

y1y2

=

� �y01y02

��11

�4�1

y1

y2

We Can Draw Direction Fields For 2D Systems

y1y2

=

� �y01y02

��11

�4�1

y1

y2

We Can Draw Direction Fields For 2D Systems

y1y2

=

� �y01y02

��11

�4�1

y1

y2

x ’ = − 1 x − 4 yy ’ = 1 x − 1 y

−3 −2 −1 0 1 2 3

−3

−2

−1

0

1

2

3

x

yWe Can Draw Direction Fields For 2D Systems

We Can Draw Direction Fields For 2D Systems

x ’ = − 1 x − 4 yy ’ = 1 x − 1 y

−3 −2 −1 0 1 2 3

−3

−2

−1

0

1

2

3

x

y

A Plot With Many Solutions Is Called a “Phase Portrait”x ’ = − 1 x − 4 y

y ’ = 1 x − 1 y

−3 −2 −1 0 1 2 3

−3

−2

−1

0

1

2

3

x

y

Phase Portraits Give Us An Idea of How Solution Behavesx ’ = − 1 x − 4 y

y ’ = 1 x − 1 y

−3 −2 −1 0 1 2 3

−3

−2

−1

0

1

2

3

x

y

Summary

• Can Use Euler’s Formula To Get General Solutions To Systems of Equations With Complex Eigenvalues

• Can Use Phase Portraits To Examine The Behavior Of Different Systems

Questions?