Complex Eigenvalues and Phase Portraits. Fundamental Set of Solutions For Linear System of ODEs With Eigenvalues and Eigenvectors and The General Solution.

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Presentation transcript:

Complex Eigenvalues and Phase Portraits

Fundamental Set of Solutions For Linear System of ODEs With Eigenvalues and Eigenvectors and The General Solution Takes The Form

Not All Matrices Have Real Eigenvalues/Eigenvectors Has Eigenvalues and Eigenvectors and, Has The General Solution

Not All Matrices Have Real Eigenvalues/Eigenvectors Has The General Solution Lots of Complex Numbers We want a Real General Solution

Not All Matrices Have Real Eigenvalues/Eigenvectors Has The General Solution Lots of Complex Numbers We want a Real General Solution Recall Euler’s Formula

Not All Matrices Have Real Eigenvalues/Eigenvectors Has The General Solution Or Just a Constant

Not All Matrices Have Real Eigenvalues/Eigenvectors Has The General Solution Or Just a Constant

Not All Matrices Have Real Eigenvalues/Eigenvectors Has The General Solution Or Just a Constant

Not All Matrices Have Real Eigenvalues/Eigenvectors Has The General Solution Or Just a Constant

Not All Matrices Have Real Eigenvalues/Eigenvectors Has The General Solution Or

Some Things To Notice Has Eigenvalues and Eigenvectors and, Are Complex Conjugates

Some Things To Notice Has Eigenvalues and Eigenvectors and, Are Complex Conjugates

This Will Always Hold If Has Complex Eigenvalues and Eigenvectors then and So In Practice, Only Need To Find and

This Will Always Hold then and So In Practice, Only Need To Find and If This Has Complex Eigenvalues and Eigenvectors

General Solution The General Solution Is If This Has Eigenvalue and Eigenvector

General Solution The General Solution Is If This Has Eigenvalue and Eigenvector

Questions?

Phase Portraits Remember Direction Fields: t y ç ç 1.0 ç ç 1.5 ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç

We Can Draw Direction Fields For 2D Systems

ç ç

ç ç

ç ç

ç ç ç ç

ç ç ç ç

ç ç ç ç

ç ç ç ç

ç ç ç ç ç ç

ç ç ç ç ç ç ç ç

ç ç ç ç ç ç ç ç ç ç

ç ç ç ç ç ç ç ç ç ç ç ç

ç ç ç ç ç ç ç ç ç ç ç ç ç ç

A Plot With Many Solutions Is Called a “Phase Portrait”

Phase Portraits Give Us An Idea of How Solution Behaves

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?