e-summary - will not tell you whether the solutions go...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
Eigenvalues, Eigenvectors, and Solution Curves Assume that J is the linearization of a two-by-two system at an equilibrium point. Case I: J has distinct, real, non-zero eigenvalues. 1. If λ 1 > λ 2 > 0: λ 1 is the dominant eigenvalue . v 1 is the dominant eigenvector . Solutions near the equilibrium will appear approxi- mately as shown at right. v 1 v 2 2. If λ 1 > 0 λ 2 : We rarely both with the label dominant . Solutions near the equilibrium will appear approxi- mately as shown at right. v 1 v 2 3. If 0 > λ 1 > λ 2 > 0: λ 1 is the dominant eigenvalue . (Note that this is true even though, in absolute value, λ 2 is bigger.) v 1 is the dominant eigenvector . Solutions near the equilibrium will appear approxi- mately as shown at right. v 1 v 2 1
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Case II: J has complex eigenvalues. In this case The eigenvalues are complex conjugates. I.e., λ = a ± bi Near the equilibrium solutions will either circle or spiral around the equilibrium.
Background image of page 2
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: will not tell you whether the solutions go clockwise or counter-clockwise. If | + 1 | < 1, then solutions near the equilibrium must spiral inward. Computing | + 1 | can be a pain. You should always seek to 1. Identify the real and imaginary parts. 2. Compute | + 1 | 2 and simplify if possible. Also, if you can prove this theorem you can save yourself future computation troubles: | + 1 | 2 = ( a + 1) 2 + b 2 , and | + 1 | 2 = tr ( J ) + det ( J ) + 1 (Recall that tr ( J ) is the sum of the diagonal entries of J .) If | + 1 | = 1, then solutions near the equilibrium will closely approximate closed loops, but closed loops are not guaranteed. If | + 1 | > 1, then solutions near the equilibrium will spiral out. 2...
View Full Document

This note was uploaded on 10/12/2011 for the course MATH 464 taught by Professor Dougbullock during the Fall '08 term at Boise State.

Page1 / 2

e-summary - will not tell you whether the solutions go...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online