MATH 20D Lecture 17

5.4 - Phase Plane of Systems

Case 1. $A$ has two real-valued eigenvalues of opposite signs.

The origin is called a saddle
”semi-stable”
2 kinds of “long-term” behavior for our solution:

1. If our initial condition starts on the direction of the eigenvector corresponds to the negative eigenvalue
   Then “stable direction”
2. If our initial condition starts everywhere else, then “unstable direction”

Case 2a. $A$ has two distinct real-valued eigenvalues that are both negative.

“fast direction”

• eigenvector of eigenvalue that is “more negative”
• controls the solution behavior away from origin
  ”slow direction”
• eigenvector of eigenvalue that is “less negative”
• controls behavior near origin
  Origin is stable, called sink/attractor

Case 2b. $A$ has two distinct real-valued eigenvalues that are both positive.

“fast direction”

• eigenvector of the eigenvalue that is “more positive”
• controls behaviors away/far from the origin
  ”slow direction”
• eigenvector of the eigenvalue that is “less positive”
• controls behavior near origin
  Origin is source/repellor/unstable

Case 3a. $A$ has complex eigenvalues ${\lambda }_{1,2}=\pm i\beta$ (pure imaginary eigenvalues)

Origin is a circle/ellipse (because solutions will orbit around circles, depend only on sin/cosine)
Stability N/A

Case 3b. $A$ has complex eigenvalues ${\lambda }_{1,2}=\pm i\beta$ with $\alpha >0$

Origin is a spiral source (because )
Unstable

Case 3c. $A$ has complex eigenvalues ${\lambda }_{1,2}=\pm i\beta$ with $\alpha <0$

Origin is a spiral sink, stable

Case 4a. $A$ has repeated eigenvalue $\lambda >0$ and two independent eigenvectors

Origin is called a star source, unstable (because )

Case 4b. $A$ has repeated eigenvalue $\lambda <0$ and two independent eigenvectors

Origin is called a star sink, stable (because )