Eigenvalues and Stability

The algebra behind every phase portrait.

12 problems · about 25 minutes · N-VM.C.11, A-REI.B.4

NameDate

Figures — use these to answer the problems

This is the system matrix. Its eigenvalues are printed below, and their signs decide the whole phase portrait.

a-2

d-2

Figure 1

Purely imaginary eigenvalues give closed orbits like these. Nothing approaches the origin and nothing escapes.

the rotation rate1

starting first quantity2

starting second quantity0

Figure 2

Straightforward practice. Get the method working first.

5 problems

Eigenvalues −5 and 2. Enter 1 stable node, 2 unstable node, 3 saddle.
Answer
2
What decides whether an equilibrium is stable?
- Whether the real parts of the eigenvalues are all negative.
- Whether the eigenvalues are large.
Eigenvalues 3 and 5. Enter 1 stable node, 2 unstable node, 3 saddle.
Answer
Eigenvalues −6 and 1. Enter 1 stable node, 2 unstable node, 3 saddle.
Answer
Complex eigenvalues with real part 0. Enter 1 stable spiral, 2 unstable spiral, 3 centre.
Answer

The same ideas with more to keep track of.

3 problems

Complex eigenvalues with real part 1. Enter 1 stable spiral, 2 unstable spiral, 3 centre.
Answer
Trace 4 and determinant 3. What is the larger eigenvalue?
Answer
A trajectory starting exactly on an eigenvector direction. Does it leave that line? 1 yes, 0 no.
Answer

Mixed problems. Work out what kind of question it is before you start.

4 problems

Sort each eigenvalue situation by whether the origin is stable.
Write each item under the heading it belongs to: Both eigenvalues negative · Both eigenvalues positive · Complex with negative real part · Opposite signs
Stable
Not stable
Eigenvalues −2 and −7. Enter 1 stable node, 2 unstable node, 3 saddle.
Answer
The Signs
Eigenvalues −4 and −1. Enter 1 stable node, 2 unstable node, 3 saddle.
Answer
The Spiral
Complex eigenvalues with real part −2. Does the spiral approach the origin? 1 yes, 0 no.
Answer