Teacher page: phase portraits and eigenvalues

The one thing to do with the animation

Do the saddle twice.

Release a particle exactly on the incoming line, then release one a hair off it. The first slides into the origin; the second swings away. Nothing else makes instability as obvious.

Then work through the five presets asking only two questions each time: does it come in or go out, and does it turn? The eigenvalue readout answers both, and the picture confirms it.

One sentence instead of six pictures

The real part decides in or out. The imaginary part decides whether it turns.

Give them that and the classification table becomes something they can rebuild rather than recall. A student who has memorised six portraits has nothing when the numbers are unfamiliar.

The answers

[[0, 1], [1, 0]]	λ = 1 and −1. Opposite signs, so a saddle.
[[−1, 0], [0, −2]]	λ = −1 and −2. Both negative, a stable node.
[[1, 0], [0, 2]]	λ = 1 and 2. Both positive, an unstable node.
[[0, 1], [−1, 0]]	λ = ±i. Purely imaginary, a centre.
[[−1, 1], [−1, −1]]	λ = −1 ± i. Complex, negative real part, a stable spiral.
1. λ = 2 and −3	B, a saddle point.
2. λ = −1 ± 2i	A, spirals inwards.
3. Purely imaginary	B, cycles forever.

Where the marks go

1 markFinding the eigenvalues. Trace and determinant is faster and safer than expanding by hand.

1 markNaming the type correctly from them.

1 markSaying what it means in context, which is the one most often dropped.

1 markOn a sketch, direction arrows pointing the right way.

What each wrong answer tells you

They give	What it means
"Stable" for a saddle	Some paths do head in, so it looks stable from one direction. Release the two nearby particles and the answer is visible.
Spiral from real eigenvalues	They have remembered that spirals exist without tying them to the complex case. The imaginary part is what turns it.
Shape right, arrows wrong	Very costly and very common. The sign of the real part sets the direction and must be checked separately from the shape.
"Spiral" as the whole answer	Incomplete. Stable or unstable, and what that means for the quantities being modelled.

Other things they will say

"What is on the axes?" The two quantities, not time. Time is the parameter along each path, which is why arrows are needed at all: without them a path has no direction.

"What does the origin mean?" The equilibrium, where both rates are zero. In a population model it is the balance point, and stability is whether the system returns to it after a nudge.

"Do I need the eigenvectors?" For the exact solution in the real distinct case, yes. For classifying and sketching, the eigenvalues alone are enough.

A possible order

	What is happening
1	Two quantities changing together. Why plotting x and y against t separately hides the structure.
2	The five presets, two questions each: in or out, turning or not.
3	Trace and determinant, and the eigenvalue formula.
4	The saddle, twice, with the near-miss release.
5	Questions 1 to 3, and classifying a system they have not seen.
6	What stability means for a real population.

Two things not to say

Do not hand out the six-row table first. Give them the one sentence about real and imaginary parts, then let them rebuild the table from it; it is a two minute exercise and it sticks.

Do not accept a sketch without arrows. A stable and an unstable node look identical on paper, and the arrows are the entire difference.

Questions to set

Three tiers, ramping the way practice should: the method on its own, then the method inside something real, then a challenge. Set the tier the class in front of you needs rather than one undifferentiated sheet. Answers are given so these can go straight onto a board.

1Fluency

The method on its own, with friendly numbers. Set these first and move on quickly once they are secure.

EigenvaluesFind the eigenvalues of [[1, 0], [0, 2]] and name the type.
1 and 2, both positive and real, so an unstable node. Everything moves away from the origin.
A stable caseDo the same for [[−1, 0], [0, −2]].
−1 and −2, both negative, so a stable node. Everything is drawn in.

2In context

The same skill inside a real situation, where the first job is working out what is being asked.

A saddleFind the eigenvalues of [[1, 0], [0, −1]] and describe the motion.
1 and −1. Opposite signs give a saddle: trajectories come in along one eigenvector and leave along the other.
Purely imaginaryFor [[0, 1], [−1, 0]] the eigenvalues are ±i. What does that mean for a population modelled by it?
A centre: closed orbits with no decay, so the two quantities cycle for ever at constant amplitude and never settle. In a real population that is suspicious, because nothing oscillates undamped indefinitely.

3Challenge

Reasoning, working backwards, or spotting an error. These are where the top grades are decided.

Classify from the signs aloneExplain how the signs of the real parts decide stability, without computing a trajectory.
Both real parts negative means every direction decays, so the origin is stable. Any positive real part means some direction grows, so it is unstable. Zero real parts with an imaginary part give neutral cycling.
Why the eigenvectors matterTwo systems both have eigenvalues 1 and −1. Explain what can still differ between their portraits.
The eigenvectors, which set the directions of the incoming and outgoing separatrices. The classification is the same saddle but the picture is rotated or sheared, so the actual paths differ.

Practicalities

Works on a phone. Nothing is loaded from any other site, so it runs behind a school firewall, and nothing a student does is saved or sent anywhere.