For x′ = Ax, try a solution shaped like an eigenvector times an exponential. The system collapses into an eigenvalue problem.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
What the eigenvectors mean
An eigenvector is a direction the system does not turn. A trajectory starting on one stays on it forever.
The signs decide stability
Both eigenvalues negative gives a stable node. Both positive gives an unstable node. Opposite signs give a saddle.
Complex eigenvalues
Complex eigenvalues mean rotation. The real part decides whether the spiral closes in, opens out, or holds a circle.
Purely imaginary
A zero real part gives a centre: closed orbits that neither approach nor escape. Undamped oscillation is exactly this case.
The whole chapter in one rule
Every phase portrait pattern is read off the eigenvalues. The picture and the algebra are the same statement.
Eigenvalues classify the equilibrium
For x′ = Ax, the eigenvalues of A determine the behaviour near the origin. Their signs and whether they are real or complex give the type and the stability directly.
Real eigenvalues
Both negative gives a stable node, both positive an unstable one, and opposite signs a saddle. Eigenvectors give the directions along which the motion is purely exponential.
Complex eigenvalues mean rotation
A complex pair produces spiralling. The real part decides whether the spiral converges or diverges, and a zero real part gives closed orbits around a centre.
Nonlinear systems are linearised
Near an equilibrium, a nonlinear system behaves like its linear approximation, so the eigenvalue analysis still classifies it. That is how the theory reaches systems it cannot solve.
Step 2: Try It Yourself
Tap and try it out.
- i-hat lands on(-2, 1)
- j-hat lands on(1, -2)
- Determinant3
- Eigenvalues-1 and -3
The dashed lines are the eigen-directions. A vector on one of those lines still points the same way after the transformation, only longer or shorter, and the eigenvalue is that stretch factor.
The shaded parallelogram is the image of the unit square, and its area is 3. That is exactly what the determinant measures.
- Released from(2, 0)
Every arrow is perpendicular to the line from the origin, so trajectories circle rather than approach.
Step 3: Watch an Example
One step at a time.
Watch Layla Classify From the Eigenvalues
Layla has a system matrix with eigenvalues −3 and 2.
- Step 1
She checks whether the eigenvalues are real, and they are.
Step 4: Your Turn
Practice makes it stick.
The Signs
Problem 1 of 2
Eigenvalues −4 and −1. Enter 1 stable node, 2 unstable node, 3 saddle.
The Spiral
Problem 2 of 2
Complex eigenvalues with real part −2. Does the spiral approach the origin? 1 yes, 0 no.
Read the Eigenvalues
1 of 8
Eigenvalues 3 and 5. Enter 1 stable node, 2 unstable node, 3 saddle.
2 of 8
Eigenvalues −6 and 1. Enter 1 stable node, 2 unstable node, 3 saddle.
3 of 8
Complex eigenvalues with real part 0. Enter 1 stable spiral, 2 unstable spiral, 3 centre.
4 of 8
Complex eigenvalues with real part 1. Enter 1 stable spiral, 2 unstable spiral, 3 centre.
5 of 8
Trace 4 and determinant 3. What is the larger eigenvalue?
6 of 8
A trajectory starting exactly on an eigenvector direction. Does it leave that line? 1 yes, 0 no.
7 of 8
Sort each eigenvalue situation by whether the origin is stable.
Tap something to move it.
- Empty
- Empty
8 of 8
Eigenvalues −2 and −7. Enter 1 stable node, 2 unstable node, 3 saddle.
Step 5: Quick Check
Show what you know.
Question 1 of 2
Eigenvalues −5 and 2. Enter 1 stable node, 2 unstable node, 3 saddle.
Question 2 of 2
What decides whether an equilibrium is stable?
What You Learned
- For x′ = Ax, the eigenvalues of A decide the entire phase portrait.
- Both negative gives a stable node, both positive an unstable one, opposite signs a saddle.
- Complex eigenvalues give spirals, and a zero real part gives closed orbits around a centre.