Coupled differential equations, the five shapes they can make, and reading which one you have straight off the eigenvalues.
Two quantities changing together, each depending on both. You cannot draw x against t and y against t and see what is going on, so instead plot x against y and draw the paths. Pick a system and tap anywhere to release a particle.
For dx/dt = ax + by and dy/dt = cx + dy, find the eigenvalues of [[a, b], [c, d]] and read the table.
| Eigenvalues | Shape | What happens |
|---|---|---|
| both real, both negative | stable node | everything is drawn into the origin |
| both real, both positive | unstable node | everything is thrown outwards |
| real, opposite signs | saddle | pulled in one way, flung out the other |
| complex, negative real part | stable spiral | spirals inwards |
| complex, positive real part | unstable spiral | spirals outwards |
| purely imaginary | centre | closed loops, going round forever |
The real part decides in or out. The imaginary part decides whether it turns. That one sentence is the entire classification, and it is quicker than memorising six pictures.
If what is under the root is negative, the eigenvalues are complex and the trajectories turn. For [[0, 1], [1, 0]] the trace is 0 and the determinant is −1, so λ = ±1: opposite signs, which is a saddle.
A saddle is never stable, even though some paths head straight for the origin. Release a particle on the incoming line in the saddle above, then release one a hair off it: the second one turns away. In a real population that means a balance that cannot survive a nudge.
The origin is the equilibrium point: both rates are zero, so nothing changes. A stable node or spiral means the system returns there after a disturbance. A saddle or an unstable node means it does not. For two interacting populations, that is the difference between a balance that holds and one that collapses.
1. A system has eigenvalues 2 and −3. The origin is:
2. Eigenvalues are −1 ± 2i. The trajectories:
3. For [[0, 1], [−1, 0]] the eigenvalues are purely imaginary. What does that mean for a population modelled by it?
Finding the eigenvalues correctly is the calculation mark, and the trace and determinant route is faster and safer than expanding the determinant by hand.
Naming the type and saying what it means is the mark most often dropped. "Complex with negative real part, so a stable spiral, so the populations settle towards the equilibrium" is a complete answer. The word "spiral" alone is not.
If asked to sketch, get the direction arrows right. A correct shape pointing the wrong way scores very little.
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