Key Idea: A coupled system shows how two quantities, like predators and prey, change together. The eigenvalues of its matrix tell you what happens near the equilibrium at the origin.
| Eigenvalues of M | Type |
|---|---|
| Same-sign real | Node: both − = sink (stable); both + = source (unstable) |
| Opposite-sign real | Saddle (unstable) |
| Complex a ± bi | Spiral: stable if a < 0, unstable if a > 0; a = 0 → centre |
Then finish
- Write the system as d/dt(x, y) = M(x, y).
- Real eigenvectors are straight-line trajectories.
- General solution: A·eλ₁ᵗ·v₁ + B·eλ₂ᵗ·v₂.
- Say what the long-term behaviour means for the two quantities.
The real part decides stable or unstable. An imaginary part makes the path turn.
M has eigenvalues −2 ± 3i. Classify. Stable spiral: the populations cycle with smaller and smaller swings.
Eigenvalues 3 and −1? A saddle, which is unstable.
Eigenvalues −1 and −4? A sink: everything moves to the origin.
Eigenvalues ±2i? A centre.