Foundations — Stability of equilibria — stable, unstable, saddle, spiral, centre

Before you can classify a "stable spiral" you must be able to read every mark on the page. Below is every symbol and idea the parent note leans on, ordered so each one is built from the one before it. Nothing is assumed.

1. A point on a plane: $(x,y)$

Picture a bird's-eye map. Give me $x$ (how far east) and $y$ (how far north) and I can put my finger on exactly one location. That finger-spot is the state.

Why the topic needs it: the whole subject is about what happens to that dot over time — so first we must agree the dot lives on a 2D plane described by two numbers.

2. Time and the dot's motion: $t$, $\dot{x}$, $\dot{y}$

The pair $(\dot{x},\dot{y})$ is the velocity vector. Think of $\dot{x}$ as "opposite of standing still in the east–west direction" — a positive $\dot{x}$ pushes the dot right, negative pushes it left.

Why the topic needs it: "stability" is a question about motion after a nudge. No notion of speed, no notion of "comes back" or "flies away".

3. The rule that sets the arrows: $f$ and $g$

The word autonomous means $f$ and $g$ depend only on where you are, not on what time it is — the wind pattern is frozen, painted onto the map once and for all.

Why the topic needs it: this pair is the differential equation. Everything downstream — equilibria, Jacobian, eigenvalues — is squeezed out of $f$ and $g$.

4. Where the wind dies: the equilibrium $(x^{\ast },y^{\ast })$

Why the topic needs it: equilibria are the only candidates for long-term rest. The entire topic asks one thing about them — is the calm spot a trap, a launchpad, or a whirlpool?

5. Small nudges: $u=x-x^{\ast }$, $v=y-y^{\ast }$

This is just re-drawing the map with the origin moved onto the calm spot. A nudge means starting at small $u,v$ (dot placed just beside the calm point) and watching whether $u,v$ shrink to zero (comes back) or grow (runs away).

Why the topic needs it: stability is a local question. By centring on the equilibrium we only ever deal with small $u,v$, which is exactly what makes the next step — linearising — legal.

6. Slopes of the wind: partial derivatives $f_x,f_y,g_x,g_y$

The reason we can use a straight-line slope at all: very close to the calm spot, the curved wind pattern looks flat, just like the Earth looks flat from your window. That's a first-order Taylor approximation — trade the true curved rule for its tangent, which is a slope times a step.

Why the topic needs it: these four numbers are the raw material of the Jacobian matrix — the local "wind rulebook".

7. Packing the slopes into a grid: the matrix $J$

Near the equilibrium the flow obeys the beautifully simple rule $\left(\begin{array}{c}\dot{u}\\ \dot{v}\end{array}\right)=J\left(\begin{array}{c}u\\ v\end{array}\right).$ Read it as: "give me your current offset $(u,v)$ and $J$ hands back your current velocity." The matrix is the linearised law of motion.

Why the topic needs it: instead of studying every point separately, one $2\times 2$ grid captures the entire local behaviour. The rest of the topic is just interrogating this grid.

Learn matrices proper in Eigenvalues and eigenvectors.

8. Two summary numbers: trace $\tau$ and determinant $\Delta$

Why the topic needs it: the parent classifies every equilibrium using only these two numbers plus one derived quantity — no eigenvector hunting required. See the whole map in the Trace–determinant plane.

9. The growth rates: eigenvalues $\lambda$

The number $i$ (with $i^2=-1$) is what lets a single formula describe turning, because $e^{i\omega t}$ traces a circle. That is the mathematical reason spirals and centres exist at all.

Eigenvalues come from $\tau ,\Delta$ by ${\lambda }_{1,2}=\frac{\tau \pm \sqrt{{\tau }^2-4\Delta }}{2}.$ The quantity under the root, $D={\tau }^2-4\Delta$, is the discriminant: $D<0$ forces the square root to be imaginary — that's precisely when spirals/centres appear.

Why the topic needs it: the sign of the real part of each $\lambda$ decides stability; the imaginary part decides rotation. Eigenvalues are the final verdict.

10. Reading the flow as a whole: phase portrait

Explore these fully in Phase portraits. When you want to prove stability without solving anything, you reach for an energy-like function — Lyapunov stability — and the physical archetype of a stable spiral is the Damped harmonic oscillator.

Prerequisite map

Equipment checklist

Test yourself — cover the right side and answer aloud.