The Fixed Points Decide the Long-Run Behavior
Until now we drew how x changes over time, plotting the graph on a time axis. But there's a more powerful picture for understanding oscillation and flow: erase time, and draw the state itself instead. The state of a swinging weight isn't captured by position x alone — at the same position it matters whether it's racing through or about to stop. So bundle position x and velocity x' into a pair and view (x, x') as a single point on a plane. This is the phase plane. As time runs, this point moves, tracing a curve on the plane, and the system's differential equation sets, at every point, an arrow for where that point goes next. Here something magical happens. Know only the kinds of those special points where the arrow becomes zero — the fixed points — and you can tell where the system ends up: whether it settles, blows up, or circles forever. And what decides those kinds is the eigenvalue we saw in linear algebra. In this lesson the whole C strand ties together through eigenvalues.
First, understand the stage called the phase plane. The horizontal axis is position x, the vertical axis is velocity x'. At any instant the system's state is fully fixed by one point on this plane, the state point. Press play. The state point of a system in damped oscillation moves and traces a curve. When the weight is pushed to one side (large x, zero velocity), the point sits at the far right of the horizontal axis; when the weight races back through the center (x zero, large velocity), the point rises up the vertical axis. Position and velocity hand off back and forth like this, and the point circles round and round the origin. Because there's damping, energy leaks away, so it traces a spiral winding ever inward, and is finally drawn into the origin — the state of rest. The curve that oscillated on the time-axis graph becomes here one clean spiral winding inward. In exchange for erasing time, the whole shape of the motion is visible at a glance.
Who decides where the state point goes? The system's equation. Remember the direction field from C1: there, each point was given a single slope. On the phase plane we go one step further and give each point a whole arrow — which way and how fast the state moves from that point. The coupled equations x'=y, y'=−x−0.4y blanket the entire plane with arrows. This is the vector field. Drag the start point anywhere. The state point follows the arrow where it stands, then follows the arrow at the next point, and so on. Faithfully stepping along the arrows traces out a trajectory. Just as a leaf dropped on a river drifts along the current, the state point flows along the vector field. Wherever you start, the flow that arrives at that point sets where it goes next. The one-dimensional flow of C1 has grown here into a two-dimensional flow.
In the whole vector field, exactly one kind of point is special: the point where the arrow's length is zero, the fixed point. There the state moves nowhere, so once placed it stays forever — an equilibrium. Yet two fixed points can have completely different characters. Take the pendulum. The spot where it hangs gently down is a fixed point; leave it there and it stays still, and a small touch only sets it circling nearby without leaving the area. This is a center, a stable fixed point. But the spot with the pendulum balanced upside-down is also a fixed point. Balance it perfectly and in theory it stays; but nudge it by a hair's width and it topples and runs far away. This is a saddle, an unstable fixed point. Use the buttons to point out the two fixed points. The fate of the system depends on which fixed point's domain the start lies in, and on what kind of fixed point it is.
The shape of the flow near a fixed point sorts into just a few kinds. Step through them with the tabs. A stable node sucks every arrow straight into the fixed point; nudge it and it returns to place without complaint. A stable spiral winds inward as it circles — returning, but oscillating on the way, exactly the damped oscillation from C2. A saddle pulls in along one direction but flings out along another; almost every start eventually runs far away, an unstable point. A center neither pulls in nor pushes out, circling the same orbit forever — undamped oscillation. An unstable spiral is the reverse, circling as it flings ever outward. These few are the entire repertoire a two-dimensional linear system can show. Even a complicated system behaves like one of this catalog near a fixed point. So what decides which system is which type?
At last everything gathers in one place. What decides a fixed point's type is the eigenvalues of the matrix that linearizes the system there — the Jacobian. In linear algebra, eigenvalues were how much a transformation stretches along which direction. Here they decide, near the fixed point, along which direction and how fast the state point is pushed away or pulled in. The eigenvalues are the roots of the quadratic λ²−τλ+Δ=0, where τ is the matrix's trace and Δ its determinant. Drag the point on the (τ, Δ) plane. The whole phase portrait changes. If both eigenvalues are negative reals (Δ>0, τ<0, positive discriminant), every direction shrinks — a stable node. If the eigenvalues are complex (negative discriminant), the rotation of A2 and C2 enters and it becomes a spiral; negative real part gives a stable spiral, positive an unstable one. Pure imaginary eigenvalues with zero real part give a center, eternal oscillation. If the eigenvalues have opposite signs (Δ<0), one direction pulls in and one flings out — a saddle. The characteristic roots of C2 were in fact the one-dimensional version of these Jacobian eigenvalues, and the complex exponential of A2 was the identity of the spiral. The eigenvalues of linear algebra, the rotation of complex numbers, and the stability of differential equations all meet in this one picture.