phase plane and fixed points
Until now we have graphed how x changes over time on a time axis. There is a more powerful picture for understanding vibration and flow, though: erase time and draw the state itself. The state of a swinging weight is more than the position x alone. At the same position, it matters whether the weight is rushing past or about to stop. So we pair the position x with the velocity x' and treat (x, x') as one point on a plane; that plane is the phase plane. As time passes, the point moves across the plane tracing a curve, and the system's differential equation sets, with an arrow at every point, where the point goes next. Here something magical happens. Know only the kinds of the special points where the arrow shrinks to 0, the fixed points, and you know where the system ends up: whether it settles, blows up, or keeps circling. And what decides those kinds is the eigenvalue from linear algebra. In this lesson the whole differential equations thread is tied together by eigenvalues.
First, understand the stage called the phase plane. The horizontal axis is the position x, and the vertical axis is the velocity x'. At any moment, the state of the system is completely fixed by one point on this plane, the state point. Press "Play" and the state point of a damped oscillator moves in gold, leaving a trail behind it. When the weight is pushed to one side (x large, velocity 0), the point sits at the right of the horizontal axis; when the weight rushes back through the middle (x 0, velocity large), the point reaches the lower part of the vertical axis, because it is moving left and its velocity is negative. As position and velocity trade back and forth, the point circles the origin clockwise. With damping, energy leaks away, so it draws a spiral winding inward and is finally drawn into the origin, the state of rest. Once it reaches the origin, the point returns to its starting place and draws again. The curve that sloshed back and forth on the time graph becomes, here, one clean inward spiral. Erasing time lets you see the whole shape of the motion at a glance.
Where the state point goes is decided by the system's equations. Recall the direction field from before: there, each point was given one slope. The phase plane goes one step further and gives each point a whole arrow: in which direction, and how fast, the state moves from there. The coupled equations x'=y, y'=−x−0.4y cover the entire plane with arrows. That is a vector field. Drag the starting point anywhere and the state point follows the arrow at that spot, then the arrow at the next spot. Step faithfully from arrow to arrow and a single trajectory is drawn: the bright gold part is the path ahead, the faint gold part the path already traveled. Just as a leaf dropped in a river floats along with the current, the state point flows along the vector field. The 1D flow we saw earlier has grown into a 2D flow here.
In the whole vector field, one kind of point is special: the point where the arrow has length 0, the fixed point. There the state does not move anywhere, so once placed it stays forever: an equilibrium. Yet fixed points can have completely different characters. The figure is the phase portrait of a pendulum, and the spot where it hangs straight down is a fixed point. Leave it there and it stays still; nudge it and it only circles around, never leaving the neighborhood. A point like this is called a center, and it counts as stable. The spot where the pendulum stands upside down is also a fixed point. Balanced perfectly, it would in theory stay put, but push it by a hair and it topples and runs far away. A point like this is called a saddle, an unstable fixed point. Light up the two fixed points in turn with the two buttons, and gold trajectories passing around each one are drawn. A system's fate depends on which fixed point's sphere of influence the starting point lies in, and what kind that fixed point is.
The shapes of flow near a fixed point fall neatly into a few types. Press the tabs one at a time to browse them like a field guide. At a stable node, every arrow is sucked straight into the fixed point, so a nudge is followed by an uncomplaining return home. A stable spiral goes in while circling around: it returns, but sloshing as it comes, and that is exactly the damped oscillation of the earlier lessons. A saddle draws things in along one direction but flings them out along another, so almost every starting point ends up running far away: an unstable point. A center neither draws in nor pushes out; it circles the same orbit forever, an undamped oscillation. The spiral on the "Unstable" tab circles the other way and is flung farther and farther outward. These few types are all the behavior a 2D linear system can show. Even a complicated system behaves like one of these entries near a fixed point. So what decides which type a given system is?
At last everything comes together. What decides the type of a fixed point is the eigenvalues of the matrix obtained by linearizing the system there, the Jacobian. In linear algebra, eigenvalues told how much a transformation stretches in which direction. Here they decide in which direction, and how quickly, the state point is pushed away from or drawn into the fixed point. The eigenvalues are the roots of the quadratic λ²−τλ+Δ=0, where τ is the trace of the matrix and Δ the determinant. Drag the point in the (τ, Δ) plane above and the phase portrait below changes completely, with the eigenvalues and the type name written alongside. The dashed parabola is the boundary where the eigenvalues turn from real to complex. 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 from the complex numbers thread enters and you get a spiral: stable if the real part is negative, unstable if it is positive. Purely imaginary eigenvalues, with real part 0, give a center, an oscillation that lasts forever. Eigenvalues of opposite sign (Δ<0) give a saddle, drawing in one way and flinging out the other. The characteristic roots of the earlier lesson were in fact exactly these Jacobian eigenvalues (one 2nd-order equation is such a 2D system), and the complex exponential was the true identity of the spiral. Eigenvalues from linear algebra, rotation from complex numbers, and stability from differential equations all meet in this one picture.