State-space · controls

A 2nd-order ODE can always be rewritten as two coupled 1st-order ones. The new variables — position and velocity, say — are the system's state. The matrix $A$ in $d o t x = A x$ is the rule that says "given where you are, here's the direction the state must move."

Reading the picture below. Each tiny arrow shows the instantaneous direction the state would move from that point. Long arrows = fast motion, short = slow. Arrows in accent are the fastest in the field. The four sliders are the entries of the $A$ matrix; changing them re-flows the entire arrow field. Tap anywhere to drop an initial condition — the trajectory is integrated forward in time and follows the arrows. Drop several to see how the whole plane behaves.

The four presets give you the canonical phase-portrait shapes you'll meet in any dynamics textbook: pure oscillation, damped spiral, saddle, and unstable spiral. The eigenvalues in the metrics row classify which shape you're in.

Tap to plant a trajectory

A₁₁ 0.00 A₁₂ 1.00 A₂₁ −1.00 A₂₂ −0.40

eigenvalues —

tr(A) —

det(A) —

type —

what to try

Press the "damped spring" preset. Then tap a few spots far from the origin. Every trajectory spirals inward and eventually dies at $(0, 0)$ . The eigenvalues in the metric row are a conjugate pair with negative real part — the classic stable-spiral signature.
Slide A₂₂ from −0.4 toward zero. Watch the spiraling slow down. At A₂₂ = 0, trajectories are perfect closed circles — no decay, no growth. The eigenvalues just crossed the imaginary axis: real part = 0. Keep going past zero, and trajectories now spiral outward — same shape, opposite time direction. The system became unstable.
Try the "saddle" preset. The eigenvalues are now one positive, one negative. There are two special directions (eigenvectors): trajectories along one come into the origin, along the other they run away. Generic trajectories follow a hyperbola-like path. This is the textbook unstable-equilibrium shape — like a ball on a saddle.
Watch the metrics as you slide. Notice that $d e t (A) = l amb d a_{1} l amb d a_{2}$ and $e x t t r (A) = l amb d a_{1} + l amb d a_{2}$ — the same numbers that decide the transfer-function pole locations.

show the math

For a 2×2 matrix $A$ , the eigenvalues are roots of

l amb d a^{2} - e x t t r (A) l amb d a + d e t (A) = 0

Real part of the eigenvalues = how fast trajectories grow or decay. Imaginary part = how fast they spin. So the same numbers that say "spiral inward" in the phase plane say "underdamped" in the transfer-function picture (Lesson 04). The two languages are the same; we just point at different parts of the elephant.