Phase Portraits & Direction Fields

Reading the arrows

At every point, the system tells you a velocity vector (ẋ, ẏ). Draw a little arrow for each and you get the direction field. A solution curve is simply a path that's tangent to those arrows everywhere — so you can trace the qualitative behaviour of any initial condition just by following the flow, without ever solving a formula. Click around the plane and watch where different starts end up.

Eigenvalues are destiny

For a linear system, the long-term fate of every trajectory is written in the matrix's eigenvalues λ. Real and opposite signs → a saddle: trajectories rush in along one direction and shoot out along another. Both negative → a stable node, everything decays to the origin. Complex with negative real part → a stable spiral, the hallmark of a damped oscillator (your spring–mass–damper lives here). Purely imaginary → a center, endless closed loops — an undamped oscillator conserving energy forever. Nudge the matrix entries and watch the classification flip in real time.

When it goes nonlinear

Real systems aren't linear, and that's where the richness is. The pendulum has a center (gentle swinging) and saddles (balanced upright) with separatrices dividing swinging from spinning. The Van der Pol oscillator pulls every trajectory onto a single limit cycle — a self-sustained rhythm, the math behind heartbeats and electronic oscillators. Predator–prey orbits in closed loops of boom and bust. Near any fixed point, though, you can linearize — and the eigenvalue rules above come right back.