General & First-year
ODE Phase Lab
Differential-equations courses hinge on one picture students rarely get to play with: the phase plane. Drag the four entries of the matrix A (or pick a preset) and the slope field, a RK4-integrated trajectory and the eigenvalues all update — click anywhere to launch a new orbit. The eigenvalues decide everything: real and same-sign give a node, opposite-sign a saddle, a complex pair a spiral, and purely imaginary a centre of closed orbits. A negative real part pulls trajectories into the origin (stable); positive pushes them away (unstable). It's the link between the algebra of eigenvalues and the geometry of solutions, made tangible.
-0.80
Trace
1.60
Determinant
λ = -0.40 ± 1.20i
Eigenvalues
Each arrow shows which way a point drifts under ẋ = A·x; the green curve is one trajectory (click anywhere to launch another). The eigenvalues of A decide the whole picture: real and same-sign → a node, opposite-sign → a saddle, complex → a spiral, and purely imaginary → closed orbits around a centre. A negative real part pulls everything into the origin; positive pushes it away.
How to use this simulation
Differential-equations courses hinge on one picture students rarely get to play with: the phase plane. Drag the four entries of the matrix A (or pick a preset) and the slope field, a RK4-integrated trajectory and the eigenvalues all update — click anywhere to launch a new orbit. The eigenvalues decide everything: real and same-sign give a node, opposite-sign a saddle, a complex pair a spiral, and purely imaginary a centre of closed orbits. A negative real part pulls trajectories into the origin (stable); positive pushes them away (unstable). It's the link between the algebra of eigenvalues and the geometry of solutions, made tangible.
Everything runs in your browser — no sign-up, no download. Change a value and the result updates instantly, so you can build a feel for how each input shapes the outcome. It pairs with Crameleon's practice exams and step sheets when you want to go from intuition to working the problems.