Vector field / phase portrait

Enter x' = f(x, y), y' = g(x, y), or a first-order equation dy/dx = g(x, y) / M dx + N dy = 0. Hover to preview the solution curve through a point and click to keep it (blue forward, orange backward); wheel to zoom, drag to pan, double-click to reset.

Presets:

The origin is a center (linearization can only say 'center or weak spiral'); orbits are circles.

Syntax: write multiplication explicitly (x*y, not xy), powers with ^, functions sin cos exp log sqrt abs …, constants pi and e.

Displayed range (equal scale): x ∈ [-4.154, 4.154], y ∈ [-3, 3] · Hover to preview a solution · click to keep it · wheel to zoom · drag to pan · double-click to reset

The results below are computed for the visible range x ∈ [-4.154, 4.154], y ∈ [-3, 3]; they are recomputed after zooming or panning, because conclusions depend on the range examined.

Equilibria

  1. (0, 0) center or weak spiral (linearization cannot tell); λ = 0 + 1i, 0 - 1i; tr = 0, det = 1. The linearization gives a purely imaginary pair of eigenvalues (real part zero to numerical precision). Linearization alone cannot distinguish a true center from an extremely slow spiral: their phase portraits are entirely different, and deciding requires a conserved quantity (such as an energy or Hamiltonian) or a higher-order nonlinear analysis.