Plot phase portraits for 2D autonomous systems. Classify equilibria as nodes, spirals, saddles, and centers with trajectory animation.
Type the two right-hand sides of a 2D autonomous system - dx/dt = f(x, y), dy/dt = g(x, y) - and this tool draws the phase portrait: a vector field of arrows, the equilibrium points where both f and g vanish, and animated trajectories from starting points you choose. Each equilibrium is classified from the eigenvalues of the Jacobian matrix of partial derivatives: node, spiral, saddle, or center, stable or unstable. The usual surprise is that the label is local and linear - it describes behavior only arbitrarily close to the equilibrium - and a 'center' label does not guarantee closed orbits once nonlinear terms are present.
A concrete input and expected output from the current implementation.
Input
dx/dt = y dy/dt = -x
Expected output
One equilibrium at (0, 0), classified as a center (Jacobian eigenvalues +i and -i, purely imaginary). Trajectories are the closed circles x^2 + y^2 = r^2, traversed counterclockwise, so nearby orbits neither approach nor escape the origin.
With x' = y and y' = -x, the squared distance from the origin has zero derivative, so every trajectory stays on a circle. Purely imaginary eigenvalues are exactly the signature the tool uses for the center label.
The tool locates the points where both right-hand sides vanish simultaneously, then evaluates the four partial derivatives of f and g at each one - the Jacobian matrix. Its two eigenvalues decide the label: both real and negative, a stable node; opposite signs, a saddle; complex with negative real part, a stable spiral; purely imaginary, a center.
At grid points it draws an arrow in the direction (f, g), the velocity of a solution passing through that point, so the field shows the flow at a glance. Picking a starting point numerically integrates the equations and traces the orbit forward and backward in time; the animation moves a marker along the trace, and arrow length conveys how fast the flow runs.
Some systems genuinely have no rest points. An equilibrium requires both right-hand sides to vanish at the same point, and for x' = x, y' = 1 the second equation is never zero, so no such point exists. The flow never settles, and the arrows are the whole picture rather than a mistake in the tool.
Probably not. Stability describes behavior arbitrarily close to the equilibrium; a trajectory started farther away can wander through other regions first, or leave the plotting window before its slow exponential approach becomes visible. Zoom in near the equilibrium and restart close to it, and the predicted convergence should appear.
The example and behavioral notes were checked against the browser implementation. Standards and primary references below define the relevant format, formula, or platform behavior.
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