A linear system's whole personality lives in one 2×2 matrix. Change its entries and watch trajectories spiral, decay, or fly apart.
State-Space Phase PortraitLive
the shape of a linear system
Controls
Presets
A linear state-space system ẋ = Ax has a character set by the eigenvalues of A. Negative real parts spiral or decay to the origin (stable); positive parts fly outward; imaginary parts add rotation. The phase portrait reveals it all at a glance. Educational tool.
Reading this result: Complex eigenvalues with negative real part: trajectories spiral inward, oscillating while decaying to the origin — a stable, ringing system.
Runs locally in your browser — free forever. Scale to the cloud when reality gets heavy.
★ Sign in to save this setupSave your tuned setup, or drop this simulation into your own site, docs, or course page.
How it works
For ẋ = Ax, the eigenvalues of A classify the behavior: negative real parts give stable nodes or spirals, positive parts give instability, imaginary parts add rotation, and a negative determinant makes a saddle. The phase portrait visualizes every trajectory at once. Educational tool.
✦
Ask the AI about this model
The math, the assumptions, real-world uses, or a code translation — explained for this exact simulation.
Is this phase portrait simulator tool really free?▾
Yes. State-Space Phase Portrait runs entirely in your browser using your device's own compute, so local use is free forever. You only pay Compute Tokens if you scale a job to the cloud.
Do I need to install anything?▾
No. Everything runs client-side in a modern browser — no downloads, no license, no account required to start.
Can I save or share my simulation?▾
Create a free account to save projects, and use a shareable embed or minted DOI to publish a live, interactive version anywhere.
How accurate are the results?▾
The solver uses established numerical methods, but results are for research and educational purposes and should be validated against experiment or professional review before you rely on them.