Configure the system
The page keeps a single-column reading flow, while the calculator fields become three columns on large screens.
Phase portrait canvas
The canvas draws the vector field, automatic trajectories, a highlighted trajectory, axes, and optional nullclines.
Tip: Increase integration steps for smoother curves, then reduce arrow density if the field becomes too crowded.
Highlighted seed coordinates
This table lists sampled points from the highlighted trajectory and supplies the downloadable CSV rows.
| Step | t | x(t) | y(t) | dx/dt | dy/dt |
|---|
Reference systems for quick testing
| System type | a | b | c | d | Trace | Determinant | Expected portrait |
|---|---|---|---|---|---|---|---|
| Stable spiral | 0 | 1 | -1 | -0.4 | -0.4 | 1 | Inward spirals to the origin |
| Saddle point | 2 | 0 | 0 | -1 | 1 | -2 | One attracting and one repelling direction |
| Stable node | -2 | 1 | 0 | -1 | -3 | 2 | Direct decay without oscillation |
| Center | 0 | 1 | -1 | 0 | 0 | 1 | Closed loops around the origin |
Mathematics behind the calculator
Linear autonomous system
x' = ax + by
y' = cx + dy
The coefficient matrix is A = [[a, b], [c, d]]. The origin is the equilibrium when the system has no constant forcing terms.
Trace and determinant test
trace = a + d
det = ad - bc
Δ = trace2 - 4det. These three values classify most planar linear equilibria.
Eigenvalues
λ = (trace ± √Δ) / 2
Real eigenvalues produce nodes or saddles. Complex eigenvalues produce spirals or centers, depending on the real part.
Trajectory integration
The portrait uses a fourth-order Runge–Kutta method. It approximates trajectories from seeded starting points by stepping forward and backward in time.
Workflow
- Choose a preset or enter custom values for the matrix coefficients
a,b,c, andd. - Set the visible x- and y-ranges to control how much of the phase plane is displayed.
- Adjust step size, integration steps, arrow density, and seed count for the desired visual detail.
- Enter a highlighted seed point to inspect one specific solution curve more closely.
- Click Plot phase portrait to place the result summary directly below the header and above the form.
- Review the classification, eigenvalues, nullclines, plot, and trajectory sample table.
- Use the export buttons to save the sampled trajectory as CSV or capture the results area as a PDF.
Frequently asked questions
1. What does this calculator plot?
It plots the phase portrait of a two-dimensional linear autonomous system. You see arrows for the vector field, many seeded trajectories, and one highlighted solution curve for closer inspection.
2. Why are trace and determinant important?
Trace controls total expansion or contraction. Determinant shows whether directions agree or split. Together with the discriminant, they classify nodes, saddles, spirals, repeated cases, and centers.
3. What is the highlighted seed point?
It is a chosen initial condition (x0, y0). The calculator tracks this solution and lists sampled coordinates so you can inspect how one path evolves over time.
4. What do nullclines tell me?
A nullcline is where one derivative becomes zero. On an x-nullcline, horizontal motion pauses. On a y-nullcline, vertical motion pauses. Their intersection helps identify equilibria and directional changes.
5. Why might the curves look jagged?
The numerical step size may be too large, or the field may change quickly. Reduce the step size or increase the integration steps to improve smoothness and accuracy.
6. Can this handle nonlinear systems?
This version is built for linear systems only. It is still useful for studying local behavior near equilibria, because nonlinear systems are often approximated by linearization.
7. What does determinant zero mean?
A zero determinant means at least one eigenvalue is zero. The equilibrium becomes degenerate, and the portrait can show non-isolated equilibria, slow drift, or semistable behavior.
8. What do the export buttons save?
The CSV contains sampled coordinates and derivatives from the highlighted trajectory. The PDF captures the summary area, including metrics and the rendered phase portrait for reporting or teaching notes.