Phase Portraits of Linear Systems

Visualize direction fields and trajectories for \(\dot{\mathbf{x}}=A\mathbf{x}\). Click the canvas to integrate trajectories from any initial condition.

The Linear System

A planar linear autonomous system has the form \(\dot{\mathbf{x}}=A\mathbf{x}\), where \(A\) is a constant \(2\times2\) matrix and \(\mathbf{x}=(x,y)^T\). The origin is always an equilibrium. The qualitative behaviour of all solutions — whether they spiral in, radiate outward, oscillate, or approach along straight lines — is determined entirely by the eigenvalues of \(A\).

\[\dot{x}=ax+by,\qquad\dot{y}=cx+dy,\qquad A=\begin{pmatrix}a&b\\c&d\end{pmatrix}\]

Classification by Eigenvalues

Let \(\tau=\operatorname{tr}A=a+d\), \(\delta=\det A=ad-bc\), and \(\Delta=\tau^2-4\delta\).

Saddle: \(\delta<0\). Two real eigenvalues of opposite sign. Stable and unstable manifolds form straight-line orbits.

Stable Node: \(\delta>0,\;\Delta\ge0,\;\tau<0\). Two negative eigenvalues; all orbits approach the origin.

Unstable Node: \(\delta>0,\;\Delta\ge0,\;\tau>0\). Two positive eigenvalues; all orbits leave the origin.

Center: \(\tau=0,\;\delta>0\). Purely imaginary eigenvalues; closed elliptical orbits. Neutrally stable.

Stable Spiral: \(\Delta<0,\;\tau<0\). Complex eigenvalues with negative real part; orbits wind inward.

Unstable Spiral: \(\Delta<0,\;\tau>0\). Complex eigenvalues with positive real part; orbits wind outward.

Defective Node: Repeated eigenvalue, single eigenvector. Parabolic-like orbits near the origin.

Star Node: \(A=\lambda I\). Repeated eigenvalue, every direction is an eigenvector; straight-line orbits everywhere.

Numerical Method

Trajectories are integrated forward and backward from the click point with a choice of one-step methods:

RK4 — classical explicit Runge–Kutta, 4th order. Good general default for non-stiff systems.
Tsit5 — Tsitouras 5(4) explicit Runge–Kutta, 6 stages, 5th-order accuracy. Smoother curves than RK4 at the same step size; same stability footprint (no benefit on stiff problems).
SDIRK2 — singly-diagonally-implicit Runge–Kutta, 2 stages, 2nd order with \(\gamma = 1-\tfrac{1}{\sqrt{2}}\). The method is L-stable, so it remains accurate on stiff linear systems where explicit methods would either blow up or require a tiny step.

In Auto mode the page picks SDIRK2 whenever the linear system is "obviously stiff" — both eigenvalues real, the same sign, with a magnitude ratio \(|\lambda_\text{fast}/\lambda_\text{slow}|\ge 50\) — and falls back to RK4 otherwise. Centers are integrated separately with a fixed-step explicit scheme to keep closed orbits closed (an implicit method would introduce numerical damping).

How to Use

Enter the matrix entries \(a,b,c,d\) in the matrix cells. The direction field and classification update automatically after a short pause.
Select an example from the dropdown to load a preset matrix with strategic trajectories illustrating the classification.
Click anywhere on the phase plane to integrate a trajectory through that initial condition (forward and backward). Trajectories are colour-coded.
Use Generate to redraw the field and clear trajectories; Clear trajectories removes curves while keeping the field.
Adjust arrow count, scale, opacity, and colour. Toggle grid and initial point markers using the checkboxes.
Open the Time Series panel to plot \(x(t)\) and/or \(y(t)\) for all current trajectories.
Download a PNG of the current portrait using Download PNG.

\(\dot{x}=ax+by,\quad\dot{y}=cx+dy\)

Matrix \(A\)

Classification

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Solver Active: RK4

Arrows

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Field colour

Eigenvec colour

Field

Grid

Init point

Eigenvectors

Load example

Phase Plane — click the canvas to draw a trajectory

Click anywhere on the plane to add a trajectory