Base initial condition

x₀ = x(0)

Comparison trajectories 1 / 8

Selected initial condition y₀

Click the portrait to add a comparison. Each one has its own variational vector from the same base trajectory. Select a comparison for the separation chart; enable all arrows to follow them together.

η(0) = y₀ − x₀

Time range

Start tEnd t

Within −50…100. Initial conditions always refer to t = 0.

Display

Vector field Full trajectories All variational arrows Actual displacement segment Linearized prediction path

HOW DOES A SMALL CHANGE EVOLVE?

Variational equations

2D system

Base x′ = f(x)Variation η′ = Df(x(t))ηPrediction y(t) ≈ x(t) + η(t)

Phase portrait

Base x(t)Comparison y(t)Variational arrow η(t)

Click to add a comparison · scroll to zoom. The orange arrow starts at the moving base point.

t = 0.00

Separation & approximation error

Selected comparison

Actual |y − x|Linearized |η|Error |y − x − η|

Along the base trajectory

Df(x(t)) · current Jacobian

Actual separation: —
Variational magnitude: —
Approximation error: —

The arrow keeps its true length. Its tip predicts the comparison point; the gap between them is the linearization error.