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Linear flow
PHASE SPACE EXPLORER

Stable spiral

Asymptotically stable
↗ Phase portrait 2 dimensions

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x₁ · x₂
t = −20.00
Initial values are at t = 0

Trace–determinant plane

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Δ = τ²/4: repeated eigenvaluesτ = 0, Δ > 0: centersΔ = 0: zero eigenvalue

Fixed samples stay in place; the tracked pair follows the stable saddle direction continuously. Each point generates one representative matrix; matrices with the same trace and determinant can differ on the boundary curves.

Solution over time

x₁x₂

System insight

EIGENVALUES

Trace Determinant

Explore your system

Every arrow shows the instantaneous direction of motion for x′ = Ax. Colored paths follow solutions from their initial conditions.

Eigenvalues describe the behavior near the origin. Negative real parts pull solutions inward; positive real parts cause growth; imaginary parts produce rotation.

This illustration models homogeneous systems with constant real coefficients. Time is in arbitrary units. Solutions are computed with a matrix exponential; paths stop beyond the plotting limit.