Phase portraits of linear systems
The eigenvalues of A determine the overall picture of all solutions, called the phase portrait. A handful of types covers every two-dimensional linear system.
Equilibrium at the origin
For \(\mathbf{x}' = A\mathbf{x}\), the origin is always an equilibrium. The eigenvalues of \(A\) decide what nearby trajectories do: approach, leave, circle, or a mixture. Classifying the origin is the first step in understanding any linear system.
Nodes
If both eigenvalues are real and negative, every trajectory approaches the origin: a stable node (sink). If both are positive, trajectories leave: an unstable node (source). Trajectories become tangent to the eigenvector of the eigenvalue closest to zero as they approach or leave.
Saddles
If the eigenvalues are real with opposite signs, the origin is a saddle point. Trajectories approach along one eigenvector and leave along the other; almost every solution eventually moves away. Saddles are always unstable, like a ball balanced on a mountain pass.
Spirals and centres
Complex eigenvalues \(\alpha \pm \beta i\) give spirals: stable if \(\alpha < 0\) (spiralling in), unstable if \(\alpha > 0\) (spiralling out). If \(\alpha = 0\), trajectories are closed ellipses around a centre, neutrally stable. The direction of rotation can be read from \(A\) at a sample point.
Trace and determinant
For a 2 by 2 matrix, the trace \(\tau = a + d\) and determinant \(\Delta = ad - bc\) classify everything. If \(\Delta < 0\): saddle. If \(\Delta > 0\): node or spiral, stable if \(\tau < 0\), unstable if \(\tau > 0\); spirals when \(\tau^2 < 4\Delta\). If \(\tau = 0\) with \(\Delta > 0\): centre.
\lambda^2 - \tau\lambda + \Delta = 0Worked example
\(A = \begin{pmatrix} -1 & 2 \\ -2 & -1 \end{pmatrix}\): \(\tau = -2\), \(\Delta = 5\), \(\tau^2 - 4\Delta = -16 < 0\). The origin is a stable spiral: eigenvalues \(-1 \pm 2i\). Solutions rotate while decaying, like a damped pendulum seen in the phase plane.
Stability summary
The origin is asymptotically stable exactly when all eigenvalues have negative real parts. Engineers design control systems so that the closed-loop matrix has eigenvalues safely in the left half of the complex plane, ensuring disturbances die out.
Common mistakes
Calling a saddle stable because some trajectories approach it. Confusing the trace and determinant tests. Forgetting that a centre is stable but not asymptotically stable. Misreading rotation direction.
Key takeaways
Real eigenvalues of one sign give nodes; opposite signs give saddles; complex eigenvalues give spirals, or centres when purely imaginary. Trace and determinant classify 2 by 2 systems quickly. Stability requires all eigenvalues to have negative real parts.
Flashcard
Both eigenvalues real and negative
Check your understanding
A 2 by 2 system has trace 0 and determinant 4. The origin is a: