Phase Portraits

Phase Portraits

A phase portrait is a geometric picture of the autonomous LTI system $\dot{x}=Ax$ in ${\mathbb{R}}^2$: at each point $x$, draw the velocity vector $\dot{x}=Ax$, and sketch representative trajectories following that vector field. For a 2D system, the portrait’s qualitative shape is determined entirely by the eigenvalues of $A$, with eigenvectors giving the invariant directions along which the dynamics reduce to scalar exponentials.

The six canonical cases, indexed by the eigenvalue pair $({\lambda }_1,{\lambda }_2)$:

Eigenvalues	Name	Stability
Real, both negative	Stable node	Asymptotically stable
Real, both positive	Unstable node	Unstable
Real, opposite sign	Saddle	Unstable (invariant stable direction exists)
Complex, $\mathrm{Re}(\lambda )<0$	Stable focus (spiral)	Asymptotically stable
Complex, $\mathrm{Re}(\lambda )>0$	Unstable focus (spiral)	Unstable
Pure imaginary	Center	Marginally stable (closed orbits)

For nodes and saddles, each eigenvector $v_i$ defines an invariant line: trajectories starting on it stay on it, evolving as $e^{{\lambda }_{i}t}{v}_i$. For foci, there are no real eigenvectors and all trajectories spiral. For centers, trajectories are closed ellipses circulating about the origin.

Phase portraits are most useful in 2D because the full state vector can be drawn on a page. The same eigenvalue-based classification extends to higher dimensions — each 2D invariant subspace of $A$ behaves like one of the cases above — but is harder to visualize directly.

Degenerate Cases defective $A$ (only one eigenvector) produce improper nodes whose trajectories bend toward the single invariant line instead of splitting into two. Zero eigenvalues produce lines of equilibria rather than isolated fixed points. These degenerate cases are structurally unstable — arbitrarily small perturbations of $A$ return the portrait to one of the six canonical cases above.

Repeated eigenvalues with a

Intuition

The vector field $\dot{x}=Ax$ says: at every point $x$, the instantaneous motion is $Ax$. Trajectories follow the vector field like water currents following a river map. The long-time shape of the flow depends on which directions are attracting (states flow toward them) and which are repelling (states flow away).

Eigenvectors are the key geometric structure. Along the eigenvector $v_i$, the system behaves like a scalar $\dot{z}={\lambda }_{i}z$ — just exponential growth or decay. Real negative $\lambda$ gives pure attraction along that line; real positive $\lambda$ gives pure repulsion. When both eigenvalues are real and the same sign, both invariant lines attract (or both repel), and the resulting flow is a node. When the eigenvalues have opposite signs, one line attracts and the other repels, producing a saddle — the most important topological feature in dynamical systems because it shows how trajectories separate.

Complex eigenvalues mean there are no real eigenvector lines at all. Instead, the eigenvalue’s real part $\sigma$ controls growth or decay, and the imaginary part $\omega$ controls rotation. Trajectories spiral — outward if $\sigma >0$, inward if $\sigma <0$, circularly if $\sigma =0$. This is the multidimensional generalization of the fact that $e^{(\sigma +i\omega )t}=e^{\sigma t}(\cos \omega t+i\sin \omega t)$.

Properties

Classification via Eigenvalues

(Real, same sign) Both eigenvalues real with the same sign produce a node.

The origin is an isolated fixed point; all trajectories approach it (stable node, both $\lambda <0$) or flee it (unstable node, both $\lambda >0$). The two eigenvector directions are invariant. Trajectories in general position are tangent at the origin to the eigenvector with the smaller $|\lambda |$, because that mode decays (or grows) more slowly and dominates near the origin.

(Real, opposite sign) Eigenvalues of opposite sign produce a saddle.

The stable eigenvector defines a one-dimensional stable manifold — trajectories starting on it approach the origin. The unstable eigenvector defines the unstable manifold — trajectories on it flee. Every other trajectory asymptotes to the unstable manifold in forward time and to the stable manifold in backward time. Saddles are the topological signature of separatrices in nonlinear systems.

(Complex conjugate) Complex eigenvalues $\sigma \pm i\omega$ produce foci (spirals) or centers.

$\sigma <0$ gives stable foci (trajectories spiral inward); $\sigma >0$ gives unstable foci (outward spirals); $\sigma =0$ gives centers (closed elliptical orbits at frequency $\omega$). The rotation direction (clockwise vs counterclockwise) depends on the off-diagonal entries of $A$ — specifically the sign of $A_{21}$ when $A_{11}=A_{22}$.

(Trace-determinant plane) $\tau =\mathrm{tr}(A)$, $\Delta =\det (A)$ encode all six cases.

For a $2\times 2$ matrix, the characteristic polynomial is ${\lambda }^2-\tau \lambda +\Delta =0$, so $\lambda =\frac{1}{2}(\tau \pm \sqrt{{\tau }^2-4\Delta })$. The discriminant ${\tau }^2-4\Delta$ distinguishes real eigenvalues ($>0$) from complex ($<0$); the signs of $\tau$ and $\Delta$ distinguish stable from unstable. A single $(\tau ,\Delta )$ diagram partitions the plane into the six regions.

Invariant Structure

(Eigenvector lines) For real distinct eigenvalues, each eigenvector spans an invariant line.

Trajectories starting on $\mathrm{span}(v_i)$ evolve as $e^{{\lambda }_{i}t}{v}_i$ and never leave it. This is because $A{v}_i={\lambda }_i{v}_i$, so the velocity at any point $c{v}_i$ is $A(c{v}_i)={\lambda }_{i}c{v}_i$, pointing along the same line. Invariant lines are the backbones around which the full portrait is organized.

(Equilibrium point) The origin is the unique equilibrium when $A$ is invertible.

$\dot{x}=0$ iff $Ax=0$, and for $\det (A)\ne 0$ the only solution is $x=0$. If $A$ is singular, the null space $\mathrm{null}(A)$ is a line (or plane) of equilibria, and the phase portrait degenerates.

(Modal decomposition) In eigenbasis coordinates, the system decouples into scalar ODEs.

Setting $z=V^{-1}x$ gives ${\dot{z}}_i={\lambda }_i{z}_i$. In the $(z_1,z_2)$ picture, eigenvector lines become the coordinate axes and the portrait is symmetric across them. Transforming back to $x$ coordinates stretches and rotates the picture but preserves its topology.

Higher Dimensions

(Invariant subspace decomposition) In ${\mathbb{R}}^n$, the state space splits into invariant subspaces, each behaving like one of the 2D cases.

Real eigenvalues give 1D invariant lines. Complex conjugate pairs give 2D invariant planes containing spiral dynamics. The full trajectory is a superposition: each mode evolves independently in its own subspace. For a 3D system with eigenvalues $\{{\lambda }_1,\sigma \pm i\omega \}$, trajectories are a 1D decay along $v_1$ plus a 2D spiral in the plane spanned by $\mathrm{Re}(v_2),\mathrm{Im}(v_2)$.

(Linearization of nonlinear systems) Near a hyperbolic equilibrium, a nonlinear system has the same portrait shape as its linearization.

The Hartman–Grobman theorem: if the linearization $A=\partial f/\partial x{|}_{x_0}$ has no eigenvalues on the imaginary axis, the nonlinear trajectories near $x_0$ are topologically equivalent to those of $\dot{x}=Ax$. This is why the linear classification is the right starting point for nonlinear phase-plane analysis — saddles stay saddles, stable foci stay stable foci. Centers and non-hyperbolic cases are the exceptions; there, higher-order terms decide.

Examples

Classifying a 2D System

Classify the phase portrait of $A=\left[\begin{array}{cc}-1& 2\\ -3& -4\end{array}\right]$.

Solution $\tau =-5$, $\Delta =10$. Discriminant: ${\tau }^2-4\Delta =25-40=-15<0$, so eigenvalues are complex.

$\lambda =\frac{1}{2}(-5\pm i\sqrt{15})$ — real part $-5/2<0$.

Stable focus (inward spiral). Trajectories spiral toward the origin at exponential rate $e^{-5t/2}$ and oscillation frequency $\sqrt{15}/2$.

Saddle Point: Inverted Pendulum

Find the phase portrait of the linearized inverted pendulum $A=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]$ (from $g/\ell =1$).

Solution $\tau =0$, $\Delta =-1<0$. Opposite-sign real eigenvalues: $\lambda =\pm 1$.

Eigenvectors: ${\lambda }_1=1$: $v_1=(1,1{)}^T$. ${\lambda }_2=-1$: $v_2=(1,-1{)}^T$.

Saddle. Stable manifold along $(1,-1{)}^T$ — falling back toward upright if the pendulum is moving backward as it passes through the peak at just the right rate. Unstable manifold along $(1,1{)}^T$. Every other initial condition eventually diverges along the unstable manifold: the pendulum falls.

Center: Undamped Oscillator

The harmonic oscillator $\ddot{y}+y=0$ becomes $A=\left[\begin{array}{cc}0& 1\\ -1& 0\end{array}\right]$ in state-space form. Classify the phase portrait and describe the trajectory from $x(0)=(1,0)$.

Solution $\tau =0$, $\Delta =1>0$, discriminant $-4<0$. Pure imaginary eigenvalues $\pm i$.

Center. No real eigenvectors; trajectories are closed orbits. Because $A$ is anti-symmetric with unit eigenvalues, the orbits are circles (with appropriate scaling, ellipses in general).

From $x(0)=(1,0)$: $x(t)=(\cos t,-\sin t)$ — a unit circle traversed clockwise with period $2\pi$. Position oscillates between $\pm 1$; velocity does the same, quarter-period out of phase. Energy is conserved, so the trajectory never approaches the origin.

Stable Node with Slow Mode

$A=\left[\begin{array}{cc}-1& 0\\ 0& -10\end{array}\right]$. Describe the trajectory from $x(0)=(1,1)$.

Solution ${\lambda }_1=-1,v_1=(1,0{)}^T$; ${\lambda }_2=-10,v_2=(0,1{)}^T$.

Already diagonal:

Stable node. Trajectory: $x(t)=(e^{-t},e^{-10t})$.

The $x_2$ component decays ten times faster than $x_1$. Early motion is dominated by the fast mode: the state dives toward the $x_1$-axis. Late motion is dominated by the slow mode: the state crawls along the $x_1$-axis toward the origin. In a general phase portrait this looks like trajectories “turning in” to become tangent to the slow eigenvector $v_1$ near the origin.

See Also

• 7.3. Solving LTI Systems — the closed-form solution whose geometry phase portraits visualize
• 7.5. Stability — eigenvalue-location criterion, same idea stated algebraically
• 7.2. Linearization — why 2D phase portraits classify nonlinear equilibria too (Hartman–Grobman)
• eigenvalues and eigenvectors — the primitive the classification rests on