Phase plane behaviour

For a 2D linear autonomous system ${\mathbf{x}}^{\prime }=A\mathbf{x}$, the phase portrait (qualitative picture of trajectories in the $xy$-plane) is determined by the eigenvalues of $A$. Six standard cases cover all the possibilities, each with its own characteristic shape and stability.

Overview of all six cases on a single whiteboard: Case 1 unstable node, Case 2 stable node (sink), Case 3 saddle, Case 4 repeated-eigenvalue nodes (proper/improper), Case 5 spirals, Case 6 center. Each case is treated in detail below.

Stability terminology

Recap of key terms (see Stability of autonomous systems for formal definitions):

• Stable: nearby trajectories stay nearby for all time.
• Asymptotically stable: nearby trajectories converge to the equilibrium.
• Unstable: at least some nearby trajectories move away.

Phase portrait classifications:

• Node: trajectories approach (or leave) along straight-line directions.
• Saddle: trajectories approach along one direction, depart along another.
• Spiral (focus): trajectories spiral in or out.
• Center: trajectories form closed orbits.

Case 1: distinct real eigenvalues, both positive ($r_1\ne r_2>0$)

Both eigenvalues positive real. Solutions $\mathbf{x}(t)=c_1{\mathbf{u}}_1{e}^{r_{1}t}+c_2{\mathbf{u}}_2{e}^{r_{2}t}$ grow exponentially in both eigenvector directions.

• Type: unstable node. (Some introductory texts call this an “improper node,” but in standard dynamical-systems terminology “improper” denotes the defective repeated-eigenvalue case in Case 4b below; with two distinct eigenvalues this is just a node.)
• Behavior: trajectories diverge exponentially from the origin.
• Equilibrium stability: unstable.

Case 2: distinct real eigenvalues, both negative ($r_1\ne r_2<0$)

Both eigenvalues negative real. Mirror of Case 1: trajectories shrink toward origin.

• Type: stable node (sink).
• Behavior: trajectories converge exponentially to origin.
• Equilibrium stability: asymptotically stable.

Case 3: distinct real eigenvalues, opposite signs ($r_1>0>r_2$)

One positive, one negative. Solutions grow in one direction, shrink in another.

• Type: saddle point.
• Behavior: trajectories approach origin along the eigenvector for the negative eigenvalue, then diverge along the positive eigenvalue’s direction.
• Equilibrium stability: unstable.

Case 4: repeated real eigenvalues ($r_1=r_2=r$)

Two subcases depending on number of eigenvectors:

Case 4a: two linearly independent eigenvectors (proper node)

Every direction is an eigenvector — the matrix is a scalar multiple of identity. Trajectories are straight lines through origin.

• $r>0$: unstable proper node.
• $r<0$: stable proper node (asymptotically stable).

Case 4b: only one eigenvector (improper node)

Defective matrix — see Repeated eigenvalues case. Trajectories curve, tangent to the eigenvector direction near origin.

• $r>0$: unstable improper node.
• $r<0$: stable improper node (asymptotically stable).

Case 5: complex conjugate eigenvalues with nonzero real part ($r=\alpha \pm i\beta$, $\alpha \ne 0$)

The system has oscillatory behavior with exponential envelope.

In polar coordinates $(\rho ,\theta )$, the radius evolves as $\rho (t)={\rho }_0{e}^{\alpha t}$ — exponential growth or decay. The angle changes at rate $\beta$.

• $\alpha >0$: trajectories spiral outward. Unstable spiral.
• $\alpha <0$: trajectories spiral inward to origin. Asymptotically stable spiral.

The direction of rotation depends on the sign of $\beta$.

The four spiral images show: $\alpha >0,\beta >0$ (unstable counterclockwise); $\alpha >0,\beta <0$ (unstable clockwise); $\alpha <0,\beta <0$ (stable clockwise); $\alpha <0,\beta >0$ (stable counterclockwise).

Case 6: pure imaginary eigenvalues ($r=\pm i\beta$, $\beta \ne 0$)

No exponential envelope ($\alpha =0$). Trajectories are closed orbits (ellipses around the origin), with period $T=2\pi /\beta$.

• Type: center.
• Behavior: bounded periodic motion forever.
• Equilibrium stability: stable but not asymptotically stable.

Summary table

Eigenvalues	Type	Stability
Real distinct, both $>0$	Node	Unstable
Real distinct, both $<0$	Node (sink)	Asymptotically stable
Real distinct, opposite signs	Saddle	Unstable
Real repeated, 2 eigenvectors	Proper node	Stable iff $r<0$
Real repeated, 1 eigenvector	Improper node	Stable iff $r<0$
Complex, $\alpha >0$	Spiral	Unstable
Complex, $\alpha <0$	Spiral	Asymptotically stable
Pure imaginary	Center	Stable, not asymptotically

Why this classification matters

For 2D linear autonomous systems, the eigenvalues of $A$ tell you everything about long-term behavior — you don’t need to solve the ODE. Each phase portrait shape corresponds to a different qualitative behavior, and the eigenvalues alone determine which one.

Furthermore, by the linearization theorem, near any equilibrium of a nonlinear system, the local phase portrait resembles one of these six cases (when the eigenvalues are not on the imaginary axis). So this classification of linear systems is the foundation for understanding nonlinear systems too.

For the linearization technique, see Locally linear system. For the formal stability theory, see Stability of autonomous systems.