The Phase Plane

What a phase plane is

A 2D autonomous ODE system has the form

ẋ = f(x, y) ẏ = g(x, y)

where the right-hand sides do not depend explicitly on t. The phase plane is the (x, y) plane treated as the state space: every point represents a possible state of the system, and the vector (f(x, y), g(x, y)) is the instantaneous velocity at that state. A solution (x(t), y(t)) traces out a trajectory — a curve tangent everywhere to the vector field.

Because the system is autonomous, the vector field is time-independent and trajectories never cross (uniqueness of solutions to ODE initial-value problems). This single fact — non-crossing trajectories — already restricts the dynamics enormously and underlies the Poincaré-Bendixson theorem.

A second-order ODE ẍ + f(x, ẋ) = 0 becomes a 2D system by setting y = ẋ: now ẋ = y and ẏ = −f(x, y). Phase-plane analysis is therefore the standard toolkit for any single-degree-of-freedom mechanical system.

Fixed points and the Jacobian

A fixed point (x*, y*) is a state where the system is at rest: f(x*, y*) = g(x*, y*) = 0. Near a fixed point, Taylor-expand the vector field to first order:

ẋ ≈ J · (x − x*, y − y*)ᵀ J = [[∂f/∂x, ∂f/∂y], [∂g/∂x, ∂g/∂y]] evaluated at (x*, y*)

The 2×2 matrix J is the Jacobian. Its eigenvalues λ₁, λ₂ classify the local linearised dynamics. For a hyperbolic fixed point (no eigenvalue on the imaginary axis), the Hartman-Grobman theorem says the nonlinear flow is topologically conjugate to the linear one — so the classification carries over to the full system.

The phase-portrait taxonomy by eigenvalues

Eigenvalues	Type	Stability	Local picture	Linear example
λ₁, λ₂ real, both < 0	Stable node	Asymptotically stable	Trajectories converge tangent to slow eigenvector	ẋ = −x, ẏ = −2y
λ₁, λ₂ real, both > 0	Unstable node	Unstable	Trajectories diverge along eigenvectors	ẋ = x, ẏ = 2y
λ₁ > 0, λ₂ < 0	Saddle	Unstable	Stable manifold + unstable manifold cross at point	ẋ = x, ẏ = −y
α ± iβ, α < 0	Stable spiral (focus)	Asymptotically stable	Trajectories spiral inward	ẋ = −x + y, ẏ = −x − y
α ± iβ, α > 0	Unstable spiral	Unstable	Trajectories spiral outward	ẋ = x + y, ẏ = −x + y
±iβ (pure imaginary)	Center	Neutrally stable (linear case)	Closed orbits — concentric ellipses	ẋ = y, ẏ = −x
Repeated λ, defective	Improper node / star	Same sign as λ	One eigendirection only; tangent convergence	ẋ = −x + y, ẏ = −y

Worked example: the simple pendulum

The undamped pendulum obeys ẍ + sin x = 0. As a 2D system:

ẋ = y ẏ = −sin x

Fixed points: y = 0 and sin x = 0, giving (nπ, 0) for every integer n. Jacobian:

J(x, y) = [[0, 1], [−cos x, 0]]

At (0, 0): J = [[0, 1], [−1, 0]], eigenvalues ±i — a center. Pendulum hangs straight down; small kicks produce closed orbits (linearly, ellipses; nonlinearly, slightly deformed ovals). At (π, 0): J = [[0, 1], [1, 0]], eigenvalues ±1 — a saddle. Pendulum balances upside-down; almost-vertical configurations either fall down to the right or to the left.

The unstable manifold leaving the saddle at (π, 0) along the eigenvector (1, 1) reconnects to the saddle at (−π, 0) along (1, −1) — a homoclinic orbit in the periodic-x covering, a closed separatrix in the cylinder phase space [0, 2π) × ℝ. This separatrix has energy E = 2 (taking the conventional normalisation H = ½y² + 1 − cos x); inside it lies oscillation, outside lies full rotation. The whole picture — three regimes meeting at a curve — is impossible to see by staring at the time series, but immediate in the phase plane.

Energy and conservative systems

For a conservative second-order ODE ẍ + V'(x) = 0, the energy H(x, y) = ½y² + V(x) is conserved along trajectories: dH/dt = y · ẏ + V'(x) · ẋ = y(−V'(x)) + V'(x) y = 0. The phase portrait is therefore the family of level sets {H = const}. Reading the potential V(x) tells you the entire qualitative dynamics — minima give centers, maxima give saddles, inflection points give degenerate cases.

For the pendulum H(x, y) = ½y² + (1 − cos x). Level sets at energy < 2 are bounded closed curves (oscillation); = 2 is the separatrix; > 2 is unbounded in x (rotation, period 2π in x). Drawing the potential V(x) = 1 − cos x — a row of wells separated by humps — and reading the level sets at each height makes the entire pendulum phase plane appear at once.

Limit cycles and van der Pol

A limit cycle is an isolated closed trajectory. The crucial word is isolated: nearby trajectories spiral toward it (stable limit cycle) or away (unstable). A linear center has a continuous family of nested closed orbits, none isolated, and so is not a limit cycle.

The canonical example is the van der Pol oscillator

ẍ − μ(1 − x²)ẋ + x = 0,

introduced by Balthazar van der Pol in 1926 to model triode-tube circuits. The damping term −μ(1 − x²) is negative for |x| < 1 (energy is pumped in) and positive for |x| > 1 (energy is dissipated). The competition produces a unique attracting limit cycle of amplitude ≈ 2 for every μ > 0. For small μ the cycle is nearly circular; for large μ it becomes a relaxation oscillation — slow drift along the cubic nullcline followed by fast jumps. Modern applications include heartbeat modelling (FitzHugh-Nagumo, a variant of van der Pol) and laser dynamics.

Poincaré-Bendixson and the 2D constraint

The Poincaré-Bendixson theorem says that in a compact, simply connected region of the plane with no fixed points, any trajectory must approach a periodic orbit. The 2D constraint is essential: in 3D and above, trajectories can wander chaotically because they can avoid both fixed points and closed orbits by going around them. Phase-plane analysis is therefore complete — you only need fixed points, periodic orbits, and connecting separatrices to describe everything. Chaos requires three or more state variables (Lorenz 1963).

Where phase-plane analysis shows up

• Mechanical oscillators. Damped and forced versions of the pendulum, Duffing oscillator, mass-spring-dashpot systems all live in the phase plane. Trajectories spiralling into a stable spiral signal damped oscillation; closed orbits encircle a center.
• Electronic circuits. The van der Pol oscillator, Liénard equations, and tunnel-diode circuits are studied directly in (current, voltage) phase planes; limit cycles are physical reality, not just abstraction.
• Neuroscience and excitability. The FitzHugh-Nagumo simplification of the Hodgkin-Huxley equations is a 2D system whose phase plane shows excitability (a kick across the separatrix triggers a full action potential before relaxation back).
• Population dynamics. Lotka-Volterra predator-prey lives in the first quadrant of phase plane (x ≥ 0, y ≥ 0). The interior fixed point is a center (linearly); the orbits are closed energy curves. Adding logistic terms breaks this into a stable spiral.
• Control theory. Phase-plane methods underpin sliding-mode control, bang-bang time-optimal control (Pontryagin), and the analysis of nonlinear feedback systems with second-order plants.
• Economics. Saddle-path stability in dynamic optimization (Ramsey growth model, Dornbusch overshooting) is read off a 2D phase plane in (capital, consumption) or (price, exchange rate).

Common pitfalls

• Treating a non-autonomous system as 2D. An equation ẍ + f(x, ẋ, t) = 0 is intrinsically 3D in the phase space (x, ẋ, t). Trying to plot it in the (x, ẋ) plane creates artefacts — trajectories appear to cross, because they actually pass at different times. The fix is to either eliminate t (impossible in general) or use a Poincaré section.
• Trusting linearisation at a center. Pure imaginary eigenvalues are not hyperbolic; Hartman-Grobman doesn't apply. Nonlinear terms can perturb a linear center into a stable or unstable spiral. The linear pendulum at the origin is a true center because the system is exactly conservative; for damped pendulums, the same calculation gives ±iβ at lowest order but the full nonlinear flow is a stable spiral.
• Confusing a center with a limit cycle. A continuous family of nested closed orbits (linear center) is not a limit cycle. A limit cycle is isolated — for any ε > 0, there are non-closed trajectories within ε of it. The van der Pol cycle is isolated; the linear oscillator's orbits are not.
• Wrong sign on the Jacobian. The Jacobian must be evaluated at the fixed point with the original sign convention. A common slip is to write ẏ = −sin x for the pendulum but forget the minus sign when computing ∂g/∂x = −cos x at x = π → cos π = −1 → −cos π = +1, giving eigenvalues ±1 (saddle). Get the sign wrong and you get ±i (center) — completely wrong qualitative answer.
• Ignoring boundary behaviour. Trajectories may run off to infinity or stop at boundaries of the physical state space (e.g. x ≥ 0 for population sizes). The Poincaré sphere compactification or careful boundary analysis fix the picture; ignoring infinity misses limit cycles at infinity in cubic systems.
• Higher-dimensional analogues. The 2D dichotomy (fixed point or periodic orbit) breaks in 3D. The Lorenz attractor lives in 3D phase space and is not a periodic orbit despite being bounded. Phase-plane intuition does not extrapolate to chaos.