Phase Plane

277. Phase Plane

A graphical method for analyzing 2-state dynamical systems. Plot the two state variables on axes; the system’s trajectory traces a curve in this plane.

277.1. For a 2-state system

\cdot 𝑥 = 𝑓 (𝑥, 𝑦), \cdot 𝑦 = 𝑔 (𝑥, 𝑦)

Each $(𝑥, 𝑦)$ point has a velocity vector $(\cdot 𝑥, \cdot 𝑦)$ . The phase portrait is the field of these vectors plus typical trajectories.

277.2. Key features

Equilibria / fixed points: $(𝑥^{*}, 𝑦^{*})$ where $𝑓 = 𝑔 = 0$ . The system rests there.
Nullclines: curves where $\cdot 𝑥 = 0$ (the $𝑥$ -nullcline) or $\cdot 𝑦 = 0$ (the $𝑦$ -nullcline). Equilibria lie at intersections.
Trajectories: integral curves of the vector field
Separatrices: trajectories that divide the plane into regions of different long-term behavior

277.3. Linearizing at an equilibrium

Near $(𝑥^{*}, 𝑦^{*})$ , Taylor-expand $𝑓, 𝑔$ :

[\begin{matrix} \cdot 𝑥 \\ \cdot 𝑦 \end{matrix}] \approx 𝐽 [\begin{matrix} 𝑥 - 𝑥^{*} \\ 𝑦 - 𝑦^{*} \end{matrix}]

where $𝐽$ is the Jacobian:

𝐽 = {[\begin{matrix} \frac{𝜕 𝑓}{𝜕 𝑥} & \frac{𝜕 𝑓}{𝜕 𝑦} \\ \frac{𝜕 𝑔}{𝜕 𝑥} & \frac{𝜕 𝑔}{𝜕 𝑦} \end{matrix}]}_{(𝑥^{*}, 𝑦^{*})}

The eigenvalues of $𝐽$ classify the equilibrium.

277.4. Classification by eigenvalues

Eigenvalues	Type	Behavior
Both real, both $< 0$	Stable node	Trajectories approach $(𝑥^{}, 𝑦^{})$ from all directions
Both real, both $> 0$	Unstable node	Trajectories diverge from $(𝑥^{}, 𝑦^{})$
Real, opposite signs	Saddle	Stable along one direction, unstable along other; separatrix
Complex, real part $< 0$	Stable spiral	Trajectories spiral inward
Complex, real part $> 0$	Unstable spiral	Trajectories spiral outward
Pure imaginary	Center	Closed orbits around equilibrium (oscillation)

The trace-determinant plane visualizes these regimes:

$𝜏 = tr (𝐽), Δ = det (𝐽)$
$Δ > 0, 𝜏 < 0$ : stable (node or spiral)
$Δ > 0, 𝜏 > 0$ : unstable (node or spiral)
$Δ < 0$ : saddle (always unstable)
$Δ > 0, 𝜏^{2} < 4 Δ$ : spiral; otherwise: node

277.5. Lotka-Volterra as example

\cdot 𝑥 = 𝛼 𝑥 - 𝛽 𝑥 𝑦, \cdot 𝑦 = 𝛿 𝑥 𝑦 - 𝛾 𝑦

Equilibrium at $(\frac{𝛾}{𝛿}, \frac{𝛼}{𝛽})$ .

Jacobian at equilibrium:

𝐽 = [\begin{matrix} 0 & - 𝛽 \frac{𝛾}{𝛿} \\ 𝛿 \frac{𝛼}{𝛽} & 0 \end{matrix}]

Trace $= 0$ , $det > 0$ . Pure imaginary eigenvalues → center → closed orbits. (Confirms Lotka-Volterra’s well-known oscillation.)

277.6. Limit cycles and chaos

For nonlinear systems with appropriate structure, isolated closed orbits (limit cycles) appear. Examples:

van der Pol oscillator
Predator-prey with Holling functional response
Belousov-Zhabotinsky chemical reaction

In dimension $\geq 3$ , chaos becomes possible (Lorenz attractor, etc.). 2-D systems can’t be chaotic by Poincaré-Bendixson theorem.

277.7. Routh-Hurwitz stability for higher dimensions

For 3+ states, eigenvalue computation generalizes. The Routh-Hurwitz criterion gives an algorithmic stability test from coefficient signs of the characteristic polynomial.

277.8. See also

Feedback Loops
Second-Order Systems
Lotka-Volterra — phase plane example
Differential Equations
Eigenvectors / Eigenvalues — for the Jacobian