275. 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.
275.1. For a 2-state system
Each point has a velocity vector . The phase portrait is the field of these vectors plus typical trajectories.
275.2. Key features
- Equilibria / fixed points: where . The system rests there.
- Nullclines: curves where (the -nullcline) or (the -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
275.3. Linearizing at an equilibrium
Near , Taylor-expand :
where is the Jacobian:
The eigenvalues of classify the equilibrium.
275.4. Classification by eigenvalues
| Eigenvalues | Type | Behavior |
|---|---|---|
| Both real, both | Stable node | Trajectories approach from all directions |
| Both real, both | Unstable node | Trajectories diverge from |
| Real, opposite signs | Saddle | Stable along one direction, unstable along other; separatrix |
| Complex, real part | Stable spiral | Trajectories spiral inward |
| Complex, real part | Unstable spiral | Trajectories spiral outward |
| Pure imaginary | Center | Closed orbits around equilibrium (oscillation) |
The trace-determinant plane visualizes these regimes:
- : stable (node or spiral)
- : unstable (node or spiral)
- : saddle (always unstable)
- : spiral; otherwise: node
275.5. Lotka-Volterra as example
Equilibrium at .
Jacobian at equilibrium:
Trace , . Pure imaginary eigenvalues → center → closed orbits. (Confirms Lotka-Volterra’s well-known oscillation.)
275.6. 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 , chaos becomes possible (Lorenz attractor, etc.). 2-D systems can’t be chaotic by Poincaré-Bendixson theorem.
275.7. 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.
275.8. See also
- Feedback Loops
- Second-Order Systems
- Lotka-Volterra — phase plane example
- Differential Equations
- Eigenvectors / Eigenvalues — for the Jacobian