This is the same Moore–Greitzer lumped-parameter model as the 3D version, but rendered natively in 2D as a dynamical-systems view: instead of a spinning rotor, you see the state point (φ, ψ) itself move across its own phase plane, together with the vector field that pushes it there and the two curves whose intersection defines the operating point:
dφ/dξ = B·(ψc(φ) − ψ)
dψ/dξ = (1/B)·(φ − φT(ψ))
ψc(φ) = ψ0 + H·[1 + 1.5(φ/W − 1) − 0.5(φ/W − 1)³] (compressor nullcline, dφ/dξ=0)
φT(ψ) = γ·√max(ψ, 0) (throttle nullcline, dψ/dξ=0)
- Phase portrait (left) — the orange curve is the compressor characteristic ψc(φ); the grey curve is the throttle law. Where they cross is the equilibrium (white diamond). Faint arrows show the actual (dφ/dξ, dψ/dξ) vector field computed at the current B and γ — the trajectory (cyan) is a real numerical solution moving through that field, not an animation.
- Unrolled cascade (top right) — the rotor blade row laid out flat instead of in a ring; each blade tints toward yellow as the local incidence deficit ψc(φ) − ψ grows, the same stall criterion used in the ODE, and the flow streaks reverse from blue to red exactly when φ crosses zero.
- φ / ψ strip chart (bottom right) — a scrolling oscilloscope trace of both state variables over the last several seconds, so a surge limit cycle reads as a repeating waveform rather than something you have to infer from a spinning 3D stage.
- B and γ — same meaning as the 3D model: high B locks a closed throttle into a deep-surge limit cycle with flow reversal; low B lets the same throttle setting settle into mild rotating stall or a stable point instead.
Real-world relevance: this instability is why jet engines have surge margins, bleed valves and variable stator vanes — an axial compressor pushed too close to its stall line at low corrected speed can surge audibly and, if sustained, cause flameout or mechanical damage from the pressure pulses. The phase-portrait view is exactly how control engineers actually study this system: as a two-state nonlinear ODE, not a 3D mechanism.