Nonlinear limit cycle oscillator for neural rhythms — Hopf bifurcation, entrainment, and frequency tuning.
Amplitude r: 0.00
Phase θ: 0.00
Freq ratio ωᴅ/ω₀: 1.00
Entrained: –
Hopf normal form: ż = (μ + iω₀)z − β|z|²z + F·e^{iωᴅt}, z = x + iy.
In polar coords: ṙ = μr − βr³ + F·cos(φ), φ̇ = ω₀ + F·sin(φ)/r.
Bifurcation: μ<0 → stable fixed point (no oscillation); μ=0 → Hopf bifurcation; μ>0 → limit cycle r* = √(μ/β).
Entrainment: With periodic drive near ω₀, the oscillator can phase-lock (Arnold tongue). Used in CPG (central pattern generator) models for locomotion and neural rhythms (gamma oscillations ~40Hz, theta ~8Hz).