Mathieu Parametric Resonance

ẍ + (δ + ε cos t)x = 0 — Ince-Strutt stability diagram, click to pick parameters

Ince-Strutt diagram — click to pick (δ,ε)
Parametric resonance occurs
near δ ≈ n²/4 (n=1,2,3,...)

Tongue boundaries:
n=1: δ = ¼ ± ½ε + O(ε²)
n=2: δ = 1 ± ε² + O(ε⁴)

Red/orange = unstable
Blue = stable

Floquet theory: solution
x(t) = e^(μt)·p(t)
μ = Floquet exponent
Unstable: Re(μ) > 0
Click diagram to select point