Floquet Parametric Resonance

Mathieu Equation Parameters

Natural frequency ω₀: 1.00 Drive frequency Ω: 2.00 Drive amplitude ε: 0.30 Damping γ: 0.05 Initial θ₀: 0.10 rad

θ̈ + 2γθ̇ + [ω₀² + ε·cos(Ωt)]θ = 0

Floquet: θ(t) = e^{μt}·p(t), p periodic

Resonance near Ω ≈ 2ω₀/n

Status: —

Max amplitude: —

Drive ratio Ω/ω₀: 2.00

Time: 0.0 s

Parametric resonance occurs when the drive frequency is near 2ω₀/n. At ε=0 the system is a harmonic oscillator; as ε grows, instability tongues appear in the Mathieu stability diagram.

Try: Ω=2, ω₀=1, ε=0.3 (primary resonance) vs. Ω=4, ω₀=1 (subharmonic).

Floquet Theory & Parametric Resonance

Phase Space Trajectory & Amplitude

Mathieu Equation Parameters

Mathieu Stability Diagram (Ince-Strutt chart)