Parametric Oscillator & Resonance
Mathieu equation: ẍ + (δ + ε cos ωt)x = 0 — Arnold tongues & parametric amplification
Natural frequency δ (= ω₀²)
1.00
Modulation ε
0.30
Drive frequency ω
2.00
Damping γ
0.020
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Parametric resonance occurs when ω ≈ 2ω₀/n. Primary tongue: ω ≈ 2ω₀. Amplitude grows exponentially inside Arnold tongue regions. Damping suppresses narrow tongues.