Parametric Resonance

Physics: Mathieu equation: ẍ + (a − 2q·cos(2t))x = 0, where a = (ω₀/Ω)² and q = ε/2. Parametric instability occurs in tongue-shaped regions in the (a, q) parameter space — the Mathieu stability diagram. Principal resonance at driving frequency Ω = 2ω₀ (first tongue at a = 1). Floquet theorem: solutions x(t+T) = λx(t); instability when |λ| > 1. Child on a swing pumps at twice the natural frequency. Parametric amplifiers in superconducting quantum circuits use this principle.