Stability tongues in a periodically driven pendulum — Mathieu equation
A pendulum whose pivot oscillates vertically obeys the Mathieu equation: θ̈ + (δ + ε·cos(ωt))·sin(θ) = 0. Stability tongues define parameter regions of resonance (parametric instability). The most prominent tongue occurs at ω ≈ 2ω₀ (Faraday resonance). The inverted pendulum can be stabilized above a critical drive frequency.