Parametric Oscillator — Mathieu Equation & Arnold Tongues

Mathieu Equation: ẍ + [δ + ε·cos(2t)]x = 0

The Mathieu equation describes a parametrically driven oscillator — a pendulum with periodically varying length, or a child pumping a swing. Stability depends on (δ, ε): Arnold tongues (resonance bands) in the parameter space mark unstable regions where small perturbations grow exponentially. The primary tongue emanates from δ=1 (2:1 resonance), secondary from δ=4 (1:1), etc. Inside a tongue: parametric amplification. Outside: bounded oscillation. Adding damping γ lifts the tongue boundaries.