Parametric Resonance — Mathieu Instability Tongues

The Mathieu equation ẍ + [δ + ε cos(t)]x = 0 describes a pendulum with periodically driven pivot. For certain (δ, ε) combinations, small perturbations grow exponentially — parametric resonance. The instability regions form characteristic "Arnold tongues" in parameter space. The lower panel shows real-time amplitude for chosen parameters.

Stable
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