Physics
The Mathieu equation appears in:
• Paul ion trap: AC voltage confines ions via parametric stability
• Parametric amplifier: pump modulates resonator
• Faraday waves: surface tension driven by oscillating gravity
• Floquet theory: periodic coefficients → Bloch waves
Stable (blue): x(t) bounded
Unstable (red): x(t) → ∞
Tongue boundaries arise from π- and 2π-periodic solutions of the Mathieu equation. The first tongue opens at a≈1, q small.