Stability diagram of ẍ + (a - 2q cos 2t)x = 0 with pendulum visualization
The Mathieu equation ẍ + (a − 2q cos 2t)x = 0 describes a parametrically driven oscillator (e.g., a pendulum with oscillating pivot). For certain (a, q) pairs the trivial solution is unstable — amplitude grows exponentially. The first instability tongue near a=1, q=0 corresponds to driving at twice the natural frequency. Used in Paul traps (Nobel 1989) and MEMS devices.