Parametric oscillator

Remarks

One speaks of parametric resonance when the periodic variation of the coefficients produces solutions that grow without bound.

The Mathieu equation $x'' + \omega_0^2 (1 + h \cos \gamma t) \, x = 0$ with $\omega_0 > 0$, $\gamma > 0$ and $h \neq 0$ satisfies the definition, with $\beta = 0$ and $q(t) = \omega_0^2 (1 + h \cos \gamma t)$, which has period $2 \pi / \gamma$. Its solutions are expressed with Mathieu functions. For small $\abs{h}$, parametric resonance occurs for $\gamma$ near the values $2 \omega_0 / n$ with $n = 1, 2, 3, \ldots$, and the instability is strongest near $\gamma = 2 \omega_0$.

A non-example is the harmonic oscillator $x'' + \omega_0^2 x = 0$: both coefficients are constant, so it fails the first clause. A forced oscillator $x'' + \omega_0^2 x = F \cos \gamma t$ is also not parametric, because the time dependence enters as a separate term and not through a coefficient.

By Floquet theory, the stability of the oscillator is decided by its two Floquet multipliers, the eigenvalues of the monodromy matrix over one period. With $\beta = 0$ their product is $1$, so the zero solution is unstable as soon as the trace of the monodromy matrix has absolute value larger than $2$.