Hill differential equation

Example

Let $T > 0$ and let $f : \R \to \R$ be continuous and $T$-periodic. Hill's equation is the second-order linear differential equation $$y'' + f(t)\, y = 0.$$ The Mathieu equation $y'' + (a - 2q \cos 2t)\, y = 0$, with real constants $a$ and $q$, is the special case $f(t) = a - 2q\cos 2t$, of period $\pi$.

Floquet analysis: let $y_1, y_2$ be the solutions with $y_1(0) = 1$, $y_1'(0) = 0$ and $y_2(0) = 0$, $y_2'(0) = 1$. The monodromy matrix of the equivalent system for $(y, y')$ is $M$ with columns $(y_1(T), y_1'(T))$ and $(y_2(T), y_2'(T))$. The system matrix has trace $0$, so by Liouville's formula $\det M = 1$. Hence the multipliers are the roots of $\mu^2 - \Delta \mu + 1 = 0$, where $\Delta = y_1(T) + y_2'(T)$ is the discriminant.

1. If $\abs{\Delta} < 2$, the multipliers are two distinct conjugate numbers of modulus $1$, and all solutions are bounded.
2. If $\abs{\Delta} > 2$, there is a real multiplier of modulus greater than $1$, and some solution is unbounded.

Check with constant $f = \omega^2$, $\omega > 0$, viewed as $T$-periodic: $y_1 = \cos \omega t$ and $y_2 = \sin(\omega t)/\omega$, so $\Delta = 2\cos \omega T$ and $\abs{\Delta} \le 2$, matching the bounded oscillations of the harmonic oscillator.