Periodic solution

Remarks

A periodic solution is a periodic function of $t$ that solves the equation. For an autonomous equation $x' = f(x)$, each constant solution is periodic with every period, and the trajectory of a nonconstant periodic solution is a periodic orbit. Example: for $x' = -x + \cos t$, the function $x(t) = \frac{1}{2}(\cos t + \sin t)$ has $x'(t) = \frac{1}{2}(\cos t - \sin t) = -x(t) + \cos t$. It is a $2\pi$-periodic solution of an equation with $2\pi$-periodic right-hand side. Non-example: the other solutions $x(t) = \frac{1}{2}(\cos t + \sin t) + c e^{-t}$ with $c \ne 0$ are not periodic, because $c e^{-t}$ is not. A maximal solution of a $T$-periodic equation with $x(t_0 + T) = x(t_0)$ for one $t_0$ is $T$-periodic when solutions are unique.