  We consider the temporal homogenization of linear ODEs of the form
$\dot{x}=Ax+\epsilon P(t)x+f(t)$, where $P(t)$ is periodic and $\epsilon$ is
small. Using a 2-scale expansion approach, we obtain the long-time
approximation $x(t)\approx \exp(At) \left( \Omega(t)+\int_0^t \exp(-A \tau)
f(\tau) \, d\tau \right)$, where $\Omega$ solves the cell problem
$\dot{\Omega}=\epsilon B \Omega + \epsilon F(t)$ with an effective matrix $B$
and an explicitly-known $F(t)$. We provide necessary and sufficient condition
for the accuracy of the approximation (over a $\mathcal{O}(\epsilon^{-1})$
time-scale), and show how $B$ can be computed (at a cost independent of
$\epsilon$). As a direct application, we investigate the possibility of using
RLC circuits to harvest the energy contained in small scale oscillations of
ambient electromagnetic fields (such as Schumann resonances). Although a RLC
circuit parametrically coupled to the field may achieve such energy extraction
via parametric resonance, its resistance $R$ needs to be smaller than a
threshold $\kappa$ proportional to the fluctuations of the field, thereby
limiting practical applications. We show that if $n$ RLC circuits are
appropriately coupled via mutual capacitances or inductances, then energy
extraction can be achieved when the resistance of each circuit is smaller than
$n\kappa$. Hence, if the resistance of each circuit has a non-zero fixed value,
energy extraction can be made possible through the coupling of a sufficiently
large number $n$ of circuits ($n\approx 1000$ for the first mode of Schumann
resonances and contemporary values of capacitances, inductances and
resistances). The theory is also applied to the control of the oscillation
amplitude of a (damped) oscillator.
