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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 $\epsilon$ issmall. Using a 2-scale expansion approach, we obtain the long-timeapproximation $x(t)\approx \exp(At) \left( \Omega(t)+\int_0^t \exp(-A \tau)f(\tau) \, d\tau \right)$, 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 conditionfor 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 usingRLC circuits to harvest the energy contained in small scale oscillations ofambient electromagnetic fields (such as Schumann resonances). Although a RLCcircuit parametrically coupled to the field may achieve such energy extractionvia parametric resonance, its resistance $R$ needs to be smaller than athreshold $\kappa$ proportional to the fluctuations of the field, therebylimiting practical applications. We show that if $n$ RLC circuits areappropriately coupled via mutual capacitances or inductances, then energyextraction 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 sufficientlylarge number $n$ of circuits ($n\approx 1000$ for the first mode of Schumannresonances and contemporary values of capacitances, inductances andresistances). The theory is also applied to the control of the oscillationamplitude of a (damped) oscillator.

Temporal homogenization of linear ODEs, with applications to parametric super-resonance and energy harvest