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

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.