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 ẋ =Ax+ϵP(t)x+f(t), where P(t) is periodic and ϵ is small. Using a 2-scale expansion approach, we obtain the long-time approximation x(t)≈exp(At) (Ω(t)+∫^t_0exp(−Aτ)f(τ)dτ), where Ω solves the cell problem Ω=ϵBΩ+ϵF(t) with an effective matrix B and an explicitly-known F(t). We provide necessary and sufficient conditions for the accuracy of the approximation (over a O(ϵ^(−1)) time-scale), and show how B can be computed (at a cost independent of ϵ). 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 κ 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 nκ. 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≈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