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• W3105687466 abstract "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 $x(t)approx exp(At) left( Omega(t)+int_0^t exp(-A tau) f(tau) , dtau right)$, 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 $nkappa$. 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 ($napprox 1000$ 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." @default.
• W3105687466 created "2020-11-23" @default.
• W3105687466 creator A5034000581 @default.
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• W3105687466 date "2015-10-01" @default.
• W3105687466 modified "2023-09-25" @default.
• W3105687466 title "Temporal Homogenization of Linear ODEs, with Applications to Parametric Super-Resonance and Energy Harvest" @default.
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• W3105687466 doi "https://doi.org/10.1007/s00205-015-0932-4" @default.
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