We consider the temporal homogenization of linear ODEs of the form ( x =Ax+ P(t)x+f(t) ), where P(t) is periodic and ( ) is small. Using a 2-scale expansion approach, we obtain the long-time approximation ( x(t) exp (At) ( (t)+ ^t exp (-A ) f( ) d ) ), where ( ) solves the cell problem ( = B + F(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 ( ^ -1 ) ) time-scale), and show how B can be computed (at a cost independent of ( )). 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 ( ) 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 ). 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.
Abstract Approximate resolution of linear systems of differential equations with varying coefficients is a recurrent problem, shared by a number of scientific and engineering areas, ranging from Quantum Mechanics to Control Theory. When formulated in operator or matrix form, the Magnus expansion furnishes an elegant setting to build up approximate exponential representations of the solution of the system. It provides a power series expansion for the corresponding exponent and is sometimes referred to as Time-Dependent Exponential Perturbation Theory . Every Magnus approximant corresponds in Perturbation Theory to a partial re-summation of infinite terms with the important additional property of preserving, at any order, certain symmetries of the exact solution. The goal of this review is threefold. First, to collect a number of developments scattered through half a century of scientific literature on Magnus expansion. They concern the methods for the generation of terms in the expansion, estimates of the radius of convergence of the series, generalizations and related non-perturbative expansions. Second, to provide a bridge with its implementation as generator of especial purpose numerical integration methods , a field of intense activity during the last decade. Third, to illustrate with examples the kind of results one can expect from Magnus expansion, in comparison with those from both perturbative schemes and standard numerical integrators. We buttress this issue with a revision of the wide range of physical applications found by Magnus expansion in the literature. Approximate solutions of matrix linear differential equations by matrix exponentials are considered. In particular, the convergence issue of Magnus and Fer expansions is treated. Upper bounds for the convergence radius in terms of the norm of the defining matrix of the system are obtained. The very few previously published bounds are improved. Bounds to the error of approximate solutions are also reported. All results are based just on algebraic manipulations of the recursive relation of the expansion generators. 