Simply improved averaging for coupled oscillators and weakly nonlinear waves

Abstract

The long time effect of nonlinear perturbation to oscillatory linear systems can be characterized by the averaging method, and we consider first-order averaging for its simplest applicability to high-dimensional problems. Instead of the classical approach, in which one uses the pullback of linear flow to isolate slow variables and then approximate the effective dynamics by averaging, we propose an alternative coordinate transform that better approximates the mean of oscillations. This leads to a simple improvement of the averaged system, which will be shown both theoretically and numerically to provide a more accurate approximation. Three examples are then provided: in the first, a new device for wireless energy transfer modeled by two coupled oscillators was analyzed, and the results provide design guidance and performance quantification for the device; the second is a classical coupled oscillator problem (Fermi-Pasta-Ulam), for which we numerically observed improved accuracy beyond the theoretically justified timescale; the third is a nonlinearly perturbed firstorder wave equation, which demonstrates the efficacy of improved averaging in an infinite dimensional setting.

Generation of Electromechanical Frequency Combs in Fluid Media with a Parametrically Driven Capacitive Microresonator

“…However, this method of solution can be both non-intuitive and time consuming, especially when performing extensive parametric studies. Alternately, the coupled ODEs can be re-written using new scaled and normalized parameters to make explicit a timescale separation, and approximated using an improved averaging theory [23,24] to obtain approximate semi-analytical solutions (details of the parameters used and the analysis are presented in the supplementary material [25]). The semi-analytical solutions provide more insight into the operation of the proposed comb generation system by providing expressions for critical drive voltage and frequency comb spacing.…”

Parametric resonance for enhancing the rate of metastable transition

“…[16]), the parametric resonant frequency also corresponds to intrinsic frequency ω 0 . This may sound inconsistent with the parametric resonant frequency of linear (e.g., qh + ω 2 0 (1 + ǫ cos(Ωt))q h = 0) or weakly nonlinear systems (e.g., [49,51]) which is Ω = 2ω 0 , but the latter is in fact, as discussed above, a special case where a 1 = a −1 = 0. As the potential of our system is in general arbitrarily nonlinear, all harmonics could exist (i.e., none of q n 's vanishes).…”

“…Consequently, we optimize over t 0 to obtain an MLP and the optimal transition rate of system (1.1). Thus (1.3) can be formally rewritten as follows [51]:…”