Higher-Order Pseudoaveraging via Harmonic Balance for Strongly Nonlinear Oscillations

Some strongly nonlinear conservative oscillators, on slight perturbation, can be studied via averaging of elliptic functions. These and many other oscillations allow harmonic balance-based averaging (HBBA), recently developed as an approximate first-order calculation. Here, we extend HBBA to higher orders. Unlike the usual higher-order averaging for weakly nonlinear oscillations, here both the dynamic variable and time are averaged with respect to an auxiliary variable. Since the harmonic balance approximations introduce technically $O(1)$ errors at each order, the higher-order results are not strictly asymptotic. Nevertheless, as we show with examples, for reasonable values of the small expansion parameter, excellent approximations are obtained.