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.

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