Efficient determination of higher-order periodic solutions using n-mode harmonic balance

This paper presents a systematic procedure to explicitly determine the algebraic equations arising from the method of harmonic balance with an arbitrary number of modes in the assumed solutions. The technique can be used for a wide variety of nonlinear oscillators (including systems of ordinary differential equations). The method is illustrated in the case of second-order differential equations with nonlinear restoring force. Although numerical methods have been employed to solve the resulting systems of algebraic equations, the general approach is analytic. As such, this study confirms independently (i.e. nonsimulation) the period-doubling cascade of an escape equation including the bifurcation universal scaling laws.