Variable-coefficient harmonic balance for periodically forced nonlinear oscillators

This paper explores the application of the method of variable-coefficient harmonic balance to nonautonomous nonlinear equations of the form XsF(X, t:λ), and in particular, a one-degree-of-freedom nonlinear oscillator equation describing escape from a cubic potential well. Each component of the solution, X(t), is expressed as a truncated Fourier series of superharmonics, subharmonics and ultrasubharmonics. Use is then made of symbolic manipulation in order to arrange the oscillator equation as a Fourier series and its coefficient are evaluated in the traditional way. The time-dependent coefficients permit the construction of a set of amplitude evolution equations with corresponding stability criteria. The technique enables detection of local bifurcations, such as saddle-node folds, period doubling flips, and parts of the Feigenbaum cascade. This representation of the periodic solution leads to local bifurcations being associated with a term in the Fourier series and, in particular, the onset of a period doubled solution can be detected by a series of superharmonics only. Its validity is such that control space bifurcation diagrams can be obtained with reasonable accuracy and large reductions in computational expense.

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