Two timescale harmonic balance. I: Application to autonomous one-dimensional nonlinear oscillators. (English) Zbl 0778.34022

Two timescale harmonic balance is a semi-analytic/numerical method for deriving periodic solutions and establishing their stability. In this first paper the method is applied to a class of nonlinear autonomous oscillators which can be described by differential equations of the type \(\ddot x+x=f(x,\dot x,\lambda,t)\), where \(\lambda\) is a control parameter. Features of both harmonic balance and multiple scales are incorporated in the method. The solution \(x(t)\) is sought as a series of superharmonics, subharmonics and ultrasubharmonics. The two timescales, associated with the amplitude and phase variations, are introduced through a parameter \(\varepsilon\). The method is applied to three versions of the van der Pol equation. Expansions in superharmonics reveal Hopf, saddle-node and homoclinic bifurcations.
Reviewer: P.Smith (Keele)


34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
34C23 Bifurcation theory for ordinary differential equations
65J99 Numerical analysis in abstract spaces
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