The role of Poincaré—Andronov—Hopf bifurcations in the application of variable-coefficient harmonic balance to periodically forced nonlinear oscillators

Abstract

The method of variable-coefficient harmonic balance (VCHB) presented in Summers (1995) is applied to the Duffing and the periodically forced van der Pol oscillator equations. A rationale is given of how Poincare-Andronov-Hopf (PAH) bifurcations in the amplitude evolution equations of VCHB relate to the various local bifurcations arising in the solution of these two oscillator equations. In order to demonstrate the crucial role played by PAH bifurcations in the amplitude evolution equations, the theory is applied in its simplest form, that is VCHB with a one-harmonic solution expansion, to the Duffing oscillator equation with a single well and softening-type nonlinearity. A single PAH bifurcation in the amplitude evolution equations is evaluated and the frequency of this bifurcation is then used to construct the curve of symmetry breaking bifurcations of the periodic solutions of the Duffing equation in (ω, F ) control space. In general, the periodic solution is represented by a truncated Fourier series with several harmonics and, therefore, by virtue of the size of the problem, numerical methods are employed to perform the algebra and the analysis. This procedure yields many various eigenvalues to track and, hence, an increased number of PAH bifurcations. The features of nonlinear resonances, period-doubling bifurcations, symmetry breaking bifurcations, subharmonic and superharmonic entrainments and NaimarkSacker bifurcations are tracked throughout control parameter space by tracing the critical eigenvalues associated with the PAH bifurcations. The local bifurcations of the two oscillator equations are classified by evaluating the imaginary part of the critical eigenvalues. In the case of the forced van der Pol equation with order one nonlinearity, the 1:1 and 3:1 superharmonic entrainment boundaries are evaluated, and the symmetry breaking and period-doubling bifurcation boundaries are derived for the first time by a semi-analytic approach.