THE ROLE OF POINCARE-ANDRONOV-HOPF BIFURCATIONS IN THE APPLICATION OFVARIABLE-COEFFICIENT HARMONIC-BALANCE TO PERIODICALLY FORCED NONLINEAR OSCILLATORS

The method of variable-coefficient harmonic balance (VCHB) presented i n Summers (1995) is applied to the Duffing and the periodically forced van der Pol oscillator equations. A rationale is given of how Poincar e-Andronov-Hopf (PAH) bifurcations in the amplitude evolution equation s of VCHB relate to the various local bifurcations arising in the solu tion of these two oscillator equations. In order to demonstrate the cr ucial role played by PAH bifurcations in the amplitude evolution equat ions, the theory is applied in its simplest form, that is VCHB with a one-harmonic solution expansion, to the Duffing oscillator equation wi th a single well and softening-type nonlinearity. A single PAH bifurca tion in the amplitude evolution equations is evaluated and the frequen cy of this bifurcation is then used to construct the curve of symmetry breaking bifurcations of the periodic solutions of the Duffing equati on in (omega, F) control space. In general, the periodic solution is r epresented by a truncated Fourier series with several harmonics and, t herefore, by virtue of the size of the problem, numerical methods are employed to perform the algebra and the analysis. This procedure yield s many various eigenvalues to track and, hence, an increased number of PAH bifurcations. The features of nonlinear resonances, period-doubli ng bifurcations, symmetry breaking bifurcations, subharmonic and super harmonic entrainments and Naimark-Sacker bifurcations are tracked thro ughout control parameter space by tracing the critical eigenvalues ass ociated with the PAH bifurcations. The local bifurcations of the two o scillator equations are classified by evaluating the imaginary part of the critical eigenvalues. In the case of the forced van der Pol equat ion with order one nonlinearity, the 1:1 and 3:1 superharmonic entrain ment boundaries are evaluated, and the symmetry breaking and period-do ubling bifurcation boundaries are derived for the first time by a semi -analytic approach.