The multi-parameter homotopy harmonic balance method for steady state problems

Abstract

The analytical solutions of a simple nonlinear equation, e.g., the Duffing equation, can be highly complicated. Homotopy continuation on just one parameter can hardly produce the whole picture, in particular, of the multiple bifurcations in multi-parameter space. This paper reports on the development of the multi-parameter homotopy harmonic balance method for the steady state solutions of a nonlinear vibration problem. The total and tangential stiffnesses with respect to the Fourier components of polynomial nonlinearity are given explicitly. New multiple solutions of the Duffing equation are given for the first time. The period doubling to chaos is interpreted in a new way. Finally, the bifurcation surfaces of folding and period doubling are constructed.