A time domain collocation method for obtaining the third superharmonic solutions to the Duffing oscillator

In this study, a simple time domain collocation method (TDC) is applied to investigate the third superharmonic solutions of the Duffing oscillator. Upon using the proposed scheme, the multivaluedness, jump phenomenon, and transitional region of the third superharmonic response are explored. The amplitude frequency response curves for various values of damping, nonlinearity, and external force are obtained and compared. In addition, instead of collocating at N points so that the resulting nonlinear algebraic system is well determined, we extend the time domain collocation method to a new version by collocating at M>N points. The resulting over determined system is solved by the least square method. The extended time domain collocation method can significantly relieve the nonphysical solution phenomenon, which may be severe in the time domain collocation method, and its equivalent high dimensional harmonic balance method. Finally, numerical examples confirm the simplicity, efficiency, and accuracy of the proposed scheme.