Complex period-1 motions in a periodically forced, quadratic nonlinear oscillator

In this paper, analytical solutions for complex period-1 motions in a periodically forced, quadratic nonlinear oscillator are presented through the Fourier series solutions with finite harmonic terms, and the corresponding stability and bifurcation analyses of the corresponding period-1 motions are carried out. Many branches of complex period-1 motions in such a quadratic nonlinear oscillator are discovered and the period-1 motion patterns changes with parameters are presented. The parameter map for excitation amplitude and frequency is developed for different complex period-1 motions. For small excitation frequency, the period-1 motion becomes more complicated. For a better understanding of complex period-1 motions in such a quadratic nonlinear oscillator, trajectories and amplitude spectrums are illustrated numerically. From stability and bifurcations analysis of the period-1 motion, the analytical bifurcation trees of period-1 motions to chaos need to be further investigated.