The response of a single-degree-of-freedom system with quadratic and cubic non-linearities to a principal parametric resonance

The response of a one-degree-of-freedom system with quadratic and cubic non-linearities to a principal parametric resonance is investigated. The method of multiple scales is used to determine the equations that describe to second order the modulation of the amplitude and phase with time about one of the foci. These equations are used to determine the fixed points and their stability. The perturbation results are verified by integrating the governing equation with use of a digital computer and an analogue computer. For small excitation amplitudes, the analytical results are in excellent agreement with the numerical solutions. The large amplitude responses are investigated by using both a digital and an analogue computer. The cases of single- and double-well potentials are investigated. Systems with double-well potentials exhibit complicated dynamic behaviors including period-multiplying and demultiplying bifurcations and chaos. In some cases, a bifurcated response coexists with another periodic attractor, and a chaotic attractor coexists with a periodic attractor. Long-time histories, phase planes, Poincaré maps, fractal basin maps, and spectra of the response are presented. A bifurcation diagram of many solutions in the excitation amplitude - excitation frequency plane is also presented.