Abstract
The method of multiple scales is used to analyze the nonlinear response of the surface of a liquid in a cylindrical container to a principal parametric resonant excitation in the presence of a two‐to‐one internal (autoparametric) resonance. Four autonomous first‐order ordinary‐differential equations are derived for the modulation of the amplitudes and phases of the two modes involved in the internal resonance when the higher mode is being excited by a principal parametric resonance. The modulation equations are used to determine the periodic oscillations and their stability. The force‐response curves exhibit the jump and saturation phenomena as well as a Hopf bifurcation, whereas the frequency‐response curves exhibit the jump phenomenon and supercritical and subcritical Hopf bifurcations. Limit‐cycle solutions of the modulation equations are found between the Hopf frequencies; they correspond to aperiodic motions of the liquid surface. All limit cycles deform and lose stability by either pitchfork or cyclic‐fold bifurcations as the excitation frequency or amplitude is varied. The pitchfork bifurcation breaks the symmetry of the limit cycles whereas the cyclic‐fold bifurcation causes cyclic jumps, which may result in a transition to chaos. Period‐three motions are found in a very narrow range of the excitation frequency.