Surface waves in closed basins under principal and autoparametric resonances

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.