Cyclic-fold Bifurcations Research Articles

Surface waves in closed basins under principal and autoparametric resonances

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.

Non-linear axisymmetric response of closed spherical shells to a radial harmonic excitation

A numerical-perturbation approach is used to study the axisymmetric dynamic response of closed spherical shells to an external harmonic excitation having a frequency near one of the natural frequencies of a flexural mode (i.e. primary resonance of a flexural mode) in the presence of a two-to-one internal (autoparametric) resonance between the excited mode and a lower flexural mode. The method of multiple-time scales is used to derive four first-order non-linear equations describing the evolution (modulation) of the amplitudes and phases of the interacting modes. A numerical scheme that combines a shooting technique and a Newton-Raphson procedure is used to calculate limit cycles of the evolution equations. The stability of these limit cycles is determined by using a Floquet analysis. As the excitation amplitude varies, the fixed-point solutions of the evolution equations exhibit the jump and saturation phenomena. They also undergo supercritical and subcritical Hopf bifurcations as the frequency of excitation varies; the supercritical Hopf-bifurcation frequency is lower than the subcritical one. As the excitation frequency increases above the supercritical Hopf-bifurcation value, the fixed-point solutions lose their stability and limit cycles are born. These limit cycles experience a symmetry-breaking (pitchfork) bifurcation followed by a cascade- of period-doubling bifurcations culminating in chaos. On the other hand, as the excitation frequency decreases below the subcritical Hopf-bifurcation value, the resulting limit cycle deforms and eventually loses its stability through a cyclic-fold bifurcation causing a transition to chaos. In the frequency range in between the above two chaotic regions the evolution equations possess limit-cycle solutions.