Three-mode interactions in harmonically excited systems with quadratic nonlinearities

An investigation is presented of the response of a three-degree-of-freedom system with quadratic nonlinearities and the autoparametric resonances ω₃≈2ω₂ and ω₂≈2ω₁ to a harmonic excitation of the third mode, where the ω_m are the linear natural frequencies of the system. The method of multiple scales is used to determine six first-order nonlinear ordinary differential equations that govern the time variation of the amplitudes and phases of the interacting modes. The fixed points of these equations are obtained and their stability is determined. For certain parameter values, the fixed points are found to lose stability due to Hopf bifurcations and consequently the system exhibits amplitude-and phase-modulated motions. Regions where the amplitudes and phases display periodic, quasiperiodic, and chaotic time variations and hence regions where the overall system motion is periodically, quasiperiodically, and chaotically modulated are determined. Using various numerical simulations, we investigated nonperiodic solutions of the modulation equations using the amplitude F of the excitation as a control parameter. As the excitation amplitude F is increased, the fixed points of the modulation equations exhibit an instability due to a Hopf bifurcation, leading to limit-cycle solutions of the modulation equations. As F is increased further, the limit cycle undergoes a period-doubling bifurcation followed by a secondary Hopf bifurcation, resulting in either a two-period quasiperiodic or a phase-locked solution. As F is increased further, there is a torus breakdown and the solution of the modulation equations becomes chaotic, resulting in a chaotically modulated motion of the system.