Period-Doubling Bifurcations and Modulated Motions in Forced Mechanical Systems

Weakly nonlinear and harmonically forced two-degree-of-freedom mechanical systems with cubic nonlinearities and exhibiting internal resonance are studied for their steady-state solutions. Using the method of averaging, the system is transformed into a four-dimensional autonomous system in amplitude and phase variables. It is shown that for low damping the constant solutions of the averaged equations are unstable over some interval in detuning. The transition in stability is due to the Hopf bifurcation and the averaged system performs limit cycle motions near the critical value of detuning. The bifurcated periodic solutions are constructed via a numerical algorithm and their stability is analyzed using Floquet theory. It is seen that the periodic branch connects two Hopf points in the steady-state response curves. For sufficiently small damping, the averaged equations, therefore, have stable limit cycles where the constant solutions are unstable. Reduction in damping results in destabilization of these periodic solutions with one Floquet multiplier leaving the inside of the unit circle through −1. This leads to period-doubling bifurcations in the averaged equations. There is, thus, an interval in detuning parameter over which the constant and the periodic solutions are unstable and the period-doubled solutions are stable. For small enough damping there are cascades of period-doubling bifurcations that ultimately lead to chaotic motions. Some of these sequences seem to be compatible with the Feigenbaum Universality Constant.