Nonlinear Rocking Motions. I: Chaos under Noisy Periodic Excitations

Abstract

The effects of low-intensity random perturbations on the stability of chaotic response of rocking objects under otherwise periodic excitations are examined analytically and via simulations. A stochastic Melnikov process is developed to identify a lower bound for the domain of possible chaos. An average phase-flux rate is computed to demonstrate noise effects on transitions from chaos to overturning. A mean Poincaré mapping technique is employed to reconstruct embedded chaotic attractors under random noise on Poincaré sections. Extensive simulations are employed to examine chaotic behaviors from an ensemble perspective. Analysis predicts that the presence of random perturbations enlarges the possible chaotic domain and bridges the domains of attraction of coexisting attractors. Numerical results indicate that overturning attractors are of the greatest strength among coexisting ones; and, because of the weak stability of chaotic attractors, the presence of random noise will eventually lead chaotic rocking responses to overturning. Existence of embedded strange attractors (reconstructed using mean Poincaré maps) indicates that rocking objects may experience transient chaos prior to overturn.