DETERMINISTIC AND STOCHASTIC ANALYSES OF CHAOTIC AND OVERTURNING RESPONSES OF A SLENDER ROCKING OBJECT

The relationship between chaos and overturning in the rocking response of a rigid object under periodic excitation is examined from both det erministic and stochastic points of view. A stochastic extension of th e deterministic Melnikov function (employed to provide a lower bound f or the possible chaotic domain in parameter space) is derived by takin g into account the presence of random noise. The associated Fokker-Pla nck equation is derived to obtain the joint probability density functi ons in state space. It is shown that global behavior of the rocking mo tion can be effectively studied via the evolution of the joint probabi lity density function. A mean Poincare mapping technique is developed to average out noise effects on the chaotic response to reconstruct th e embedded strange attractor on the Poincare section. The close relati onship between chaos and overturning is demonstrated by examining the structure of the invariant manifolds. It is found that the presence of noise enlarges the boundary of possible chaotic domains in parameter space and bridges the domains of attraction of coexisting responses. N umerical results consistent with the Foguel alternative theorem, which discerns asymptotic stabilities of responses, indicate that the overt urning attracting domain is of the greatest strength. The presence of an embedded strange attractor (reconstructed using the mean Poincare m apping technique) indicates the existence of transient chaotic rocking response.