Effects of small random uncertainties on non-linear systems studied by the generalized cell mapping method

Abstract

The effects of small random uncertainties on certain non-linear systems under parametric and external excitation are studied by carrying out global analyses using the generalized cell mapping (GCM) method. The solutions to some fundamental problems in non-linear systems such as bifurcations, steady state solutions, limiting probability distribution of steady state solutions, domains of attraction and basin boundaries are obtained. It is found that system uncertainties increase the unpredictability of the system, and blur the geometrical structure of solutions of the deterministic system by making more multipledomicile cells near the boundaries of domains of attraction, and producing larger persistent groups representing stable steady solutions of the system. Furthermore, the noise fluctuations of the system induce early transition to chaos for a system undergoing a sequence of period doubling bifurcations to chaos, and destabilize first the solution with “weaker protection” for a system having multiple stable steady state solutions. This result is of importance to the reliability design of mechanical and structural systems that have multiple stable steady state solutions and are expected to endure uncertainties.