Response analysis of a forced duffing oscillator with fuzzy uncertainty

Abstract

The transient and steady-state membership distribution functions (MDFs) of fuzzy response of a forced Duffing oscillator with fuzzy uncertainty are studied by means of the Fuzzy Generalized Cell Mapping (FGCM) method. The FGCMmethod is first introduced. A rigorous mathematical foundation of the FGCM is established with a discrete representation of the fuzzy master equation for the possibility transition of continuous fuzzy processes. The FGCM offers a very effective approach for solutions to the fuzzy master equation based on the min-max operator of fuzzy logic. Fuzzy response is characterized by its topology in the state space and its possibility measure of MDFs. The response topology is obtained based on the qualitative analysis of the FGCM involving the Boolean operation of 0 and 1. The evolutionary process of transient and steady-stateMDFs is determined by the quantitative analysis of the FGCM with the min-max calculations. It is found that the evolutionary orientation of MDFs is in accordance with invariant manifolds leading to invariant sets. In the evolutionary process of a steady-state fuzzy response with an increase of the intensity of fuzzy noise, a merging bifurcation is observed in a sudden change of the MDFs from two sharp peaks of most possibility to one peak band around unstable manifolds.