Response analysis of fuzzy nonlinear dynamical systems

The transient and steady-state membership distribution functions (MDFs) of fuzzy response of a Duffing–Van der Pol oscillator with fuzzy uncertainty are studied by means of the fuzzy generalized cell mapping (FGCM) method. 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. Fuzzy response is characterized by its topology in the state space and its possibility measure of MDFs. The evolutionary orientation of MDFs is in accordance with invariant manifolds toward 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 MDFs from two sharp peaks of maximum possibility to one peak band around unstable manifolds.