Nonlinear Dyn (2014) 78:1221–1232 DOI 10.1007/s11071-014-1509-8 ORIGINAL PAPER Response analysis of fuzzy nonlinear dynamical systems Ling Hong · Jun Jiang · Jian-Qiao Sun Received: 23 January 2014 / Accepted: 29 May 2014 / Published online: 15 July 2014 © Springer Science+Business Media Dordrecht 2014 Abstract 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 map- ping (FGCM) method. A rigorous mathematical foun- dation of the FGCM is established with a discrete rep- resentation of the fuzzy master equation for the possi- bility transition of continuous fuzzy processes. Fuzzy response is characterized by its topology in the state space and its possibility measure of MDFs. The evo- lutionary orientation of MDFs is in accordance with invariant manifolds toward invariant sets. In the evolu- tionary 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. Keywords Fuzzy uncertainty · Fuzzy response · Possibility measure · Membership distribution function · Generalized cell mapping L. Hong (B ) · J. Jiang State Key Lab for Strength and Vibration, Xi’an Jiaotong University, Xi’an 710049, China e-mail: hongling@mail.xjtu.edu.cn J.-Q. Sun School of Engineering, University of California at Merced, Merced, CA 95344, USA 1 Introduction Engineering systems are often subjected to uncertain- ties (noise) that are associated with the lack of precise knowledge of system parameters and operating condi- tions and that are originated from variability in man- ufacturing processes [1, 2]. The uncertainty can have significant influence on the dynamic response and the reliability of the system, which has no analogue in the deterministic counterpart, even under small noise uncertainties. For example, noise in nonlinear systems can induce chaos [3, 4], attractor and basin hopping [5, 6], complexity [7], bifurcations [8, 9]. In general, uncertainty is often mathematically modeled as a ran- dom variable or a fuzzy set leading to the two cate- gories of stochastic and fuzzy dynamics. Their major goals are to deal with the response analysis for dynam- ical systems with stochastic and fuzzy uncertainties [10–12]. In the theory of fuzzy dynamics, the main method of system response (solution) is to find the membership distribution functions (MDFs) as a func- tion of time using fuzzy master equation (FME) [12]. FME tells how the MDF evolves in time similarly to how the Fokker–Planck equation (FPE) gives the time evolution of the probability density functions (PDFs) for stochastic dynamics [10]. This paper proposes the fuzzy generalized cell-mapping (FGCM) method to analyze the response of nonlinear dynamical systems with fuzzy uncertainties. Specifically, we are interested in a nonlinear dynamical system whose response is a fuzzy process, and study the transient and steady-state 123