Simulating continuous fuzzy systems for fuzzy solution surfaces

Abstract

Our approach to simulation of a fuzzy ODE system is to evaluate the system with triangular fuzzy parameters, evaluating at the left/vertex/right supports and values in between. Solutions are presented as a graph(s) of the variable(s) of interest, with respect to time. With multiple fuzzy parameters, a solution graph is likely to become overloaded with superfluous information. Because of the cost of processing and plotting the superfluous information, critical support values may be skipped. We implement a reduction algorithm for determining solution boundaries, as trajectories are computed. Keeping aware of membership values, quantization considerations, and solution neighborhoods, we are able to collect a fraction of the data collected by brute force methods, and yet provide better coverage of fuzzy parameters. Three dimensional fuzzy solution trajectories are the second improvement we investigate. By including membership computation in our simulations, we produce fuzzy solution surface boundaries. These surfaces enclose the possible solution space. To make this process manageable, we combine this with boundary determination. In this paper we describe our method and present 3D solutions of two classical systems of ODE.