FUZZY CONTINUOUS DYNAMICAL SYSTEM: A MULTIVARIATE OPTIMIZATION TECHNIQUE

Subject Applied Mathematics
Title FUZZY CONTINUOUS DYNAMICAL SYSTEM: A MULTIVARIATE OPTIMIZATION TECHNIQUE
Author(s) Abhirup Bandyopadhyay and Samarjit Kar
Keywords Fuzzy continuous differential system, Numerical solution, Fuzzy initial value problem (FIVP), Multivariate optimization, Property of sufficiency of vertices (PSV), Ecological degradation model.
Abstract This paper presents a multivariate optimization technique for the numerical simulation of continuous dynamical systems whose parameters, functional forms and/or initial conditions are modeled by fuzzy distributions. Fuzzy differential equation (FDE) is interpreted by using the strongly generalized differentiability concept and is shown that by this concept any FDE can be transformed to a system of ordinary differential equations (ODEs). By solving the associate ODEs one can find solutions for FDE. This approach has an inherited drawback of increasing uncertainty at each instance of time generally with nonlinear functional forms. Here we present a methodology to numerically simulate interval calculus and implements a new approach to the numerical integration of fuzzy dynamical systems, where the propagation of imprecision as a fuzzy distribution in the phase space is solved by a constrained multivariate optimization technique. Numerical simulations of some fuzzy dynamical systems (viz. Lotka Volterra model, Lorenz model) are also reported. Finally ecological degradation in wetlands of India is modeled by fuzzy initial value problem and some sustainable solution is proposed. Let us consider the FDE with initial value condition: