Differential Equations based on Fuzzy Rules

The purpose of this paper is to use the Fuzzy Set Theory, specially fuzzy rule-based systems, as a mathematical tool for modeling dynamical systems given by differential equation, whose direction field is f (IVP-f ). As fuzzy systems methodology produces input-output systems, that is, it produces m outputs from n inputs, one can see it as fr : R → R, where r is the number of rules of the rule base. Under certain assumptions, one can prove that the functions (fr) have good analytical properties. Beyond it, they can approximate theoretical functions (f). The main result of this paper states that the solutions x of an ODE whose direction field is f (IVP-f ) can be approximated by the solutions xr of an ODE whose direction field is fr (IVP-fr). By utilizing important theorems as the Lebesgue’s Dominated Convergence Theorem it was possible to prove the proposed theorem. Finally we use a numerical method (Runge-Kutta) to simulate solutions xr that approximates xr when n increases. The solutions xr approximates the solution x of IVP-f also provided {xr } n→∞ −→ xr r→∞ −→ x. Keywords— approximation theory, dynamical systems, fuzzy systems, initial value problem, numerical solutions.