Chaotic behavior of Fuzzy Recurrent Models

We often use recurrent fuzzy rule bases for describing of weak-formalized dynamic processes. There exist Mamdani and TakagiSugeno (TS) types that are distinguished with consequents organization. We consider in this paper 0 and 1 order TS models and Mamdani model in respect to investigation of chaotic behavior. It is well known that chaotic properties of dynamic system don’t allow to realize of long-period predictions and to organize effective control of such systems. That is why important to answer if fuzzy model is chaotic or not using special properties of such model. We propose new approach for identification of chaotic behavior based on consequents properties. 1 Chaos in 0 order TS model In simplest case TS model has the following form R1: If 1 L xk = then ( ) k k x f x 1 1 = + , R2: If 2 L xk = then ( ) k k x f x 2 1 = + (1) .... RN: If 1 L xk = then ( ) k N k x f x = +1 , where x is a scalar state variable, i L are linguistic variables (terms), and the ( ) x f i are real-valued functions. If ( ) i i A x f = with constant Ai we have a TS model of 0 order. In this case the transition function 1 : + → k k x x f (2) of rule base (1) can be written as ∑ ∑