Evolution of systems under fuzzy dynamic laws (Q1267483)

The paper deals with a description of the global behavior of fuzzy systems governed by the relationship (sup-min convolution) \[ \mu(x,t+ \delta)= \sup_x \biggl(\min\bigl( P_\delta (x'\to x),\mu(x',t) \bigr)\biggr). \] Where \(\mu(x,t)\) and \(\mu(x,t+\delta)\) are fuzzy states of the system at two specific time instants \((t\) and \(t+\delta\), respectively) while \(P_\delta (x'\to x)\) is the transition possibility. In parallel to the master equations of stochastic dynamics, the authors introduce equations of fuzzy dynamics where \[ P_\delta (x'\to x)= p(v,x-v \delta,t) \] with \(v=(x-x')/ \delta\). Subsequently they propose a description of fuzzy systems with the use of flows and vector field. Two classes of fuzzy systems are defined and analyzed in detail.