Uncertain dynamical systems defined by pseudomeasures

Abstract

This paper deals with uncertain dynamical systems in which predictions about the future state of a system are assessed by so called pseudomeasures. Two special cases are stochastic dynamical systems, where the pseudomeasure is the conventional probability measure, and fuzzy dynamical systems in which the pseudomeasure is a so called possibility measure.New results about possibilistic systems and their relation to deterministic and to stochastic systems are derived by using idempotent pseudolinear algebra.By expressing large deviation estimates for stochastic perturbations in terms of possibility measures, we obtain a new interpretation of the Freidlin-Wentzell quasipotentials for stochastic perturbations of dynamical systems as invariant possibility densities.

Large deviations from the thermodynamic limit in globally coupled maps

“…(A. 12) In the light of [49] this can be called a stochastic possibilistic dynamics. If we start from V [N |ψ] N = h(ψ) and iterate eq.…”

“…Instead of going into the details of these arguments again with an adjusted set of preconditions, we pragmatically content ourselves here with showing how to find a quasipotential if it exists, which can be done by studying the algebraic consequences of Theorem 3 . The proofs of the following statements can be found in [49] Define the least n-action (in [49] called the n-step transition pseudodensity) between χ 0 ∈ P(M) and χ n ∈ P(M) as S n (χ n |χ 0 ) := inf…”

“…The set R s F is related to and in many cases identical to the union of attractors of the map F ; for details about this relation see [49].…”

Large deviation principle with generated pseudo measures

“…A powerful approach to LDP is through the convergence of the family of probability measures to idempotent sup-measure as it was given in [20]. In the area of fuzzy sets, idempotent measures are known as possibility measures (see [24,7,17,12]). …”

Idempotent version of the Fréchet contingency array problem

“…[7], [8], [9], [1], [28] and references therein) and for particular classes of uncertain dynamical systems (see e.g. [17]). Last but not least it is proved in [33] that the Fréchet array problem is max-plus linear which means that it is linear when addition is max and multiplication is +.…”