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On potential kernels associated with random dynamical systems

Hmissi M., Mokchaha F., Hmissi A.

Opuscula Mathematica

2015

Vol. 35, no. 4

499--515

Let (Θ, φ) be a continuous random dynamical system defined on a probability space (Ω, F, P) and taking values on a locally compact Hausdorff space E. The associated potential kernel V is given by [formula]. In this paper, we prove the equivalence of the following statements: 1. The potential kernel of (Θ, φ) is proper, i.e. V ƒ is x-continuous for each bounded, x-continuous function with uniformly random compact support. 2. (Θ, φ) has a global Lyapunov function, i.e. a function L : Ω x E → (0, ∞) which is x-continuous and L,(Θ tω, φ(t, ω)x) ↓0 as t ↑ ∞. In particular, we provide a constructive method for global Lyapunov functions for gradient-like random dynamical systems. This result generalizes an analogous theorem known for deterministic dynamical systems.

Random models of coupled inductors

Walczak J., Mazurkiewicz S.

Computer Applications in Electrical Engineering

2013

Vol. 11

321--327

This paper describes the method of determining the first two moments of the state equation solution. The presented method is applied to the analysis of the coupled coils system. Parameters of coupled coils are random variables. There are supplied by the ideal voltage sources. It is assumed that the forces are stochastic processes. The results are illustrated by the example.