Janusz Traple

Invariant measures related with Poisson driven stochastic differential equation

A Poisson driven stochastic differential equation generates a semigroup of operators (Pt)t[greater-or-equal, slanted]0 describing the evolution of measures along trajectories and a Markov operator P corresponding to the change of measures from a jump to jump. We show that the semigroup (Pt)t[greater-or-equal, slanted]0 has a finite invariant measure if and only if the operator P has the same property. The main result is applied to problems related with the existence and the dimension of invariant measures.

Weak almost periodic solutions of differential equations

A very important question in many practical situations is to known if there exists a mean value of a bounded solution x of a differential equation x’ =f(t, x), By the mean value we understand the limit lim,,, (l/T) ]lx(f) dr if it exists. Eberlein in [l] has found a class of functions which have the mean value-the weak almost periodic (wap) functions. A bounded continuous function x is weak almost periodic if the set of its translations (x,: s E R} is relatively compact in the weak topology of the space of bounded continuous functions on R. The purpose of this paper is to examine the properties of wap solutions of the differential equation. We demonstrate theorems on the existence of wap solution of the differential equation with the weak almost periodic right-hand side.

Differential equations with dynamical perturbations

Abstract A perturbation theorem is proved for ordinary differential equations whose righthand side depends on the trace of a dynamical system. This theorem unifies some classical results concerning equations with periodic and almost periodic right-hand sides. It is also applied to systems with pseudo-random perturbations.

Discrete processes with dynamical and semidynamical perturbations

Abstract An existence theorem is proved for a difference system governed by a dynamical system. This theorem unifies and generalizes some results concerning equations with periodic, almost periodic, weakly almost periodic, and pseudo-random right-hand sides. The notion of the natural extension due to Rohlin is used to obtain some results in the case of semidynamical systems.

Modelling of the reversed compound pendulum with a nonperiodically oscillating suspension

Abstract The vertical position is stable for a reversed compound pendulum provided its suspension is submitted to properly changing vibrations which can have periodic or weak almost periodic character. An abstract mathematical model describing phenomena of this kind by means of differential equations with weak almost periodic terms (wap functions) is considered. The results are of the homogenization type in the sense that the solutions of some nonautonomous systems of differential equations converge to the solutions of the homogenized autonomous (with respect to periodic variable) systems. The abstract general theorem is specified in some particular cases and some physical interpretation is given.