On existence of periodic and almost periodic motions in nonlinear dynamical systems over locally compact Abelian groups

摘要

The existence of periodic and almost periodic solutions of abstract random equations describing dynamical systems over locally compact Abelian groups(LCAG) is considered. First some notation and terminology to be used later is outlined, mainly in the case of periodic stochastic processes over LCAG's. Let G be a topological LCAG, H any closed, normal subgroup of G. H-periodic stochastic processes in weak sense are introduced i.e. stochastic processes ξ(t,ω), t∈G, ω∈Ω, which are constant on each coset of H (i.e. P(τX)=P(τ) V τ∈G, x∈H). Our systems are described by a class of nonlinear integral equations on LCAG G with a Haar measure denoted by μ(·). A nonlinear random integral equation of Volterra-Stieltjes type on G is an equation of the form x(t,ω)=z(t,ω)+∫_Gk(dτ,ω))f(x(tτ~(-1)ω),)), t∈G, ω∈Ω The existence of H-periodic solutions of random nonlinear systems, subjected to H-periodic forcing function is studied. We use the fixed-point theorems for contracting maps and the Leray-Schauder fixed-point index in the case of compact operators perturbed by contracting ones to solve the equations in some reflexive Banach spaces of stochastic processes. Our technique is similar to the describing function method. The setting is general enough to allow continuous systems as well as to discrete ones. The present results are satisfactory for many applications.To demonstrate the applicability of the method advanced, a specific example of a mechanical dynamic system with discrete time is considered.