Parametric topological entropy and differential equations with time–dependent impulses

Abstract

Stimulated by a multiple solvability of periodic boundary value problems to differential equations with time–dependent impulses, we recall a related definition of parametric topological entropy for a sequence of continuous self–maps on a compact metric space. The main simple idea consists in replacing the iterates of a single map by the compositions of several various maps. For an equicontinuous countable family of self–maps on a compact connected polyhedron, we develop a lower estimate of this entropy in terms of the asymptotic Nielsen numbers of their compositions. This Ivanov–type equality is then applied, via the associated Poincaré translation operators, to differential equations with time–dependent impulses on tori. If the supporting space differs from a homotopy type of tori, then the situation becomes more delicate. Nevertheless, on compact connected punctured surfaces, we are still able to apply in a similar way the Artin braid group theory to planar differential equations with a finite number of homeomorphic impulses. Some further possibilities are commented in remarks and several illustrative examples are supplied.