Topological entropy of composition and impulsive differential equations satisfying a uniqueness condition

Abstract

The impulsive differential equations are examined via the associated Poincaré translation operators in terms of topological entropy. The crucial role is played by the entropy analysis of the compositions of Poincaré’s operators with the impulsive maps. For the scalar (one-dimensional) problems, the lower entropy estimations can be effectively obtained by means of horseshoes. For the vector (higher-dimensional) problems, the situation becomes more delicate and requires rather sophisticated techniques. Five main theorems are presented about a positive topological entropy (i.e. topological chaos) for given impulsive problems. For vector linear homogeneous differential equations with constant coefficients and isometric impulses, the zero entropy is deduced under commutativity restrictions imposed on the components of a mentioned composition. Several illustrative examples and numerical simulations are supplied.