Unpredictable sequences and Poincar\'e chaos

Abstract

To make research of chaos more friendly with discrete equations, we introduce the concept of an unpredictable sequence as a specific unpredictable function on the set of integers. It is convenient to be verified as a solution of a discrete equation. This is rigorously proved in this paper for quasilinear systems, and we demonstrate the result numerically for linear systems in the critical case with respect to the stability of the origin. The completed research contributes to the theory of chaos as well as to the theory of discrete equations, considering unpredictable solutions.