# Unpredictable sequences and Poincar\'e chaos

## TL;DR

This paper introduces the concept of unpredictable sequences in discrete equations, rigorously proves their existence in quasilinear systems, and demonstrates their numerical occurrence in linear systems, advancing chaos theory and discrete equations.

## Contribution

It defines unpredictable sequences and proves their solutions exist in quasilinear systems, linking chaos theory with discrete equations.

## Key findings

- Unpredictable sequences are solutions to certain discrete equations.
- Existence of unpredictable solutions is rigorously proved for quasilinear systems.
- Numerical demonstrations show unpredictable solutions in linear systems.

## 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.