Poincar\'e chaos and unpredictable functions

TL;DR

This paper advances the theory of Poincaré chaos by constructing continuous unpredictable functions and exploring their role in differential and discrete systems, supported by symbolic dynamics and logistic map analysis.

Contribution

It introduces the construction of continuous unpredictable functions and initiates the development of unpredictable solutions theory for differential and discrete equations.

Findings

Unpredictable orbits for symbolic dynamics and logistic map are obtained.

Construction of unpredictable functions and Poisson functions is demonstrated.

Preliminary results on chaos existence in differential and hybrid systems are presented.

Abstract

The results of this study are continuation of the research of Poincar\'e chaos initiated in papers (Akhmet M, Fen MO. Commun Nonlinear Sci Numer Simulat 2016;40:1-5; Akhmet M, Fen MO. Turk J Math, doi:10.3906/mat-1603-51, accepted). We focus on the construction of an unpredictable function, continuous on the real axis. As auxiliary results, unpredictable orbits for the symbolic dynamics and the logistic map are obtained. By shaping the unpredictable function as well as Poisson function we have performed the first step in the development of the theory of unpredictable solutions for differential and discrete equations. The results are preliminary ones for deep analysis of chaos existence in differential and hybrid systems. Illustrative examples concerning unpredictable solutions of differential equations are provided.