Poincaré chaos and unpredictable functions Marat Akhmet a, $^{∗}$, Mehmet Onur Fen b $^{a}$Department of Mathematics, Middle East Technical University, 06800 Ankara, Turkey $^{b}$Basic Sciences Unit, TED University, 06420 Ankara, Turkey The results of this study are continuation of the research of Poincaré 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, con- tinuous 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. Keywords: Poincaré chaos; Unpredictable function; Unpredictable solutions; Unpredictable sequence. It is useless to say that the theory of dynamical systems is a research of oscillations, and the latest motion of the theory is the Poisson stable trajectory [1]. In paper [2] inspired by chaos investigation we have introduced a new type of oscillation, next to the Poisson stable one, and called the initial point for it the unpredictable point and the trajectory itself the unpredictable orbit . These novelties make a connection of the homoclinic chaos and the latest types of chaos possible through the concept of unpredictability , that is sensitivity assigned to a single orbit. Thus, the Poincaré chaos concept has been eventually shaped in our paper [2]. We have also determined the unpredictable function on the real axis as an unpredictable point of the Bebutov dynamics in paper [3] to involve widely differential and discrete equations to the chaos investigation. Nonetheless, we need a more precise description of what one understands as unpredictable function. The present research is devoted to this constructive duty. One can say that the analysis became productive, since we have learnt that the unpredictable functions can be bounded, and this is also true for Poisson functions, newly introduced in this paper. 1 Introduction Abstract arXiv:1608.07301v1 [nlin.CD] 25 Aug 2016 ∗ Corresponding Author Tel.: +90 312 210 5355, Fax: +90 312 210 2972, E-mail: marat@metu.edu.tr 1