Unpredictable sequences and Poincaré chaos Marat Akhmet 1 , $^{∗}$, Mehmet Onur Fen 2 $^{1}$Department of Mathematics, Middle East Technical University, 06800 Ankara, Turkey $^{2}$Basic Sciences Unit, TED University, 06420 Ankara, Turkey 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. Keywords: Unpredictable sequences; Unpredictable solutions; Quasilinear systems; Poincaré chaos. In the instrumental sense, discreteness has been the main object in chaos investigation. To check this, it is sufficient to recall definitions of chaos [1]-[3]. They are based on sequences and maps. One can say that stroboscopic observation of a motion was the single way to indicate the irregularity in a continuous dynamics. Embedding the research to the theory of differential equations requests definitions of chaos for continuous dynamics, which are not related to discreteness [4]-[7]. The research as well as the origin of the chaos [8] gave us strong arguments for the development of motions in classical dynamical systems theory [9] by proceeding behind Poisson stable points to unpredictable points [10]. Then, the dynamics has been specified such that a function that is bounded on the real axis is an unpredictable point [11]- [13]. In the papers [11]-[13], we have demonstrated that unpredictable functions are easy to be analyzed as solutions of differential equations. This paradigm is not completed, if one does not consider discrete equations. This is the reason why in the present paper we deliver discrete analogues for unpredictable functions calling them unpredictable sequences, and prove for the first time in the literature assertions on the existence and uniqueness of unpredictable solutions of recurrent equations. The role of the novelty cannot be underestimated for applications as well as for theoretical analysis, exceptionally for the modern development of computer technologies, software, and robotics [14, 15]. ∗ Corresponding Author. E-mail: marat@metu.edu.tr, Tel: +90 312 210 5355 1 1 Introduction Abstract arXiv:1704.06963v1 [nlin.CD] 23 Apr 2017