Unpredictable solutions of differential equations 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 We apply the topology of convergence on compact sets to define unpredictable functions [5, 6]. The topology is metrizable and easy for applications with integral operators. To demonstrate the effectiveness of the approach, the existence and uniqueness of the unpredictable solution for a delay differential equation is proved. As a corollary of the theorem, a similar assertion for a quasilinear ordinary differential equation is formulated. Examples with simulations illustrate the obtained results. Keywords: Unpredictable solutions; Unpredictable sequences; Topology of convergence on compact subsets; Poincaré chaos; Delay differential equations; Ordinary differential equations From the applications point of view, the theory of differential equations focuses on equilibria, periodic, and almost periodic oscillations. They meet the needs of any real world problem related to mechanics, electronics, economics, biology, etc., if one searches for regular and stable dynamics of an isolated motion. However, they are not sufficient for many modern and perspective demands of robotics, computer techniques, and the internet, and chaotic dynamics comprise constructive properties for applications. This is the reason why it is important to join the power of deterministic chaos with the immensely rich source of methods for differential equations. We contributed to this in our papers [1, 2] and the book [3], where a method of replication of chaos has been developed. It consists of the verification of ingredients of chaos such as sensitivity, transitivity, proximality and existence of infinitely many unstable regular motions [7, 12, 13, 17] for solutions of an equation with chaotic perturbation. This approach gives a very effective instrument for application of the accumulated knowledge in chaos research. Nevertheless, we are not glad with the necessity to check the presence of several ingredients. Therefore, in our opinion, unpredictable functions have become a panacea for simplification of chaos analysis through differential equations. In this paper, another step in adaptation of unpredictable functions to the theory of differential equations has been made. We apply convergence on compact subsets of the real axis to determine unpredictable functions for two reasons. The first reason is that the topology is easily metrizable, in particular, to the metric for ∗ Corresponding Author Tel.: +90 312 210 5355, Fax: +90 312 210 2972, E-mail: marat@metu.edu.tr 1 arXiv:1611.05084v1 [nlin.CD] 15 Nov 2016 Abstract 1 Introduction and preliminaries