Existence of Unpredictable Solutions and Chaos Marat Akhmet a, ∗ and Mehmet Onur Fen a a Department of Mathematics, Middle East Technical University, 06800, Ankara, Turkey $^{∗}$Corresponding Author Tel.: +90 312 210 5355, Fax: +90 312 210 2972, E-mail: marat@metu.edu.tr In paper [1] unpredictable points were introduced based on Poisson stability, and this gives rise to the existence of chaos in the quasi-minimal set. This time, an unpredictable function is determined as an unpredictable point in the Bebutov dynamical system. The existence of an unpredictable solution and consequently chaos of a quasi-linear system of ordinary differential equations are verified. This is the first time that the description of chaos is initiated from a single function, but not on a collection of them. The results can be easily extended to different types of differential equations. An application of the main theorem for Duffing equations is provided. Keywords: Unpredictable point; Unpredictable function; Poisson stability; Bebutov dynamical system; Quasi-linear differential equation; Unpredictable solution; Chaotic dynamics; Chaos control The row of periodic, quasi-periodic, almost periodic, recurrent, Poisson stable motions had been succes- sively developed in the theory of dynamical systems. Then, chaotic dynamics started to be considered, which is not a single motion phenomenon, since a prescribed set of motions is required for a definition [2, 3]. Our manuscript serves for proceeding the row and involving chaos as a purely functional object in nonlinear dynamics. In our previous paper [1], we introduced unpredictable motions based on Poisson stability. This time, we introduce the concept of an unpredictable function as an unpredictable point in the Bebutov dynamics [4]. It was proved in [1] that an unpredictable point gives rise to the existence of chaos in the quasi-minimal set. Thus, if one shows the existence of an unpredictable solution of an equation, then the chaos exists. The result of the present study as well as our previous paper concerning replication of chaos [5] support the opinion of Holmes [6] that the theory of chaos has to be a part of the theory of differential equations. Since the main body of the results on chaotic motions have been formulated in terms of differential and difference equations, we may suggest that all these achievements have to be embedded and developed ∗ Corresponding Author Tel.: +90 312 210 5355, Fax: +90 312 210 2972, E-mail: marat@metu.edu.tr 1 arXiv:1602.02614v1 [nlin.CD] 8 Feb 2016 1 Introduction Abstract