Recently we introduced the concept of Poincar´e chaos. In the present paper, by means of the Bebutov dynamical system, an unpredictable solution is considered as a generator of the chaos in a quasilinear system. The results can be easily extended to different types of differential equations. | Turk J Math (2017) 41: 254 – 266 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics doi: Research Article Existence of unpredictable solutions and chaos 1 Marat AKHMET1,∗, Mehmet Onur FEN2 Department of Mathematics, Middle East Technical University, Ankara, Turkey 2 Basic Sciences Unit, TED University, Ankara, Turkey Received: • Accepted/Published Online: • Final Version: Abstract: Recently we introduced the concept of Poincar´e chaos. In the present paper, by means of the Bebutov dynamical system, an unpredictable solution is considered as a generator of the chaos in a quasilinear system. The results can be easily extended to different types of differential equations. An example of an unpredictable function is provided. A proper irregular behavior in coupled Duffing equations is observed through simulations. Key words: Poincar´e chaos, unpredictable function, Poisson stability, Bebutov dynamical system, quasilinear differential equation, chaos control 1. Introduction The row of periodic, quasiperiodic, almost periodic, recurrent, and Poisson stable motions was successively 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 [16, 22, 31]. Our manuscript serves for proceeding the row and involving chaos as a purely functional object in nonlinear dynamics. In our previous paper [13], we introduced unpredictable motions based on Poisson stability. This time, we introduce the concept of an unpredictable function as an unpredictable point in Bebutov dynamics [29]. It was proved in [13] that an unpredictable point gives rise to the existence of chaos in the quasiminimal set. Thus, if one shows the existence of an unpredictable solution of an equation, then the chaos exists. The present study as well as our previous results concerning .
