different, and this reveals the presence of periodic motions in the quasi-minimal set. The unpredictable function as an unpredictable point of the Bebutov dynamics has been defined, and chaos in the quasi-minimal set of the function is verified. This is the first time in the literature that the existence of an unpredictable solution for a quasi-linear ordinary differential equation is proved. Moreover, through simulations it is demonstrated that cycles and non-cyclic Poisson stable orbits can coexist in a quasi-minimal set. The concept of unpredictable solutions can be useful for finding more delicate features of chaos in systems with complicated dynamics. Researches based on unpredictable functions may pave the way for the functional analysis of chaos to involve the operator theory results. Hopefully, our approach will give a basis for a deeper comprehension and possibility to unite different appearances of chaos. In this framework, the results can be developed for partial differential equations, integro-differential equations, functional differential equations, evolution systems, etc. 13 [4] G.R. Sell, Topological Dynamics and Ordinary Differential Equations, Van Nostrand Reinhold Com- pany, London, 1971. [3] R.L. Devaney, An Introduction to Chaotic Dynamical Systems, Addison-Wesley, United States of America, 1989. [2] T.Y. Li, J.A. Yorke, Period three implies chaos, Amer. Math. Monthly 87 (1975) 985-992. [1] M. Akhmet, M.O. Fen, Unpredictable points and chaos, arXiv:1602.00315 (submitted). 6 Conclusions References Figure 4: The stabilization of the 2 − periodic solution of (5.15) corresponding to the 2 − periodic orbit κ (1) ≈ 0 . 34459 , κ (2) ≈ 0 . 90497 of (5.11). The value ε = 0 . 072 is used and the control is switched on at t = ζ$_{25}$. 0 50 100 150 −0.5 0 0.5 1 t z 1 0 50 100 150 −1.5 −1 −0.5 0 0.5 t z 2