Figure 3: The orbit of system (5.10) with u$_{0}$ = 0 , x$_{0}$ = 1 , and y$_{0}$ = 1 . The figure manifests that the orbit behaves chaotically near the 14 -periodic orbit of (5.9). 6 Conclusion The starting point for the present research is the unpredictable point [10], a new object for the dynamical systems theory founded by Poincaré and Birkhoff [8, 9]. In the paper [10], we developed the Poisson stability of a motion to unpredictability such that a new type of chaos, the Poincaré chaos, has been obtained. It has become clear that the concept can be easily extended to the object of analysis in the theory of differential equations, considering unpredictable functions as points moving by shifts of the time argument [11]-[13]. Therefore, in our opinion, a new field to analyze in the theory of differential equations has been discovered. Since many results of differential equations have their counterparts in discrete equations, it is easy to suppose that theorems on the existence of unpredictable solutions can be proved for discrete equations. The present paper is the first one to realize the paradigm. The existence and uniqueness theorem for quasilinear difference equations has been proved, when the perturbation is an unpredictable sequence. This is visualized as Poincaré chaos in simulations. References [1] T.Y. Li, J.A. Yorke, Period three implies chaos, Am. Math. Mon. 82 (1975) 985–992. [2] R.L. Devaney, An Introduction to Chaotic Dynamical Systems, Addison-Wesley Publishing Com- pany, United States of America, 1987. [3] S. Wiggins, Global Bifurcation and Chaos: Analytical Methods, Springer-Verlag, New York, Berlin, 1988. [4] M. Akhmet, Devaney chaos of a relay system, Commun. Nonlinear Sci. Numer. Simulat. 14 (2009) 1486–1493. 9