Strongly unpredictable solutions of difference equations

It so happens that the line of oscillations in the classical theory of dynamical systems, which is founded by H.Poincar'e and G.Birkhoff was broken at Poisson stable motions. The next oscillations were considered as actors of chaotic processes. This article discusses the new type of oscillations, unpredictable sequences, the presence of which proves the existence of Poincare chaos. The sequence is defined as an unpredictable function on the set of integers. The results continue the description of chaos which isinitiated from a single motion, an unpredictable one. To demonstrate the effectiveness of the approach, the existence and uniqueness of the unpredictable solution for a quasilinear difference equation are proved. An example with numerical simulations is presented to illustrate the theoretical results. Since unpredictability is request for all coordinates of solutions, the concept of strong unpredictability can be useful for investigation of neural networks, brain activity, robotics, where complexity is related to optimization and effectiveness.