M. Akhmet has been supported by a grant (118F161) from TUB¨ ˙ITAK, the Scientific and Technological Research Council of Turkey. M. Tleubergenova and A. Zhamanshin have been supported in parts by the MES RK grant (AP05132573) ”Cellular neural networks with continuous/discrete time and singular perturbations” (2018-2020). Republic of Kazakhstan.

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In this study, the existence and uniqueness of the unpredictable solution for a non-homogeneous linear system of ordinary differential equations is considered. The hyperbolic case is under discussion. New properties of unpredictable functions are discovered. The presence of the solutions confirms the existence of Poincaré chaos. Simulations illustrating the chaos are provided. existence of unpredictable solutions simultaneously means the presence of Poincaré chaos, i.e., unpredictable solutions are "irregular". This makes the subject attractive for applications. Finally, we consider new properties of the functions. Sufficient conditions are provided such that a linear transformation of an unpredictable function is unpredictable, and it is proved that the sum of an unpredictable function and a periodic function is an unpredictable function.Ke α(t−s) g + (s + t n ) − g + (s) ds ≤ d t Ke α(t−s) ξds+

The existence of unpredictable motions in systems of quasilinear differential equations with hyperbolic linear part is rigorously proved. We make use of the topology of uniform convergence on compact sets and the contraction mapping principle to prove the existence of unpredictable motions. Appropriate examples with simulations that support the theoretical results are provided.

In this paper, modulo periodic Poisson stable functions have been newly introduced. Quasilinear differential equations with modulo periodic Poisson stable coefficients are under investigation. The existence and uniqueness of asymptotically stable modulo periodic Poisson stable solutions have been proved. Numerical simulations, which illustrate the theoretical results are provided.