Quasilinear differential equations with strongly unpredictable solutions

Abstract

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.

Unpredictable Solutions of Linear Impulsive Systems

“…We have already obtained the essential results for ordinary linear and quasilinear systems, as well as discrete equations [17,18]. One of the main contributions of the present study is the presentation of the novel definitions of discontinuous unpredictable function and unpredictable discrete set.…”

“…For instance, some topological properties of Poincaré chaos were considered by Miller [14], and Thakur and Das [15]. Various types of differential equations with unpredictable solutions were considered in the papers [2,12,[16][17][18] and in the book [19]. Recently it is proved in paper [20] that unpredictable motions take place also in random processes.…”

Modulo Periodic Poisson Stable Solutions of Quasilinear Differential Equations

“…The unpredictable point of the Bebutov dynamics is the unpredictable function. In papers [7][8][9][10][11][12][13][14][15], we provided a dynamical method, how to construct Poisson stable functions. Deterministic and stochastic dynamics have been used.…”

“…Deterministic and stochastic dynamics have been used. Deterministically unpredictable functions have been constructed as solutions of hybrid systems, consisting of discrete and differential equations [9,13,14], and randomly they are results of the Bernoulli process inserted into a linear differential equation [7,10,16]. Unpredictable oscillations in neural networks have been researched in [7,13,[17][18][19].…”

“…In papers [8][9][10]14] and books [7,13], discussing existence of unpredictable solutions, we have developed a new method how to approve Poisson stable solutions, since unpredictable functions are a subset of Poisson stable functions, and to verify the unpredictability one must check, if the Poisson stability is valid. The method is distinctly different than the comparability method by character of recurrence, which was introduced in [20] and later has been realized in several articles [21][22][23][24][25][26][27].…”