Fractals: Dynamics in the Geometry

Modulo Periodic Poisson Stable Solutions of Quasilinear Differential Equations

“…In papers [21][22][23][24], quasilinear systems are with constant matrices of coefficients, and in our case, we research systems with periodic and, even with Poisson stable coefficients. Another significant novelty is the numerical simulation of the Poisson stable functions and solutions [7,9,13,14]. We believe that altogether, the present suggestions can shape a new interesting science direction, not only in the theoretical study of differential equations, but also they provide rich opportunities for applications in mechanics, electronics, artificial neural networks, neuroscience.…”

“…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].…”

“…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]. Unlike papers [7][8][9][10][13][14][15][16][17][18][19], the present research is busy with the new type of Poisson stable functions. Correspondingly, it is the first time in literature, when quasilinear equations with Poisson stable coefficients are under investigation.…”