Strongly Unpredictable Oscillations of Hopfield-Type Neural Networks

Abstract

In this paper, unpredictable oscillations in Hopfield-type neural networks is under investigation. The motion strongly relates to Poincaré chaos. Thus, the importance of the dynamics is indisputable for those problems of artificial intelligence, brain activity and robotics, which rely on chaos. Sufficient conditions for the existence and uniqueness of exponentially stable unpredictable solutions are determined. The oscillations continue the line of periodic and almost periodic motions, which already are verified as effective instruments of analysis and applications for image recognition, information processing and other areas of neuroscience. The concept of strongly unpredictable oscillations is a significant novelty of the present research, since the presence of chaos in each coordinate of the space state provides new opportunities in applications. Additionally to the theoretical analysis, we have provided strong simulation arguments, considering that all of the assumed conditions are fulfilled.

Modulo Periodic Poisson Stable Solutions of Quasilinear Differential Equations

“…For this reason, we will consider the solution x(t) of the system (19), with initial values x 1 (0) = 1, x 2 (0) = 1 and x 3 (0) = 1. Using the inequality (18) one can obtain that x(t) − x(t) ≤ e −1.54 x(0) − x(0) for t ≥ 0. The last inequality shows that x(t) − x(t) decreases exponentially.…”

“…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].…”

“…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.…”

Modulo periodic Poisson stable solutions of quasilinear differential equations

“…For this reason, we will consider the solution x(t) of the system (19), with initial values x1 (0) = 1, x2 (0) = 1 and x3 (0) = 1. Using the inequality (18) one can obtain that x(t) − x(t) ≤ e −1.54 x(0) − x(0) for t ≥ 0. The last inequality shows that x(t) −…”

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