Unpredictable Oscillations for Hopfield-Type Neural Networks with Delayed and Advanced Arguments

Abstract

This is the first time that the method for the investigation of unpredictable solutions of differential equations has been extended to unpredictable oscillations of neural networks with a generalized piecewise constant argument, which is delayed and advanced. The existence and exponential stability of the unique unpredictable oscillation are proven. According to the theory, the presence of unpredictable oscillations is strong evidence for Poincaré chaos. Consequently, the paper is a contribution to chaos applications in neuroscience. The model is inspired by chaotic time-varying stimuli, which allow studying the distribution of chaotic signals in neural networks. Unpredictable inputs create an excitation wave of neurons that transmit chaotic signals. The technique of analysis includes the ideas used for differential equations with a piecewise constant argument. The results are illustrated by examples and simulations. They are carried out in MATLAB Simulink to demonstrate the simplicity of the diagrammatic approaches.

Modulo Periodic Poisson Stable Solutions of Quasilinear Differential Equations

“…The last inequality shows that x(t) − x(t) decreases exponentially. Consequently, the graph of the solution x(t) asymptotically approaches the Poisson stable solution x(t) of the system (19), as time increases. The Figure 5 demonstrates the coordinates of the solution x(t), which illustrate the Poisson stability of the system (19).…”

“…Consequently, the graph of the solution x(t) asymptotically approaches the Poisson stable solution x(t) of the system (19), as time increases. The Figure 5 demonstrates the coordinates of the solution x(t), which illustrate the Poisson stability of the system (19). In the Figure 6 the trajectory of the function x(t) is depicted.…”

“…It is worth noting that the simulation of the Poisson stable solution, x(t), is not possible, since the initial value is not known precisely. 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.…”

“…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.…”