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.
Highlights
There are hybrid neural networks, which are neither continuous-time nor purely discrete-time, and among them are dynamical systems with impulses and models with piecewise constant arguments [1,2,3,4,5,6,7,8,9,10]
Hopfield-type neural networks are effective at adaptive pattern recognition and vision and image processing [20,21,22]
It is known that unpredictable oscillations cause chaotic behavior [29,30,31,32,33,34,35,36,37,38,39,40,41]
Summary
There are hybrid neural networks, which are neither continuous-time nor purely discrete-time, and among them are dynamical systems with impulses and models with piecewise constant arguments [1,2,3,4,5,6,7,8,9,10]. A new model of Hopfield-type neural networks with an unpredictable input-output, as well as a delayed and advanced generalized piecewise constant argument is proposed. The most developed are unpredictable oscillations, which were introduced and developed in [31,32,33,34,35,36,37,38,39] This is the first time unpredictable oscillations have been considered for neural networks with a generalized-type piecewise constant argument. To the best of our knowledge, there have been very few results on the dynamical behavior of Hopfield-type neural networks with piecewise constant arguments [16,17,18,19,26,27]. We improve on previous methods by considering unpredictable inputs, which allow studying the distribution of chaotic signals in neural networks
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