Inertial Neural Networks with Unpredictable Oscillations

Abstract

In this paper, inertial neural networks are under investigation, that is, the second order differential equations. The recently introduced new type of motions, unpredictable oscillations, are considered for the models. The motions continue a line of periodic and almost periodic oscillations. The research is of very strong importance for neuroscience, since the existence of unpredictable solutions proves Poincaré chaos. Sufficient conditions have been determined for the existence, uniqueness, and exponential stability of unpredictable solutions. The results can significantly extend the role of oscillations for artificial neural networks exploitation, since they provide strong new theoretical and practical opportunities for implementation of methods of chaos extension, synchronization, stabilization, and control of periodic motions in various types of neural networks. Numerical simulations are presented to demonstrate the validity of the theoretical results.

Dynamics of Hopfield-Type Neural Networks with Modulo Periodic Unpredictable Synaptic Connections, Rates and Inputs

“…The method of included intervals , which was introduced in paper [ 26 ] and has been developed in [ 27 , 28 , 29 , 33 ], is a powerful instrument for verifying convergence properties. This technique has been applied in the study of continuous unpredictable solutions of Hopfield-type neural networks with delayed and advanced arguments [ 30 ] and in the study of discontinuous unpredictable solutions of impulsive neural networks with Hopfield structures [ 31 ].…”

“…Unpredictable functions were defined as unpredictable points in the Bebutov dynamical system [ 25 ], where the topology of convergence on compact sets of the real axis is used instead of the metric space. The use of such convergence significantly simplifies the problem of proving the existence of unpredictable solutions for differential equations and neural networks, and a new method of included intervals has been introduced and developed in several papers [ 26 , 27 , 28 , 29 , 30 , 31 ].…”

Unpredictable and Poisson Stable Oscillations of Inertial Neural Networks with Generalized Piecewise Constant Argument

“…Nevertheless, to provide complete information of the results, we prove two main assertions on both unpredictable and Poisson stable motions in the neural networks. Poisson stability is proved by the method of included intervals , which was introduced and developed in [ 27 , 28 , 29 ]. Note that, currently, this method remains the main way to prove convergence, due to its efficiency in theory differential equations.…”

“…For the first time, unpredictable motions were investigated for inertial neural networks in [ 29 ], that is, unpredictable functions were used as input data in system ( 1 ). The article considers the case when the reduced-order transformation formula includes all parameters, unlike the articles [ 10 , 14 , 15 , 18 , 19 ].…”

Unpredictable solutions of quasilinear differential equations with generalized piecewise constant arguments of mixed type

“…It can be utilized for various types of dynamical equations in the future. Moreover, we introduced and developed an entirely new method that shows how to verify the unpredictability property for solutions of differential equations and oscillations in neural networks [9,17,18,19,20,21,22,23]. It promises to be universal and can be applied for various types of differential equations.…”