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

Akhmet, Marat
Tleubergenova, Madina
Nugayeva, Zakhira
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 Poincare 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.


Approximation of Abstract Differential Equations
Karasözen, Bülent; Guidetti, David (Springer , 2004-02-01)
In this work we study the approximation of solutions of abstract retarded functional differential equations (ARFDE) with unbounded delay by means of solutions of ARFDE with bounded delay. As consequence we establish some results of stability and existence of periodic solutions for the first one.
Unpredictable solutions of linear differential and discrete equations
Akhmet, Marat; Tleubergenova, Madina; Zhamanshin, Akylbek (2019-01-01)
The existence and uniqueness of unpredictable solutions in the dynamics of nonhomogeneous linear systems of differential and discrete equations are investigated. The hyperbolic cases are under discussion. The presence of unpredictable solutions confirms the existence of Poincare chaos. Simulations illustrating the chaos are provided.
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Belenli Akbaş, Mine; Kaya Merdan, Songül; Rebholz, Leo G.; Department of Mathematics (2016)
In this dissertation, efficient and reliable numerical algorithms for approximating solutions of multiphysics flow problems are investigated by using numerical methods. The interaction of multiple physical processes makes the systems complex, and two fundamental difficulties arise when attempting to obtain numerical solutions of these problems: the need for algorithms that reduce the problems into smaller pieces in a stable and accurate way and for large (sometimes intractable) amount of computational resou...
Unpredictable Solutions of Linear Impulsive Systems
Akhmet, Marat; Fen, Mehmet Onur; Nugayeva, Zakhira (2020-10-01)
We consider a new type of oscillations of discontinuous unpredictable solutions for linear impulsive nonhomogeneous systems. The models under investigation are with unpredictable perturbations. The definition of a piecewise continuous unpredictable function is provided. The moments of impulses constitute a newly determined unpredictable discrete set. Theoretical results on the existence, uniqueness, and stability of discontinuous unpredictable solutions for linear impulsive differential equations are provid...
Stability analysis of recurrent neural networks with piecewise constant argument of generalized type
Akhmet, Marat; Yılmaz, Elanur (2010-09-01)
In this paper, we apply the method of Lyapunov functions for differential equations with piecewise constant argument of generalized type to a model of recurrent neural networks (RNNs). The model involves both advanced and delayed arguments. Sufficient conditions are obtained for global exponential stability of the equilibrium point. Examples with numerical simulations are presented to illustrate the results.
Citation Formats
M. Akhmet, M. Tleubergenova, and Z. Nugayeva, “Unpredictable Oscillations for Hopfield-Type Neural Networks with Delayed and Advanced Arguments,” MATHEMATICS, pp. 0–0, 2021, Accessed: 00, 2021. [Online]. Available: https://hdl.handle.net/11511/89536.