Neural networks with piecewise constant argument and impact activation

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Tezin Türü: Doktora

Tezin Yürütüldüğü Kurum: Orta Doğu Teknik Üniversitesi, Fen Edebiyat Fakültesi, Matematik Bölümü, Türkiye

Tezin Onay Tarihi: 2011

Öğrenci: ENES YILMAZ

Danışman: MARAT AKHMET

Özet:

This dissertation addresses the new models in mathematical neuroscience: artificial neural networks, which have many similarities with the structure of human brain and the functions of cells by electronic circuits. The networks have been investigated due to their extensive applications in classification of patterns, associative memories, image processing, artificial intelligence, signal processing and optimization problems. These applications depend crucially on the dynamical behaviors of the networks. In this thesis the dynamics are presented by differential equations with discontinuities: differential equations with piecewise constant argument of generalized type, and both impulses at fixed moments and piecewise constant argument. A discussion of the models, which are appropriate for the proposed applications, are also provided. Qualitative analysis of existence and uniqueness of solutions, global asymptotic stability, uniform asymptotic stability and global exponential stability of equilibria, existence of periodic solutions and their global asymptotic stability for these networks are obtained. Examples with numerical simulations are given to validate the theoretical results. All the properties are rigorously approved by using methods for differential equations with discontinuities: existence and uniqueness theorems; stability analysis through the Second Lyapunov method and linearization. It is the first time that the problem of stability with the method of Lyapunov functions for differential equations with piecewise constant argument of generalized type is investigated. Despite the fact that these equations are with deviating argument, stability criteria are merely found in terms of Lyapunov functions.