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Sigmoidal approximations of a nonautonomous neural network with infinite delay and Heaviside function

Peter E. Kloeden, V. M. Villarragut

TL;DR

This paper approximate a nonautonomous neural network with infinite delay and a Heaviside signal function by neural networks with sigmoidal signal functions and proves the existence of pullback attractors in both cases, and the convergence of the attractors of the sigmoid models to those of the Heavisid inclusion.

Abstract

In this paper, we approximate a nonautonomous neural network with infinite delay and a Heaviside signal function by neural networks with sigmoidal signal functions. We show that the solutions of the sigmoidal models converge to those of the Heaviside inclusion as the sigmoidal parameter vanishes. In addition, we prove the existence of pullback attractors in both cases, and the convergence of the attractors of the sigmoidal models to those of the Heaviside inclusion.

Sigmoidal approximations of a nonautonomous neural network with infinite delay and Heaviside function

TL;DR

This paper approximate a nonautonomous neural network with infinite delay and a Heaviside signal function by neural networks with sigmoidal signal functions and proves the existence of pullback attractors in both cases, and the convergence of the attractors of the sigmoid models to those of the Heavisid inclusion.

Abstract

In this paper, we approximate a nonautonomous neural network with infinite delay and a Heaviside signal function by neural networks with sigmoidal signal functions. We show that the solutions of the sigmoidal models converge to those of the Heaviside inclusion as the sigmoidal parameter vanishes. In addition, we prove the existence of pullback attractors in both cases, and the convergence of the attractors of the sigmoidal models to those of the Heaviside inclusion.
Paper Structure (12 sections, 26 theorems, 124 equations)

This paper contains 12 sections, 26 theorems, 124 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Set-valued topology and integrals
  5. Set-valued dynamical systems
  6. Existence and uniqueness of solutions
  7. The sigmoidal systems
  8. The differential inclusions
  9. Asymptotic behavior
  10. The differential inclusions
  11. The sigmoidal systems
  12. Comparison of attractors

Key Result

lemma thmcounterlemma

If $\{u^m\}_m\subset C_\gamma$ converges to $u\in C_\gamma$, then $\{u^m\}_m$ converges to $u$ uniformly on compact sets.

Theorems & Definitions (59)

  • lemma thmcounterlemma
  • proof
  • definition thmcounterdefinition
  • lemma thmcounterlemma
  • definition thmcounterdefinition
  • definition thmcounterdefinition
  • definition thmcounterdefinition
  • definition thmcounterdefinition
  • definition thmcounterdefinition
  • definition thmcounterdefinition
  • ...and 49 more