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Nonlinear Hybrid Continuous/Discrete-Time Models

  • Book
  • © 2011

Overview

Authors:
  1. Marat Akhmet

    1. Dept. Mathematics, Middle East Technical University (METU), Ankara, Turkey

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  • Most general differential equations with piece-wise constant argument are investigated
  • New technique of analysis is used. A new model of blood pressure distribution is introduced with complete analysis Synchronization of quite identical integrate-and-fire oscillators is proved
  • General model of quite identical integrate-and-fire oscillators is constructed
  • Model of recurrent neural networks with continuous/discrete-time is investigated
  • Includes supplementary material: sn.pub/extras

Part of the book series: Atlantis Studies in Mathematics for Engineering and Science (ASMES, volume 8)

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Table of contents (10 chapters)

  1. Front Matter

    Pages i-xiii
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  2. Introduction

    • Marat Akhmet
    Pages 1-15

  3. Linear and Quasi-Linear Systems with Piecewise Constant Argument

    • Marat Akhmet
    Pages 17-32

  4. The Reduction Principle for Systems with Piecewise Constant Argument

    • Marat Akhmet
    Pages 33-48

  5. The Small Parameter and Differential Equations with Piecewise Constant Argument

    • Marat Akhmet
    Pages 49-69

  6. Stability

    • Marat Akhmet
    Pages 71-88

  7. The State-Dependent Piecewise Constant Argument

    • Marat Akhmet
    Pages 89-104

  8. Almost Periodic Solutions

    • Marat Akhmet
    Pages 105-119

  9. Stability of Neural Networks

    • Marat Akhmet
    Pages 121-132

  10. The Blood Pressure Distribution

    • Marat Akhmet
    Pages 133-174

  11. Integrate-and-Fire Biological Oscillators

    • Marat Akhmet
    Pages 175-199

  12. Back Matter

    Pages 203-216
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Keywords

About this book

The book is mainly about hybrid systems with continuous/discrete-time dynamics. The major part of the book consists of the theory of equations with piece-wise constant argument of generalized type. The systems as well as technique of investigation were introduced by the author very recently. They both generalized known theory about differential equations with piece-wise constant argument, introduced by K. Cook and J. Wiener in the 1980s. Moreover, differential equations with fixed and variable moments of impulses are used to model real world problems. We consider models of neural networks, blood pressure distribution and a generalized model of the cardiac pacemaker. All the results of the manuscript have not been published in any book, yet. They are very recent and united with the presence of the continuous/discrete dynamics of time. It is of big interest for specialists in biology, medicine, engineering sciences, electronics. Theoretical aspects of the book meet very strong expectations of mathematicians who investigate differential equations with discontinuities of any type.

Reviews

From the reviews:

“Hybrid systems are a recent concept in dynamical systems theory and have important and extensive applications. … The present book generalizes the concept of differential equations with piecewise constant argument by considering a more general type of equations and develops new methodological approaches to explore those challenging systems. … This book is well written and readable and will be extremely useful not only to expert readers but also to graduate and advanced undergraduate students.” (Meng Fan, Mathematical Reviews, January, 2013)

Authors and Affiliations

  • Dept. Mathematics, Middle East Technical University (METU), Ankara, Turkey

    Marat Akhmet

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