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Density Evolution Under Delayed Dynamics

An Open Problem

  • Book
  • © 2020

Overview

Authors:
  1. Jérôme Losson

    1. BC Partners, London, UK

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  2. Michael C. Mackey

    1. Department of Physiology, McGill University, Montreal, Canada

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  3. Richard Taylor

    1. Department of Mathematics and Statistics, Thompson Rivers University, Kamloops, Canada

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  4. Marta Tyran-Kamińska

    1. Institute of Mathematics, University of Silesia, Katowice, Poland

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  • Describes an open and difficult mathematical problem, offering a new research field for young investigators
  • Gives a brief but unified contrasting description of examination of dynamics from a trajectory or an ensemble point of view
  • Highlights all of the problems attendant to the development of an appropriate measure to examine ergodic behavior in infinite-dimensional dynamical systems
  • Presents possible applications of functional calculus to infinite dimensional dynamical systems
  • Offers a motivated area of research combining different fields of mathematics

Part of the book series: Fields Institute Monographs (FIM, volume 38)

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Table of contents (9 chapters)

  1. Front Matter

    Pages i-ix
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  2. Introduction and Background to Density Evolution Problems

    1. Front Matter

      Pages 1-1
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    2. Introduction and Motivation

      • Jérôme Losson, Michael C. Mackey, Richard Taylor, Marta Tyran-Kamińska
      Pages 3-8

    3. Density Evolution in Systems with Finite-Dimensional Dynamics

      • Jérôme Losson, Michael C. Mackey, Richard Taylor, Marta Tyran-Kamińska
      Pages 9-16

  3. Illustrating the Problem and Making It Precise for Differential Delay Equations

    1. Front Matter

      Pages 17-17
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    2. Dynamics in Ensembles of Differential Delay Equations

      • Jérôme Losson, Michael C. Mackey, Richard Taylor, Marta Tyran-Kamińska
      Pages 19-27

    3. The Problem

      • Jérôme Losson, Michael C. Mackey, Richard Taylor, Marta Tyran-Kamińska
      Pages 29-41

  4. Possible Analytical Approaches

    1. Front Matter

      Pages 43-43
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    2. The Hopf Functional Approach

      • Jérôme Losson, Michael C. Mackey, Richard Taylor, Marta Tyran-Kamińska
      Pages 45-78

    3. The Method of Steps

      • Jérôme Losson, Michael C. Mackey, Richard Taylor, Marta Tyran-Kamińska
      Pages 79-95

  5. Part IV

    1. Front Matter

      Pages 97-97
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    2. Turning a Differential Delay Equation into a High-Dimensional Map

      • Jérôme Losson, Michael C. Mackey, Richard Taylor, Marta Tyran-Kamińska
      Pages 99-114

    3. Approximate “Liouville-Like” Equation and Invariant Densities for Delay Differential Equations

      • Jérôme Losson, Michael C. Mackey, Richard Taylor, Marta Tyran-Kamińska
      Pages 115-130

    4. Summary and Conclusions

      • Jérôme Losson, Michael C. Mackey, Richard Taylor, Marta Tyran-Kamińska
      Pages 131-131

  6. Back Matter

    Pages 133-138
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Keywords

About this book

This monograph has arisen out of a number of attempts spanning almost five decades to understand how one might examine the evolution of densities in systems whose dynamics are described by differential delay equations. Though the authors have no definitive solution to the problem, they offer this contribution in an attempt to define the problem as they see it, and to sketch out several obvious attempts that have been suggested to solve the problem and which seem to have failed. They hope that by being available to the general mathematical community, they will inspire others to consider–and hopefully solve–the problem. Serious attempts have been made by all of the authors over the years and they have made reference to these where appropriate. 

Reviews

“The book is devoted to the problem of evolution of densities in dynamical systems described by delay differential equations. … The book is interesting and introduces the reader to the problem of the evolution of the densities. Indeed, formulating (if possible) such an evolution one is able to guess the asymptotic behavior of the solutions with known density of their initial values. The examples in the book are carefully selected in order to illustrate the main theoretical results.” (George Karakostas, zbMATH 1462.37001, 2021)

Authors and Affiliations

  • BC Partners, London, UK

    Jérôme Losson

  • Department of Physiology, McGill University, Montreal, Canada

    Michael C. Mackey

  • Department of Mathematics and Statistics, Thompson Rivers University, Kamloops, Canada

    Richard Taylor

  • Institute of Mathematics, University of Silesia, Katowice, Poland

    Marta Tyran-Kamińska

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