Density Evolution Under Delayed Dynamics

1. Front Matter

   Pages i-ix

   PDF

2. Introduction and Background to Density Evolution Problems

   1. Front Matter

      Pages 1-1

      PDF

   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

      PDF

   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

      PDF

   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

      PDF

   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

   PDF

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. 

• ordinary differential equations
• ergodic theory
• Hopf characteristic functionals
• applications of functional calculus
• Markov processes
• infinite-dimensional spaces

“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)

• 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

Policies and ethics