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2020 | Buch

Introduction and Motivation Kapitel lesen Erstes Kapitel lesen

Density Evolution Under Delayed Dynamics

An Open Problem

verfasst von: Jérôme Losson, Michael C. Mackey, Richard Taylor, Marta Tyran-Kamińska

Verlag: Springer US

Buchreihe : Fields Institute Monographs

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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SUCHEN

Über dieses Buch

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.

Inhaltsverzeichnis

Frontmatter

Introduction and Background to Density Evolution Problems

Frontmatter
Chapter 1. Introduction and Motivation
Abstract
In examining the dynamical behavior of a system there are fundamentally two options available to the experimentalist.
1.
In the first option s/he will examine the dynamical trajectories of individuals, be they fundamental particles in a cloud chamber or cells in a petri dish or animals in an ecological experiment. In this case the experimentalist may be interested in replicating the experiment many times, and building up a statistical description of the observed behavior under the assumption (among others) that the trajectory behavior will be replicated between trials given the same initial conditions.
 
2.
In the second option this approach will be forsaken for one in which the evolving statistics of large populations are examined. This is, of course, most familiar in statistical mechanics, but is also important in many other areas. The advantage of this approach is that if one can understand the dynamics of density evolution, then many interesting statistical quantities can be computed, and the results compared with experimental results.
 
Jérôme Losson, Michael C. Mackey, Richard Taylor, Marta Tyran-Kamińska
Chapter 2. Density Evolution in Systems with Finite-Dimensional Dynamics
Abstract
For background material see Lasota and Mackey (Chaos, fractals, and noise: Stochastic aspects of dynamics, Applied Mathematical Sciences, vol. 97, Springer-Verlag, New York, 1994).
Jérôme Losson, Michael C. Mackey, Richard Taylor, Marta Tyran-Kamińska

Illustrating the Problem and Making It Precise for Differential Delay Equations

Frontmatter
Chapter 3. Dynamics in Ensembles of Differential Delay Equations
Abstract
As pointed out in Chap. 1, there are fundamentally two types of data that are taken in experimental situations, and one is related to statistical properties of large ensembles of “units” that are typically assumed to have the same dynamics. If their dynamics are described by a differential delay equation of the form in Eq. (1.​2), then we must consider what is likely to be measured. The first figure of this chapter will aid in this.
Jérôme Losson, Michael C. Mackey, Richard Taylor, Marta Tyran-Kamińska
Chapter 4. The Problem
Abstract
The central aim of this work is to try to understand how to apply probabilistic concepts, e.g., from ergodic theory, to the dynamics of delay differential equations. Before such a project can proceed, a number of foundational questions must be addressed. For instance,
  • In what sense can a differential delay equation be interpreted as a dynamical system, i.e., with a corresponding evolution semigroup?
  • What is the phase space for such a system?
  • What semigroup of transformations governs the phase space dynamics of a differential delay equation?
Jérôme Losson, Michael C. Mackey, Richard Taylor, Marta Tyran-Kamińska

Possible Analytical Approaches

Frontmatter
Chapter 5. The Hopf Functional Approach
Abstract
In this chapter we examine the possibility of extending techniques developed for the study of turbulent fluid flows to the statistical study of the dynamics of differential delay equations. Because the phase spaces of differential delay equations are infinite-dimensional, phase-space densities for these systems should be thought of as functionals.
Jérôme Losson, Michael C. Mackey, Richard Taylor, Marta Tyran-Kamińska
Chapter 6. The Method of Steps
Abstract
Consider the augmented differential delay equation initial value problem (with τ ≡ 1)
$$\displaystyle \begin {aligned}{} &x'(t) = \begin {cases} \mathcal {G}\big ( x(t) \big ) & t \in [0,1) \\ \mathcal {F}\big ( x(t), x(t-1) \big ) & t \geq 1 \end {cases} \\ &x(0) = x_0, \end {aligned} $$
with \(x(t) \in \mathbb {R}\), and suppose that an ensemble of initial values x 0 is specified with density f 0. We would like to derive an evolution equation for the density f(x, t) of the corresponding ensemble of solutions x(t).
Jérôme Losson, Michael C. Mackey, Richard Taylor, Marta Tyran-Kamińska

Part IV

Frontmatter
Chapter 7. Turning a Differential Delay Equation into a High-Dimensional Map
Abstract
This entire section was originally published in Losson and Mackey (Phys Rev E 52(1):115–128, 1995). In this chapter we examine the potential use of a variety of approximations, or reductions, of a differential delay equation to a system of ordinary differential equations in the first instance, and reducing the delay differential equation to a high-dimensional map in the second.
Jérôme Losson, Michael C. Mackey, Richard Taylor, Marta Tyran-Kamińska
Chapter 8. Approximate “Liouville-Like” Equation and Invariant Densities for Delay Differential Equations
Abstract
This entire chapter is taken from an unpublished manuscript (S.R. Taylor, Liouville-like equations and invariant densities for delay differential equations, 2011).
Jérôme Losson, Michael C. Mackey, Richard Taylor, Marta Tyran-Kamińska
Chapter 9. Summary and Conclusions
Abstract
This is a weird monograph to say the least. Rather than presenting a body of finished work with theorems and proofs and examples and applications, we have presented the problem of how to treat the evolution of densities under that action of delayed dynamics and given no solutions! We have simply illustrated the problem in Chaps. 1–3 and then given all of the reasons why it is difficult mathematically in Chap. 4. Following this we have presented a series of chapters in which we detail various attempts that have been made to solve the problem, all of which have led to naught. This is definitely NOT your standard mathematical monograph! So why have we done this? Simply to lay out a map of what we think are blind alleys for the unwary neophyte starting out in search of a solution.
Jérôme Losson, Michael C. Mackey, Richard Taylor, Marta Tyran-Kamińska
Backmatter
Metadaten
Titel
Density Evolution Under Delayed Dynamics
verfasst von
Jérôme Losson
Michael C. Mackey
Richard Taylor
Marta Tyran-Kamińska
Copyright-Jahr
2020
Verlag
Springer US
Electronic ISBN
978-1-0716-1072-5
Print ISBN
978-1-0716-1071-8
DOI
https://doi.org/10.1007/978-1-0716-1072-5