How Can We Describe Density Evolution Under Delayed Dynamics?

TL;DR

This paper reviews the current state of density evolution in delayed dynamical systems, highlighting challenges and introducing a new analytical approach demonstrated through a simple example.

Contribution

It presents a novel method for analyzing density evolution in delay differential equations, addressing gaps in existing analytical techniques.

Findings

Numerical studies reveal complex dynamics in delay systems.

Existing analytical methods are largely ineffective for these systems.

A new approach offers promising insights into density evolution with delays.

Abstract

Although the theory of density evolution in maps and ordinary differential equations is well developed, the situation is far from satisfactory in continuous time systems with delay. This paper reviews some of the work that has been done numerically and the interesting dynamics that have emerged, and the largely unsuccessful attempts that have been made to analytically treat the evolution of densities in differential delay equations. We also present a new approach to the problem and illustrate it with a simple example.