The idea of using the output of a system as an input
is not a new one (much of the world of modern electronics
in fact depends upon it). *Feedback* refers to this effect.
In this case, *Delayed* means we wish to explore
whether retaining a history of the past of the system will
have any effect on its behavior. In general terms, we wish
to modify (1) in order to account for
its history:

will be seen to be a function of in order to maintain important features of the system (namely its fixed points).

The simplest case for (2) would be

In other words, a history of one iteration will be kept. A simple form for will be constructed, with the following in mind: it is desired to maintain the fixed points of the original system, since the behavior of a system in general depends in a large part on its fixed points, their stability, and the stable and unstable manifolds of those fixed points.

"Maintain the fixed points of the original system" means that (3) should equal (1) at a fixed point. This means that at a fixed point, it is desirable that . Since at a fixed point, one way to do this is for to have a factor of . We choose proportional to this factor:

(4) |

so that

In order to perform analysis on this system (5), it is convenient to introduce a new variable . Then . (5) can be rewritten as a two dimensional system