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Next: Analysis Up: Control of the Quadratic Previous: Introduction

Delayed Feedback

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:


$\displaystyle x_{n+1} = a - x_n^2 + f(x_n, x_{n-1}, x_{n-2}, ...)$     (2)

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

The simplest case for (2) would be


$\displaystyle x_{n+1} = a - x_n^2 + f(x_n, x_{n-1})$     (3)

In other words, a history of one iteration will be kept. A simple form for $f(x_n,x_{n-1})$ 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 $f(x_n,x_{n-1}) = 0$. Since $x_{n-1}=x_n$ at a fixed point, one way to do this is for $f$ to have a factor of $x - x_{n-1}$. We choose $f$ proportional to this factor:


$\displaystyle f = b(x_n - x_{n-1}) \qquad b \in \mathbb{R}$     (4)

so that
$\displaystyle x_{n+1} = a - x_n^2 + b (x_n - x_{n-1})$     (5)

In order to perform analysis on this system (5), it is convenient to introduce a new variable $y_n = x_{n-1}$. Then $y_{n+1} = x_n$. (5) can be rewritten as a two dimensional system


$\displaystyle \left(
\begin{array}{cc}
x_{n+1} \\
y_{n+1}
\end{array}\right)
=
\left(
\begin{array}{cc}
a - x_n^2 + b (x_n - y_n) \\
x_n
\end{array}\right)$     (6)


next up previous
Next: Analysis Up: Control of the Quadratic Previous: Introduction
URA Program Website 2004-12-02