Dynamical systems and ergodic theory Definition

Cobweb plot

Definition

Let $I \subseteq \R$ be an interval, let $f \colon I \to I$ be a map, and let $x_0 \in I$ . Let $x_n = f^n(x_0)$ for $n \ge 0$ , where $f^n$ is the $n$ -th iterate of $f$ , so that $x_{n+1} = f(x_n)$ . Consider the points $P_{2n} = (x_n, x_n)$ and $P_{2n+1} = (x_n, x_{n+1})$ for $n \ge 0$ . The cobweb plot of the orbit of $x_0$ is the polygonal path $P_0 P_1 P_2 \cdots$ . It is drawn together with the graph of $f$ and the diagonal $\set{(x, x) : x \in I}$ .

Remarks

The points $(x_n, x_{n+1})$ lie on the graph of $f$ and the points $(x_n, x_n)$ lie on the diagonal. So the path alternates vertical segments to the graph and horizontal segments to the diagonal. A fixed point $p$ of $f$ is a point where the graph meets the diagonal, and the path starting at $x_0 = p$ stays at $(p, p)$ .

Example: let $f(x) = x/2 + 1$ on $\R$ and $x_0 = 0$ . The orbit is $0, 1, 3/2, 7/4, \dots$ , with $x_n = 2 - 2^{1-n}$ for $n \ge 0$ . The cobweb path goes from $(0, 0)$ up to $(0, 1)$ , across to $(1, 1)$ , up to $(1, 3/2)$ , and so on, a staircase that climbs to the fixed point $(2, 2)$ .

Non-example: the plot of the points $(n, x_n)$ against the time $n$ is a time series plot. It is not a cobweb plot, because it does not use the graph of $f$ and the diagonal.

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Fixed point concept
Iterated function concept
Cobweb plot definition

Tag	`9GWGW4YRD9`
URI	`m2://9GWGW4YRD9`
Kind	definition
Writing	W0 stub
MSC	37E05, 26A18, 37M99
Defines	Cobweb plot
Uses	Iterated function, Fixed point

Cobweb plot

Definition

Remarks

Reading path

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