Cobweb Diagrams

When iterating the recursive equation

\[ \left\{ \begin{align*} a_0 &= c \\ a_{n+1} &= 1+\frac{1}{a_n},n\geq0 \end{align*} \right. \]

you:

1. have a current number; starting with a number \(a_0\)
2. find the function value of the current number; the first time you find \(f(a_0)\)
3. get a new current number from the function value; \(f(a_0)\) will become the new current number \(a_1\)
4. repeat from 1

You can illustrate the steps graphically by using a cobweb diagram.

Move the slider steps!

Repelling and attracting fixpoints using a cobweb diagram

Move the point a0 to see that one of the fixpoints is repelling and the other one attracting.

Slope of the Function

Any point that is "closes enough" to an attracting fixpoint will end up at the fixpoint when iterated. The slope of the function at the fixpoint decides whether the fixpoint is repelling or attracting (or neither).

Find out for what slopes a fixpoint is attracting and for which it is repelling.

Exercise 1

The recursive equation

\[ \left\{ \begin{align*} a_0 &= c \\ a_{n+1} &= 1+\frac{1}{a_n},n\geq0 \end{align*} \right. \]

has two fixpoints \(x_1=\dfrac{1+\sqrt{5}}{2}\) and \(x_1=\dfrac{1-\sqrt{5}}{2}\).

Show that one of them is repelling and one is attracting by differentiating the function.

Exercise 2

The equation \(x=\cos (x)\) can not be solved algebraically.

From looking at the cob web diagram one can see that the fixpoint will attract any point, you do not need to start close to the fixpoint.

Since the fixpoint will also be the limit of corresponding recursive equation, one can use the recursive equation to solve the equation.

Use Octave to iterate the corresponding recursive equation and hence solve the equation \(x=\cos (x)\) numerically.