Fixpoints

A recursive equation can be described in a general way like this

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

If an has a limit as \(n\rightarrow \infty\) , applying the function f on this limit will yield the same number again. The limit is called a fixpoint. If we call the limit x0, this equality is true:

\[f(x_0)=x_0\]

You find the fixpoints to a recursive equation by solving the equation \(f(x)=x\).

When iterating the recursive equation you apply the same function on itself over and over again.

\[x,f(x),f(f(x)),f(f(f(x))),\ldots \]

If the recursive equation is

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

the function of a function of a ..., becomes a continued fraction

\[1+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{1+\cdots}}}\]

If the recursive equation is

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

the function of a function of a ..., becomes a continued square root

\[\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\cdots}}}}\]


Exercises

  1. Find the fixpoints to the continued fraction above
  2. Find the fixpoints to the continued square root above

Attracting or repelling fixpoints

A fixpoint can be attracting or repelling.

Damped pendulum

A damped pendulum has two fixpoints. One is when the pendulum is pointing down, at the angle 0, and one is when the pendulum is pointing up, at the angle π. In theory (not considering the Heisenberg uncertainty principle) if the pendulum starts at exactly the angle π it will stay there, this is a repelling fixpoint, but if you move it slightly it will eventually end up at the attracting fixpoint, the angle 0.

Change the speed or the angle. Move the red point.

In the Scratch-model below, the Ladybug traces out a phase diagram. The y-coordinate of the Ladybug is given by the angular speed, the x-coordinate is given by the angle.

Click anywhere to start the pendulum.
Click c to erase the trace of the Ladybug.
Adjust the sliders and click the green flag to start pendulum again.

Repelling fixpoints

If the fixpoints are irrational numbers you will never find the repelling fixpoints when iterating a recursive equation. Even if you start at a repelling fixpoint, the iterations will result in values getting further and further away from the fixpoint. Regardless of how many correct decimals your electronic device can handle, it cannot handle infinitely many correct decimals; there will always be an error. In order to find the repelling fixpoints you need some other method.

For a recursive equation

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

solve the equation

\[f(x)=x\]

The fixpoints are the intersections between the line and the graph of the function.

image

You can use a picture like the one above to make a cobweb diagram. Using a cobweb diagram it is easy to see if a fixpoint is repelling or attracting.

further info:

Continued fractions

by Malin Christersson under a Creative Commons Attribution-Noncommercial-Share Alike 2.5 Sweden License

www.malinc.se