Period doubling: the logistic map
You should now be used to the idea that the way in which a mechanical system evolves in time is described through a differential equation, of the form
![\[ \frac{d^2x}{dt^2} = f(x, \dot{x}) \]](mastermathpng-0.png)

But there are other ways of modelling the way in which some systems evolve in time. The simplest alternative is an iterative scheme which specifies a series of discrete values x1,x2,x3… through a equation of the form
Equations like this are very easy to explore numerically. And they show a kind of chaos...through the way in which the sequence of xn values evolves
The best studied of all the equations of this type is the logistic map in which
The smooth curve is the function y=f(x). The straight line is y=x.
The horizontal slider allows you to set the initial value of x
The zig-zag-line which starts at that point picks out the sequence of x values that follow from repeated application of the mapping (they are the points at which the zig-zag line intersects the y=x line; but you will need to think about this).
Finally the vertical slider bar allows you to control the value of a.
This simple system displays an astonishing richness.
For some values of a the sequence of x values evolves simply to a stable limiting value.
But if you vary a carefully you will find a region where the sequence settles down to an oscillation between two values...or four..or eight or... [These oscillations show up in red.]
The behaviour at the end of this spectrum is chaos; the route by which it is reached is known as period doubling.
Play and see..
You will find a little more by way of help at the source site below