The Period Doubling Route to Chaos

This is an applet of the period doubling route to chaos, demonstrating the ideas of iterating one dimensional maps, the universal scaling of this route to chaos, and the renormalization group theory.

One Dimensional Maps

Perhaps the simplest "dynamical system" is the iteration of a "one dimensional map": you give me a value x between 0 and 1, and I give you back y=f(x), also between 0 and 1, where f is a simple function e.g. a quadratic function. This value of y becomes the next input value for the iteration. Symbolically

xn+1 = f(xn)

We can think of n as a discrete "time" variable. Even for simple functions f the sequence of numbers generated has a rich behavior, for example periodic (repeating) with an arbitrary repeat period that may be chosen by varying a parameter in f, or apparently random, which (loosely) we call chaos.

The iteration of the map is conveniently shown graphically from the plot of y=f(x). For the input value x=xn the output is the value y i.e moving vertically to the graph of the function to f(x). To get the next input value xn+1 we need the value x=y given by moving horizontally to the diagonal. The next iteration xn+2 is given by again moving vertically to f(x), etc.

This process is demonstrated in the accompanying applet. To start, take the default values (hit the Reset button), and iterate with the Step button.